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Dissertations / Theses on the topic 'Hyperbolic dynamical systems'
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Ponce, Gabriel. "Fine ergodic properties of partially hyperbolic dynamical systems." Universidade de São Paulo, 2014. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-20032015-113539/.
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Let f : T3 → T3 be a C2 volume preserving partially hyperbolic diffeomorphism homotopic to a linear Anosov automorphism A : T3 → T3. We prove that if f is Kolmogorov, then f is Bernoulli. We study the characteristics of atomic disintegration of the volume measure whenever it occurs. We prove that if the volume measure m has atomic disintegration on the center leaves then the disintegration has one atom per center leaf. We give a condition, depending only on the center Lyapunov exponent of the diffeomorphism, that guarantees atomic disintegration of the volume measure on center leaves. We construct an open family of diffeomorphisms satisfying this condition which generates the first examples of foliations which are both measurable and minimal. In this same construction we give the first examples of partially hyperbolic diffeomorphisms with zero center Lyapunov exponent and homotopic to a linear Anosov.Seja f : T3 → T3 um difeomorfismo C2 parcialmente hiperbólico, homotópico a um automorfismo de Anosov linear e preservando a medida de volume m. Provamos que se f é Kolmogorov então f é Bernoulli. Estudamos as características da desintegração atômica da medida de volume quando esta ocorre. Provamos que se a medida de volume m tem desintegração atômica nas folhas centrais então a desintegração tem um átomo por folha central. Apresentamos uma condição, a qual depende apenas do expoente de Lyapunov central do difeomorfismo, que garante desintegração atômica da medida de volume. Construímos uma família aberta de difeomorfismos satisfazendo esta condição, o que gerou os primeiros exemplos de folheações que são mensuráveis e ao mesmo tempo minimais. Nesta mesma construção damos os primeiros exemplos de difeomorfismos parcialmente hiperbólicos com expoente de Lyapunov central nulo e homotópico a um Anosov linear.
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Petty, Taylor Michael. "Nonlocally Maximal Hyperbolic Sets for Flows." BYU ScholarsArchive, 2015. https://scholarsarchive.byu.edu/etd/5558.
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In 2004, Fisher constructed a map on a 2-disc that admitted a hyperbolic set not contained in any locally maximal hyperbolic set. Furthermore, it was shown that this was an open property, and that it was embeddable into any smooth manifold of dimension greater than one. In the present work we show that analogous results hold for flows. Specifically, on any smooth manifold with dimension greater than or equal to three there exists an open set of flows such that each flow in the open set contains a hyperbolic set that is not contained in a locally maximal one.
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Al-Nayef, Anwar Ali Bayer, and mikewood@deakin edu au. "Semi-hyperbolic mappings in Banach spaces." Deakin University. School of Computing and Mathematics, 1997. http://tux.lib.deakin.edu.au./adt-VDU/public/adt-VDU20051208.110247.
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The definition of semi-hyperbolic dynamical systems generated by Lipschitz continuous and not necessarily invertible mappings in Banach spaces is presented in this thesis. Like hyperbolic mappings, they involve a splitting into stable and unstable spaces, but a slight leakage from the strict invariance of the spaces is possible and the unstable subspaces are assumed to be finite dimensional.
Bi-shadowing is a combination of the concepts of shadowing and inverse shadowing and is usually used to compare pseudo-trajectories calculated by a computer with the true trajectories. In this thesis, the concept of bi-shadowing in a Banach space is defined and proved for semi-hyperbolic dynamical systems generated by Lipschitz mappings. As an application to the concept of bishadowing, linear delay differential equations are shown to be bi-shadowing with respect to pseudo-trajectories generated by nonlinear small perturbations of the linear delay equation. This shows robustness of solutions of the linear delay equation with respect to small nonlinear perturbations.
Complicated dynamical behaviour is often a consequence of the expansivity of a dynamical system. Semi-hyperbolic dynamical systems generated by Lipschitz mappings on a Banach space are shown to be exponentially expansive, and explicit rates of expansion are determined. The result is applied to a nonsmooth noninvertible system generated by delay differential equation.
It is shown that semi-hyperbolic mappings are locally φ-contracting, where -0 is the Hausdorff measure of noncompactness, and that a linear operator is semi-hyperbolic if and only if it is φ-contracting and has no spectral values on the unit circle. The definition of φ-bi-shadowing is given and it is shown that semi-hyperbolic mappings in Banach spaces are φ-bi-shadowing with respect to locally condensing continuous comparison mappings. The result is applied to linear delay differential equations of neutral type with nonsmooth perturbations.
Finally, it is shown that a small delay perturbation of an ordinary differential equation with a homoclinic trajectory is chaotic.
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Gaito, Stephen Thomas. "Shadowing of weakly pseudo-hyperbolic pseudo-orbits in discrete dynamical systems." Thesis, University of Warwick, 1992. http://wrap.warwick.ac.uk/109461/.
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We consider Cr (r ≥ 1 +γ) diffeomorphisms of compact Riemannian manifolds. Our aim is to develop the analytic machinery required to describe the topological symbolic dynamics of sets of weakly hyperbolic orbits. The Pesin set is an example of such a set. For Axiom-A dynamical systems, that is, for diffeomorphisms which have a uniformly hyperbolic nonwandering set which is the closure of the periodic orbits, this analytic machinery is provided by the Shadowing Lemma. This lemma is a consequence of the Stable Manifold Theorem, and the local product structure of the nonwandering set of an Axiom-A diffeomorphism. Weakly hyperbolic invariant sets, such as the Pesin set, do not, in general, have local product structure. We can however, prove a generalization of the Shadowing Lemma by combining Anosov’s Stability Lemma with the Stable Manifold Theorem. In essence we prove a perturbed Stable Manifold Theorem. In order to deal with weakly hyperbolic orbits we use Pugh and Shub’s graph transform version of Pesin’s Stable Manifold Theorem. Normally, the contraction required to prove either Anosov’s Stability Lemma or the Stable Manifold Theorem, is derived from the hyperbolicity of a “supporting” invariant set. In fact neither of these proofs require this invariance; hyperbolic, or even pseudo-hyperbolic, families of pseudo-orbits are all that they require. This allows us to conclude the existence of shadowing orbits in the neighbourhood of “hyperbolic invariant sets” of numerical simulations of lowdimensional dynamical systems. In particular corresponding to any such numerical “hyperbolic invariant set”, there is a uniformly hyperbolic invariant set of the dynamical system itself.
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Waddington, Simon. "Prime orbit theorems for closed orbits and knots in hyperbolic dynamical systems." Thesis, University of Warwick, 1992. http://wrap.warwick.ac.uk/109425/.
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This thesis consists of four chapters, each with its own notation and references. Chapters 1, 2 and 3 are independent pieces of research. Chapter 0 is an introduction which sets out the definitions and results needed in the main part of the thesis. In Chapter 1, we derive asymptotic formulae for the number of closed orbits of a toral automorphism which is ergodic, but not necessarily hyperbolic. Previously, such formulae were known only in the hyperbolic case. The proof uses an analogy with the Prime Number Theorem. We also give a new proof of the uniform distribution of periodic points. In Chapter 2, we derive various asymptotic formulae for the numbers of closed orbits in the Lorenz and Smale horseshoe templates with given knot invariants, (specifically braid index and genus). We indicate how these estimates can be applied to more complicated flows by giving a bound for the genus of knotted periodic orbits in the ' figure of eight template'. In Chapter 3, we prove a dynamical version of the Chebotarev density theorem for group extensions of geodesic flows on compact manifolds of variable negative curvature. Specifically, the group is taken to be the weak direct sum of a finite abelian group. We outline an application to twisted orbits.
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Canestrari, Giovanni. "On the Kolmogorov property of a class of infinite measure hyperbolic dynamical systems." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2021. http://amslaurea.unibo.it/22352/.
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Smooth maps with singularities describe important physical phenomena such as the collisions of rigid spheres among them and/or with the walls of a container. Questions about the ergodic properties of these models (which can be mapped into billiard models) were first raised by Boltzmann in the nineteenth century and lie at the foundation of Statistical Mechanics. Billiard models also describe the diffusive motion of electrons bouncing off positive nuclei (Lorentz gas models) and in this situation the physical measure can be considered infinite. It is therefore of great importance to study the ergodic properties of maps when the measure they preserves is infinite.
The aim of this thesis is to present an original result on smooth maps with singularities which preserve an infinite measure. Such result establishes the atomicity of the tail $\sigma$-algebra (and hence strong chaotic properties) in the presence of a totally conservative behavior.
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Leclerc, Gaétan. "Nonlinearity, fractals, Fourier decay - harmonic analysis of equilibrium states for hyperbolic dynamical systems." Electronic Thesis or Diss., Sorbonne université, 2024. http://www.theses.fr/2024SORUS264.
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Ce doctorat se situe à l'intersection entre le domaine de la géométrie fractale et des systèmes dynamique hyperbolique. Étant donné un système dynamique hyperbolique dans un espace euclidien (de petite dimension), considérons un sous-ensemble fractal compact invariant, ainsi qu'une mesure de probabilité invariante supportée sur cet ensemble fractal, avec de bonnes propriétés statistiques, telle que la mesure d'entropie maximale. La question est la suivante : la transformée de Fourier de la mesure tends elle vers zéro a la vitesse d'une puissance de xi ? Notre objectif principal est de montrer que, pour plusieurs familles de systèmes dynamiques hyperboliques, la non-linéarité de la dynamique suffit à démontrer de tels résultats de décroissance. Ces énoncés seront obtenus en utilisant un outil puissant du domaine de la combinatoire additive : le phénomène de somme-produitThis PhD lies at the intersection between fractal geometry and hyperbolic dynamics. Being given a (low dimensional) hyperbolic dynamical system in some euclidean space, let us consider a fractal compact invariant subset, and an invariant probability measure supported on this fractal set with good statistical properties, such as the measure of maximal entropy. The question is the following: does the Fourier transform of the measure exhibit power decay ? Our main goal is to give evidence, for several families of hyperbolic dynamical systems, that nonlinearity of the dynamics is enough to prove such decay results. These statements will be obtained using a powerful tool from the field of additive combinatorics: the sum-product phenomenon
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Canalias, Vila Elisabet. "Contributions to Libration Orbit Mission Design using Hyperbolic Invariant Manifolds." Doctoral thesis, Universitat Politècnica de Catalunya, 2007. http://hdl.handle.net/10803/5927.
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Aquesta tesi doctoral està emmarcada en el camp de l'astrodinàmica. Presenta solucions a problemes identificats en el disseny de missions que utilitzen òrbites entorn dels punts de libració, fent servir la teoria de sistemes dinàmics.El problema restringit de tres cossos és un model per estudiar el moviment d'un cos de massa infinitessimal sota l'atracció gravitatòria de dos cossos molt massius. Els cinc punts d'equilibri d'aquest model, en especial L1 i L2, han estat motiu de nombrosos estudis per aplicacions pràctiques en les últimes dècades (SOHO, Genesis...).
Genèricament, qualsevol missió en òrbita al voltant del punt L2 del sistema Terra-Sol es veu afectat per ocultacions degudes a l'ombra de la Terra. Si l'òrbita és al voltant de L1, els eclipsis són deguts a la forta influència electromagnètica del Sol. D'entre els diferents tipus d'òrbites de libració, les òrbites de Lissajous resulten de la combinació de dues oscil.lacions perpendiculars. El seu principal avantatge és que les amplituds de les oscil.lacions poden ser escollides independentment i això les fa adapatables als requeriments de cada missió. La necessitat d'estratègies per evitar eclipsis en òrbites de Lissajous entorn dels punts L1 i L2 motivaren la primera part de la tesi. En aquesta part es presenta una eina per la planificació de maniobres en òrbites de Lissajous que no només serveix per solucionar el problema d'evitar els eclipsis, sinó també per trobar trajectòries de transferència entre òrbites d'amplituds diferents i planificar rendez-vous.
Per altra banda, existeixen canals de baix cost que uneixen els punts L1 i L2 d'un sistema donat i representen una manera natural de transferir d'una regió de libració a l'altra. Gràcies al seu caràcter hiperbòlic, una òrbita de libració té uns objectes invariants associats: les varietats estable i inestable. Si tenim present que la varietat estable està formada per trajectòries que tendeixen cap a l'òrbita a la qual estan associades quan el temps avança, i que la varietat inestable fa el mateix però enrera en el temps, una intersecció entre una varietat estable i una d'inestable proporciona un camí asimptòtic entre les òrbites corresponents. Un mètode per trobar connexions d'aquest tipus entre òrbites planes entorn de L1 i L2 es presenta a la segona part de la tesi, i s'hi inclouen els resultats d'aplicar aquest mètode als casos dels problemes restringits Sol Terra i Terra-Lluna.
La idea d'intersecar varietats hiperbòliques es pot aplicar també en la cerca de camins de baix cost entre les regions de libració del sistema Sol-Terra i Terra-Lluna. Si existissin camins naturals de les òrbites de libració solars cap a les lunars, s'obtindria una manera barata d'anar a la Lluna fent servir varietats invariants, cosa que no es pot fer de manera directa. I a l'inversa, un camí de les regions de libració lunars cap a les solars permetria, per exemple, que una estació fos col.locada en òrbita entorn del punt L2 lunar i servís com a base per donar servei a les missions que operen en òrbites de libració del sistema Sol-Terra. A la tercera part de la tesi es presenten mètodes per trobar trajectòries de baix cost que uneixen la regió L2 del sistema Terra-Lluna amb la regió L2 del sistema Sol-Terra, primer per òrbites planes i més endavant per òrbites de Lissajous, fent servir dos problemes de tres cossos acoblats. Un cop trobades les trajectòries en aquest model simplificat, convé refinar-les per fer-les més realistes. Una metodologia per obtenir trajectòries en efemèrides reals JPL a partir de les trobades entre òrbites de Lissajous en el model acoblat es presenta a la part final de la tesi. Aquestes trajectòries necessiten una maniobra en el punt d'acoblament, que és reduïda en el procés de refinat, arribant a obtenir trajectòries de cost zero quan això és possible.
This PhD. thesis lies within the field of astrodynamics. It provides solutions to problems which have been identified in mission design near libration points, by using dynamical systems theory.
The restricted three body problem is a well known model to study the motion of an infinitesimal mass under the gravitational attraction of two massive bodies. Its five equilibrium points, specially L1 and L2, have been the object of several studies aimed at practical applications in the last decades (SOHO, Genesis...).
In general, any mission in orbit around L2 of the Sun-Earth system is affected by occultations due to the shadow of the Earth. When the orbit is around L1, the eclipses are caused by the strong electromagnetic influence of the Sun. Among all different types of libration orbits, Lissajous type ones are the combination of two perpendicular oscillations. Its main advantage is that the amplitudes of the oscillations can be chosen independently and this fact makes Lissajous orbits more adaptable to the requirements of each particular mission than other kinds of libration motions. The need for eclipse avoidance strategies in Lissajous orbits around L1 and L2 motivated the first part of the thesis. It is in this part where a tool for planning maneuvers in Lissajous orbits is presented, which not only solves the eclipse avoidance problem, but can also be used for transferring between orbits having different amplitudes and for planning rendez-vous strategies.
On the other hand, there exist low cost channels joining the L1 and L2 points of a given sistem, which represent a natural way of transferring from one libration region to the other one. Furthermore, there exist hyperbolic invariant objects, called stable and unstable manifolds, which are associated with libration orbits due to their hyperbolic character. If we bear in mind that the stable manifold of a libration orbit consists of trajectories which tend to the orbit as time goes by, and that the unstable manifold does so but backwards in time, any intersection between a stable and an unstable manifold will provide an asymptotic path between the corresponding libration orbits. A methodology for finding such asymptotic connecting paths between planar orbits around L1 and L2 is presented in the second part of the dissertation, including results for the particular cases of the Sun-Earth and Earth-Moon problems.
Moreover, the idea of intersecting hyperbolic manifolds can be applied in the search for low cost paths joining the libration regions of different problems, such as the Sun-Earth and the Earth-Moon ones. If natural paths from the solar libration regions to the lunar ones was found, it would provide a cheap way of transferring to the Moon from the vicinity of the Earth, which is not possible in a direct way using invariant manifolds. And the other way round, paths from the lunar libration regions to the solar ones would allow for the placement of a station in orbit around the lunar L2, providing services to solar libration missions, for instance. In the third part of the thesis, a methodology for finding low cost trajectories joining the lunar L2 region and the solar L2 region is presented. This methodology was developed in a first step for planar orbits and in a further step for Lissajous type orbits, using in both cases two coupled restricted three body problems to model the Sun-Earth-Moon spacecraft four body problem. Once trajectories have been found in this simplified model, it is convenient to refine them to more realistic models. A methodology for obtaining JPL real ephemeris trajectories from the initial ones found in the coupled models is presented in the last part of the dissertation. These trajectories need a maneuver at the coupling point, which can be reduced in the refinement process until low cost connecting trajectories in real ephemeris are obtained (even zero cost, when possible).
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Högele, Michael, and Ilya Pavlyukevich. "Metastability of Morse-Smale dynamical systems perturbed by heavy-tailed Lévy type noise." Universität Potsdam, 2014. http://opus.kobv.de/ubp/volltexte/2014/7063/.
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We consider a general class of finite dimensional deterministic dynamical systems with finitely many local attractors each of which supports a unique ergodic probability measure, which includes in particular the class of Morse–Smale systems in any finite dimension. The dynamical system is perturbed by a multiplicative non-Gaussian heavytailed Lévy type noise of small intensity
ε > 0. Specifically we consider perturbations leading to a Itô, Stratonovich and canonical (Marcus) stochastic differential equation. The respective asymptotic first exit time and location problem from each of the domains
of attractions in case of inward pointing vector fields in the limit of ε-> 0 has been investigated by the authors. We extend these results to domains with characteristic boundaries and show that the perturbed system exhibits a metastable behavior in the sense that there exits a unique ε-dependent time scale on which the random system converges to a continuous time Markov chain switching between the invariant measures. As examples we consider α-stable perturbations of the Duffing equation and a chemical system exhibiting a birhythmic behavior.
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Canadell, Cano Marta. "Computation of Normally Hyperbolic Invariant Manifolds." Doctoral thesis, Universitat de Barcelona, 2014. http://hdl.handle.net/10803/277215.
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The subject of the theory of Dynamical Systems is the evolution of systems with respect to time. Hence, it has many applications to other areas of science, such as Physics, Biology, Economics, etc. and it also has interactions with other parts of Mathematics.
The global behavior of a dynamical system is organized by its invariant objects, the simplest ones are equilibria and periodic orbits (and related invariant manifolds). Normally hyperbolic invariant manifolds (NHIM for short) are some of these invariant objects. They have the property to persist under small perturbations of the system. These NHIM are characterized by the fact that the directions on the points of the manifold split into stable, unstable and tangent components. The growth rate of stable directions (for which forward evolution of the system goes to zero) and unstable directions (for which backward evolution goes to zero) dominate the growth rate of the tangent directions. The robustness of normally hyperbolic invariant manifolds makes them very useful to understand the global dynamics. Both the theory and the computation of these objects are important for the general understanding of a dynamical system.
The main goal of my thesis is to develop efficient algorithms for the computation of normally hyperbolic invariant manifolds, give a rigorous mathematical theory and implement them to explore new mathematical phenomena. For simplicity, we consider the problem for discrete dynamical systems, since it is known that the discrete case implies the continuous case using time one flow. We consider a diffeomorphism F : Rm → Rm and a d-torus parameterized by K : Td → Rm which is invariant under F. This means that there exists a diffeomorphism f : Td → Td (the internal dynamics) such that it satisfies F ◦ K = K ◦ f, (0.3) called the invariance equation.
Our goal is to solve this invariance equation considering two different scenarios: one in which we do not know the internal dynamics of the invariant torus (where K and f are our unknowns), see Chapter 4, and the other in which we impose that the internal dynamics is a rigid rotation with a quasi-periodic frequency (where K is the unknown and f is the rigid rotation), for which we also need to add an adjusting parameter to equation (0.3), see Chapters 2 and 3. Additionally, in both cases we are also interested in computing the invariant tangent and normal bundles.L’objecte d’estudi dels Sistemes Dinàmics és l’evolució dels sistemes respecte del temps. Per aquesta raó, els Sistemes Dinàmics presenten moltes aplicacions en altres àrees de la Ciència, com ara la Física, Biologia, Economia, etc. i tenen nombroses interaccions amb altres parts de les Matemàtiques. Els objectes invariants organitzen el comportament global d’un sistema dinàmic, els més simples dels quals són els punts fixos i les òrbites periòdiques (així com les seves corresponents varietats invariants). Les Varietats Invariants Normalment Hiperbòliques (NHIM forma abreviada provinent de l’anglès) són alguns d’aquests objectes invariants. Aquests objectes posseeixen la propietat de persistir sota petites pertorbacions del sistema. Les NHIM estan caracteritzades pel fet que les direccions en els punts de la varietat presenten una divisió en components tangent, estable i inestable. L’índex de creixement de les direccions estables (per les quals la iteració endavant del sistema tendeix cap a zero) i inestables (per les quals la iteració enrere del sistema tendeix cap a zero) domina l’índex de creixement de les direccions tangents. La robustesa de les varietats invariants normalment hiperbòliques les fa de gran utilitat a l’hora d’estudiar la dinàmica global. Per aquesta raó, tant la teoria com el càlcul d’aquests objectes sós molt importants per al coneixement general d’un sistema dinàmic. L’objectiu principal d’aquesta tesi és desenvolupar algoritmes eficients pel càlcul de varietats invariants normalment hiperbòliques, donar-ne resultats teòrics rigorosos i implementar-los per a explorar nous fenòmens matemàtics. Per simplicitat, considerarem el problema per a sistemes dinàmics discrets, ja que és ben conegut que el cas discret implica el cas continu usant operadors d’evolució. Considerem així difeomorfismes donats per F : Rm → Rm i un d-tor F-invariant parametritzat per K : Td → Rm. És a dir, existeix un difeomorfisme f : Td → Td (la dinàmica interna) tal que satisfà l’equació F ◦ K = K ◦ f, (0.1) anomenada equació d’invariància. La nostra finalitat és solucionar aquesta equació d’invariància considerant dos possibles escenaris: un en el qual no coneixem quina és la dinàmica interna del tor (on K i f són les nostres incògnites), veure Capítol 4, i un altre en el qual imposem que la dinàmica interna sigui una rotació rígida amb freqüència quasi-periòdica (on K és una incògnita i f és la rotació rígida), pel qual necessitarem, a més a més, afegir un paràmetre ajustador a l’equació (0.1), veure Capítols 2 i 3. En ambdós casos també estarem interessats en el càlcul dels fibrats invariants tangent i normals.
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Simmons, Skyler C. "Topological Properties of Invariant Sets for Anosov Maps with Holes." BYU ScholarsArchive, 2011. https://scholarsarchive.byu.edu/etd/3101.
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We begin by studying various topological properties of invariant sets of hyperbolic toral automorphisms in the linear case. Results related to cardinality, local maximality, entropy, and dimension are presented. Where possible, we extend the results to the case of hyperbolic toral automorphisms in higher dimensions, and further to general Anosov maps.
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Hemmingsson, Nils. "On the dynamics of a family of critical circle endomorphisms." Thesis, KTH, Matematik (Avd.), 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-259743.
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In this thesis we study two seperate yet related three parameter-families of continuously differentiable maps from the unit circle to unit circle which have a single critical point. For one of the families we show that there is a set of positive measure of parameters such that there is a set of positive measure for which all points in the latter set, the derivative experiences exponential growth. We do so by applying a similar methodology to what Michael Benedicks and Lennart Carleson used to study the quadratic family. For the other family we attempt to show a similar but weaker result using a similar method, but do not manage to do so. We expound on what difficulties the latter family provides and what features Benedicks and Carleson used for the quadratic family that we do not have available.I den här uppsatsen studerar vi två olika men relaterede treparameterfamiljer av kontinuerligt differentierbara avbildningar från enhetscirkeln till enhetscirkeln som har exakt en kritisk punkt. For den ena familjen visar vi att det finns en mängd av positivt mått av parametrar sådana att det finns en mängd av positivt mått så att för varje punkt i den senarenämnde mängden erfar derivatan exponentiell tillväxt. Vi uppnår detta genom att använda en metod som liknar den som Michael Benedicks och Lennart Carleson använde för att studera den kvadratiska familjen. För den andra familjen försöker vi visa ett liknande men svagare resultat genom att använda en liknande metodik men misslyckas. Vi diskuterar och förklarar vilka svårigheter den senare familjen ger och vilka egenskaper som Benedicks och Carleson använder sig av hos den kvadratiska familjen som vår familj saknar
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Dutilleul, Tom. "Dynamique chaotique des espaces-temps spatialement homogènes." Thesis, Paris 13, 2019. http://www.theses.fr/2019PA131019.
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En 1963, Belinsky, Khalatnikov et Lifshitz ont proposé une description conjecturale de la géométrie asymptotique des modèles cosmologiques au voisinage de leur singularité initiale. En particulier, il y est avancé que la géométrie asymptotique des espaces-temps spatialement homogènes « génériques » devrait avoir un comportement oscillatoire chaotique modelé sur la dynamique d’une application discrète : l’application de Kasner. Nous démontrons que cette conjecture est vraie au moins pour un ensemble d’espaces-temps de mesure de Lebesgue strictement positive. Dans le contexte des espaces-temps spatialement homogènes, l’équation d’Einstein de la relativité générale se réduit à un système d’équations différentielles sur un espace des phases de dimension finie : les équations de Wainwright-Hsu. La dynamique de ces équations encode l’évolution de la géométrie des hypersurfaces spatiales dans les espaces-temps spatialement homogènes. Notre preuve est basée sur l’hyperbolicité non-uniforme des équations de Wainwright-Hsu. Nous considérons l’application de Poincaré associée aux solutions de ces équations sur une section transverse au flot et nous démontrons qu’il s’agit d’une application non-uniformément hyperbolique avec singularités. Ceci nous permet de construire des variétés stables locales « à la Pesin » pour cette application et de montrer que la réunion des orbites passant par ces variétés stables locales recouvre une partie de l’espace des phases de mesure de Lebesgue strictement positive. Le comportement oscillatoire chaotique des espaces-temps correspondant à ces orbites est une conséquence de cette construction. Du point de vue des systèmes dynamiques, les équations de Wainwright-Hsu se révèlent être très riches et posent un certain nombre de défis. Pour comprendre le comportement asymptotique d’un nombre conséquent de solutions de ces équations, nous serons amenés à : • faire une analyse fine de la dynamique locale d’un champ de vecteurs au voisinage d’une singularité partiellement hyperbolique dégénérée et non linéarisable, • travailler avec des applications non-uniformément hyperboliques ayant des singularités, pour lesquelles la théorie usuelle (due à Pesin et Katok-Strelcyn) ne s’applique pas à cause de la faible régularité de ces applications, • considérer des conditions arithmétiques exotiques exprimées en termes de fractions continues et utiliser des propriétés ergodiques quelque peu sophistiquées de l’application de Gauss pour montrer que ces propriétés sont génériques, etcIn 1963, Belinsky, Khalatnikov and Lifshitz have proposed a conjectural description of the asymptotic geometry of cosmological models in the vicinity of their initial singularity. In particular, it is believed that the asymptotic geometry of generic spatially homogeneous spacetimes should display an oscillatory chaotic behaviour modeled on a discrete map’s dynamics (the so-called Kasner map). We prove that this conjecture holds true, if not for generic spacetimes, at least for a positive Lebesgue measure set of spacetimes. In the context of spatially homogeneous spacetimes, the Einstein field equations can be reduced to a system of differential equations on a finite dimensional phase space: the Wainwright-Hsu equations. The dynamics of these equations encodes the evolution of the geometry of spacelike slices in spatially homogeneous spacetimes. Our proof is based on the non-uniform hyperbolicity of the Wainwright-Hsu equations. Indeed, we consider the return map of the solutions of these equations on a transverse section and prove that it is a non-uniformly hyperbolic map with singularities. This allows us to construct some local stable manifolds à la Pesin for this map and to prove that the union of the orbits starting in these local stable manifolds cover a positive Lebesgue measure set in the phase space. The chaotic oscillatory behaviour of the corresponding spacetimes follows. The Wainwright-Hsu equations turn out to be quite interesting and challenging from a purely dynamical system viewpoint. In order to understand the asymptotic behaviour of (many of) the solutions of these equations, we will in particular be led to: • carry a detailed analysis of the local dynamics of a vector field in the neighborhood of degenerate nonlinearizable partially hyperbolic singularities, • deal with non-uniformly hyperbolic maps with singularities for which the usual theory (due to Pesin and Katok-Strelcyn) is not relevant due to the poor regularity of the maps, • consider some unusual arithmetic conditions expressed in terms of continued fractions and use some rather sophisticated ergodic properties of the Gauss map to prove that these properties are generic
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Chaves, Daniel Pedro Bezerra. "Sistemas dinâmicos de eventos discretos com aplicação ao fluxo geodésico em superfícies hiperbólicas." [s.n.], 2011. http://repositorio.unicamp.br/jspui/handle/REPOSIP/260471.
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Orientador: Reginaldo Palazzo JúniorTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação
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Resumo: Neste trabalho apresentamos um método de descrição combinatorial para o fluxo geodesico sobre uma região hiperbólica compacta, tendo como objetivo associar a seqüências de codificação, parâmetros topologicos oriundos destas superfícies. Isto permite conjugar conceitos topologicos e combinatoriais oriundos das superfícies estudadas com conceitos de teoria da informação e codificação. Demonstramos como a propriedade de completude de um sistema dinâmico de eventos discretos invariantes no tempo se reflete na topologia do espaço de trajetórias do sistema, quando especificadas por seqüências bi-infinitas e descritas sobre um alfabeto finito. A mesma estrutura obtida pelo processo de codificação do fluxo geodesico, e a qual passamos a chamar de sistema simbólico fechado (ssf). Identificamos como um ssf pode ser caracterizado globalmente, através do seu conjunto de restrições irredutíveis, ou localmente, por conjuntos de restrições dependentes do contexto. Ambas derivadas de relações de ordem parcial. Disto determinamos métodos de representação do ssf. Através da relação entre os métodos de codificação aritmético e geométrico, propomos processos de codificação sobre superfícies hiperbólicas, determinando como as representações mínimas das seqüências código do fluxo geodesico podem ser construídas a partir das propriedades topológicas e combinatoriais da superfície
Abstract: In this work we present methods for a combinatorial description of the geodesic flow on a hyperbolic compact surface, with the intent of identifying how the topological parameters of the surface may be associated with discrete sequences. This approach allows to conjugate the topological and combinatorial properties of a surface with concepts of information theory and coding. We determine the intrinsic topological property of complete and time-invariant discrete dynamical systems whose trajectories are bi-infinite sequences over a finite alphabet. The same structure generated by the geodesic flow coding methods, that we call shift space. We show how a shift space can be completely characterized by the irreducible forbidden set and locally by the constraint sets, and how both can be obtained through partial order relations. As consequence of these results, some constructions to represent the shift spaces are proposed. Methods for coding source sequences on hyperbolic surfaces are proposed, based on T-piecewise and common-sets relations that exist between these methods. We conclude by specifying a construction procedure for presentations of arithmetic codes that is related with the topological and combinatorial properties of the hyperbolic surface
Doutorado
Telecomunicações e Telemática
Doutor em Engenharia Elétrica
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Leguil, Martin. "Cocycle dynamics and problems of ergodicity." Thesis, Sorbonne Paris Cité, 2017. http://www.theses.fr/2017USPCC159/document.
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Le travail qui suit comporte quatre chapitres : le premier est centré autour de la propriété de mélange faible pour les échanges d'intervalles et flots de translation. On y présente des résultats obtenus avec Artur Avila qui renforcent des résultats précédents dus à Artur Avila et Giovanni Forni. Le deuxième chapitre est consacré à un travail en commun avec Zhiyuan Zhang et concerne les propriétés d'ergodicité et d'accessibilité stables pour des systèmes partiellement hyperboliques de dimension centrale au moins égale à deux. On montre que sous des hypothèses de cohérence dynamique, center bunching et pincement fort, la propriété d'accessibilité stable est dense en topologie C^r, r>1, et même prévalente au sens de Kolmogorov. Dans le troisième chapitre, on expose les résultats d'un travail réalisé en collaboration avec Julie Déserti, consacré à l'étude d'une famille à un paramètre d'automorphismes polynomiaux de C^3 ; on montre que de nouveaux phénomènes apparaissent par rapport à ce qui était connu dans le cas de la dimension deux. En particulier, on étudie les vitesses d'échappement à l'infini, en montrant qu'une transition s'opère pour une certaine valeur du paramètre. Le dernier chapitre est issu d'un travail en collaboration avec Jiangong You, Zhiyan Zhao et Qi Zhou ; on s'intéresse à des estimées asymptotiques sur la taille des trous spectraux des opérateurs de Schrödinger quasi-périodiques dans le cadre analytique. On obtient des bornes supérieures exponentielles dans le régime sous-critique, ce qui renforce un résultat précédent de Sana Ben Hadj Amor. Dans le cas particulier des opérateurs presque Mathieu, on montre également des bornes inférieures exponentielles, qui donnent des estimées quantitatives en lien avec le problème dit "des dix Martinis". Comme conséquences de nos résultats, on présente des applications à l'homogénéité du spectre de tels opérateurs ainsi qu'à la conjecture de DeiftThe following work contains four chapters: the first one is centered around the weak mixing property for interval exchange transformations and translation flows. It is based on the results obtained together with Artur Avila which strengthen previous results due to Artur Avila and Giovanni Forni. The second chapter is dedicated to a joint work with Zhiyuan Zhang, in which we study the properties of stable ergodicity and accessibility for partially hyperbolic systems with center dimension at least two. We show that for dynamically coherent partially hyperbolic diffeomorphisms and under certain assumptions of center bunching and strong pinching, the property of stable accessibility is dense in C^r topology, r>1, and even prevalent in the sense of Kolmogorov. In the third chapter, we explain the results obtained together with Julie Déserti on the properties of a one-parameter family of polynomial automorphisms of C^3; we show that new behaviours can be observed in comparison with the two-dimensional case. In particular, we study the escape speed of points to infinity and show that a transition exists for a certain value of the parameter. The last chapter is based on a joint work with Jiangong You, Zhiyan Zhao and Qi Zhou; we get asymptotic estimates on the size of spectral gaps for quasi-periodic Schrödinger operators in the analytic case. We obtain exponential upper bounds in the subcritical regime, which strengthens a previous result due to Sana Ben Hadj Amor. In the particular case of almost Mathieu operators, we also show exponential lower bounds, which provides quantitative estimates in connection with the so-called "Dry ten Martinis problem". As consequences of our results, we show applications to the homogeneity of the spectrum of such operators, and to Deift's conjecture
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Svanström, Fredrik. "Properties of a generalized Arnold’s discrete cat map." Thesis, Linnéuniversitetet, Institutionen för matematik (MA), 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-35209.
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After reviewing some properties of the two dimensional hyperbolic toral automorphism called Arnold's discrete cat map, including its generalizations with matrices having positive unit determinant, this thesis contains a definition of a novel cat map where the elements of the matrix are found in the sequence of Pell numbers. This mapping is therefore denoted as Pell's cat map. The main result of this thesis is a theorem determining the upper bound for the minimal period of Pell's cat map. From numerical results four conjectures regarding properties of Pell's cat map are also stated. A brief exposition of some applications of Arnold's discrete cat map is found in the last part of the thesis.
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DISCENDENTI, MARCO. "Secondary elliptic islands in a dynamical system close to hyperbolic." Doctoral thesis, Università degli Studi di Roma "Tor Vergata", 2010. http://hdl.handle.net/2108/1402.
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Consideriamo una famiglia a un parametro di diffeomorfismi simplettici del toro bidimensionale con un punto fisso che passa dall'iperbolicità all'ellitticità. Dimostriamo che esiste un insieme di valori per il parametro per cui la mappa deve avere delle isole ellittiche secondarie.We consider a one parameter family of symplectic maps that cross a non-uniformly hyperbolic situation into an elliptic one. We prove that there exists a set of values of the parameter such that the map has secondary elliptic islands.
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Glaister, P. "Approximate Riemann solvers for systems of hyperbolic conservation laws." Thesis, University of Reading, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.382211.
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Fall, Djiby. "Longtime dynamics of hyperbolic evolutionary equations in ubounded domains and lattice systems." [Tampa, Fla.] : University of South Florida, 2005. http://purl.fcla.edu/fcla/etd/SFE0001053.
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Monclair, Daniel. "Dynamique lorentzienne et groupes de difféomorphismes du cercle." Phd thesis, Ecole normale supérieure de lyon - ENS LYON, 2014. http://tel.archives-ouvertes.fr/tel-01061010.
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Cette thèse comporte deux parties, axées sur des aspects différents de la géométrie lorentzienne. La première partie porte sur les groupes d'isométries de surfaces lorentziennes globalement hyperboliques spatialement compactes, particulièrement lorsque le groupe exhibe une dynamique non triviale (action non propre). Le groupe d'isométries agit naturellement sur le cercle par difféomorphismes, et les résultats principaux portent sur la classification de ces représentations. Sous une hypothèse sur le bord conforme, on obtient une conjugaison par homéomorphisme avec l'action projective d'un sous-groupe de PSL(2,R) ou de l'un de ses revêtements finis. La différentiabilité de la conjuguante est étudiée, avec des résultats qui garantissent une conjugaison dans le groupe de difféomorphismes du cercle dans certains cas. On donne également des contre-exemples à l'existence d'une conjugaison différentiable, y compris pour des groupes ayant une dynamique riche. Ces constructions s'appuient sur l'étude de flots hyperboliques en dimension trois. Sans l'hypothèse sur le bord conforme, on obtient une semi conjugaison et un isomorphisme de groupes. On construit également des exemples pour lesquels il n'existe pas de conjugaison topologique. La seconde partie de cette thèse étudie un espace-temps vu comme un système dynamique multi-valuée : à un point on associe sont futur causal. Cette approche, déjà présente dans les travaux de Fathi et Siconolfi, permet de concrétiser le lien entre fonctions de Lyapunov en systèmes dynamiques et fonctions temps. Le résultat principal est une version lorentzienne du Théorème de Conley : on peut définir l'ensemble récurrent par chaînes d'un espace-temps, et il existe une fonction continue croissante le long de toute courbe causale orientée vers le futur, strictement croissante si le point de départ de la courbe n'est pas dans l'ensemble récurrent par chaînes. Ces techniques s'adaptent aussi dans un espace-temps stablement causal, ce qui permet de donner une nouvelle preuve d'une partie du Théorème d'Hawking.
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Barril, Basil Carles. "Semilinear hyperbolic equations and the dynamics of gut bacteria." Doctoral thesis, Universitat Autònoma de Barcelona, 2018. http://hdl.handle.net/10803/643304.
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En aquesta tesi proposem un marc matemàtic amb el qual analitzar la dinàmica dels microorganismes que creixen als intestins dels animals. Aquest marc consisteix en un sistema d’EDPs hiperbòliques amb termes de reacció no lineals i certes condicions de frontera que relacionen els microbis de l’ambient amb els que es troben dins els hostes.
Al capítol 1 solucionem el Problema Abstracte de Cauchy associat al model considerant la seva formulació semilineal en un determinat espai de Banach X. L’estructura semilineal del sistema obtingut és especial perquè, d’una banda, la llei d’evolució es pot expressar com la suma d’un operador lineal però no acotat i una funció Lipschitz no lineal (situació habitual), però, d’altra banda, la pertorbació no lineal pren valors no en X sinó en un espai més gran Y relacionat amb X (situació atípica). Per tal de tractar el problema utilitzem la teoria de semigrups duals. També estudiem l’estabilitat del sistema al voltant d’equilibris quan la pertorbació no lineal és Fréchet diferenciable. Aquests resultats es basen en dues propietats: la primera relaciona la dinàmica del semiflux amb el semigrup linealitzat al voltant de l’equilibri, i la segona relaciona el comportament asimptòtic del semigrup lineal amb l’espectre del seu generador. La darrera es prova mostrant que el “Teorema de l’Aplicació Espectral” sempre es compleix en els semigrups obtinguts en linealitzar el semiflux.
Al capítol 2 es presenta i s’analitza un sistema semilineal d’EDPs hiperbòliques autònom que representa la proliferació de bacteris en un grup heterogeni d’animals. S’assumeix que els bacteris que creixen a l’intestí poden trobar-se suspesos a la llum o adherits a l’epiteli. Donem una condició en funció de paràmetres ecològics que determina l’existència d’equilibris endèmics així com llur estabilitat. Plantegem algunes implicacions relacionades amb la teràpia amb bacteriòfags. Al capítol 3 donem, com a funció de paràmetres del model, el número reproductiu bàsic associat a la població bacteriana, és a dir, el nombre esperat de cèl·lules filles que produeix un bacteri. Addicionalment, introduïm una quantitat alternativa que es basa en el número de bacteris que es produeixen a l’intestí a partir d’un bacteri de l’ambient. La fórmula associada a aquesta segona quantitat és més simple que la primera, la qual cosa permet abordar qüestions sobre la biologia del sistema amb més facilitat. Ambdues quantitats coincideixen i són iguals a 1 al llindar que determina l’extinció, per sota del qual la població bacteriana s’extingeix. També obtenim valors òptims de les dues quantitats sota certes relacions entre els paràmetres del model.In this thesis we propose a mathematical framework to analyse the dynamics of microorganisms growing within the guts of animals. Such a framework consists of a hyperbolic system of PDEs with non-linear reaction terms and certain boundary conditions that link the microbes in the environment with those inside the hosts. In chapter 1 we solve the Abstract Cauchy Problem associated to the model by considering the semilinear formulation on a certain Banach space X. The semilinear structure of the system obtained in this way is special because, on the one hand, the evolution law can be expressed as the sum of a linear unbounded operator and a non-linear Lipschitz function (which is typical) but, on the other hand, the non-linear perturbation takes values not in X but on a larger space Y which is related to X (which is atypical). In order to deal with this situation we use the theory of dual semigroups. Stability results around steady states are also given when the nonlinear perturbation is Fréchet differentiable. These results are based on two propositions: one relating the local dynamics of the non-linear semiow with the linearised semigroup around the equilibrium, and a second relating the dynamical properties of the linearised semigroup with the spectral values of its generator. The later is proven by showing that the Spectral Mapping Theorem always applies to the semigroups one obtains when the semiow is linearised. In chapter 2 an autonomous semi-linear hyperbolic pde system for the proliferation of bacteria within a heterogeneous population of animals is presented and analysed. It is assumed that bacteria grow inside the intestines and that they can be either attached to the epithelial wall or as free particles in the lumen. A condition involving ecological parameters is given, which can be used to decide the existence of endemic equilibria as well as local stability properties of the non-endemic one. Some implications on phage therapy are addressed. In chapter 3 the basic reproduction number associated to the bacterial population, i.e. the expected number of daughter cells per bacterium, is given explicitly in terms of biological parameters. In addition, an alternative quantity is introduced based on the number of bacteria produced within the intestine by one bacterium originally in the external media. The latter depends on the parameters in a simpler way and provides more biological insight than the standard reproduction number, allowing the design of experimental procedures. Both quantities coincide and are equal to one at the extinction threshold, below which the bacterial population becomes extinct. Optimal values of both reproduction numbers are derived assuming parameter trade-offs.
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Unterweger, Kristof Gregor [Verfasser]. "High-Performance Coupling of Dynamically Adaptive Grids and Hyperbolic Equation Systems / Kristof Gregor Unterweger." München : Verlag Dr. Hut, 2017. http://d-nb.info/1126296031/34.
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Purdy, Daniel S. "An application of the hyperbolic navigation radio system for automated position and control." Thesis, Virginia Tech, 1989. http://hdl.handle.net/10919/46061.
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As automation in the construction site of the future becomes a reality, position location systems are necessary to provide real-time data to an operator. This thesis addresses problems associated with development of a real time automated position location system using a method similar to hyperbolic navigation methods. The Automated Position and Control (APAC) project is a joint effort between the Civil and Electrical Engineering departments at Virginia Polytechnic and State University and Bechtel Eastern Power Corporation.
Master of Science
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Schnellmann, Daniel. "Viana maps and limit distributions of sums of point measures." Phd thesis, KTH, Matematik (Inst.), 2009. http://tel.archives-ouvertes.fr/tel-00694201.
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This thesis consists of five articles mainly devoted to problems in dynamical systems and ergodic theory. We consider non-uniformly hyperbolic two dimensional systems and limit distributions of point measures which are absolutely continuous with respect to the Lebesgue measure. Let $f_{a_0}(x)=a_0-x^2$ be a quadratic map where the parameter $a_0\in(1,2)$ is chosen such that the critical point $0$ is pre-periodic (but not periodic). In Papers A and B we study skew-products $(\th,x)\mapsto F(\th,x)=(g(\th),f_{a_0}(x)+\al s(\th))$, $(\th,x)\in S^1\times\real$. The functions $g:S^1\to S^1$ and $s:S^1\to[-1,1]$ are the base dynamics and the coupling functions, respectively, and $\al$ is a small, positive constant. Such quadratic skew-products are also called Viana maps. In Papers A and B we show for several choices of the base dynamics and the coupling function that the map $F$ has two positive Lyapunov exponents and for some cases we further show that $F$ admits also an absolutely continuous invariant probability measure. In Paper C we consider certain Bernoulli convolutions. By showing that a specific transversality property is satisfied, we deduce absolute continuity of the to these Bernoulli convolutions associated distributions. In Papers D and E we consider sequences of real numbers in the unit interval and study how they are distributed. The sequences in Paper D are given by the forward iterations of a point $x\in[0,1]$ under a piecewise expanding map $T_a:[0,1]\to[0,1]$ depending on a parameter $a$ contained in an interval $I$. Under the assumption that each $T_a$ admits a unique absolutely continuous invariant probability measure $\mu_a$ and that some technical conditions are satisfied, we show that the distribution of the forward orbit $T_a^j(x)$, $j\ge1$, is described by the distribution $\mu_a$ for Lebesgue almost every parameter $a\in I$. In Paper E we apply the ideas in Paper D to certain sequences which are equidistributed in the unit interval and give a geometrical proof of an old result by Koksma.
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Gersbacher, Christoph [Verfasser], and Dietmar [Akademischer Betreuer] Kröner. "Higher-order discontinuous finite element methods and dynamic model adaptation for hyperbolic systems of conservation laws." Freiburg : Universität, 2017. http://d-nb.info/1136263853/34.
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Lichtner, Mark. "Exponential dichotomy and smooth invariant center manifolds for semilinear hyperbolic systems." Doctoral thesis, [S.l.] : [s.n.], 2006. http://deposit.ddb.de/cgi-bin/dokserv?idn=981306659.
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Dalal, Abdulsalam Elmabruk Daw. "Shadow Wave Solutions for Some Balance Law Systems." Phd thesis, Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu, 2017. https://www.cris.uns.ac.rs/record.jsf?recordId=104976&source=NDLTD&language=en.
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In the first part, the pressureless gas dynamic system with source (body force) is examined and solved by using Shadow Waves. The source represents gravity and Shadow Wave solution (containing the delta function) shows acceleration (contrary to shocks, for example). In the second part, one will nd numerical calculations that conrms the above results.Rad je posvecen analizi modela gasa bez pritiska uz dodatak izvora. Model je resen koriscenjem senka talasa. U ovom slucaju, izvor predstavlja uticaj gravitacije na cestice u modelu. Za razliku od udarnih talasa, talasi senke koje sadrze delta funkciju, krecu se ubrzano pod gravitacionim uticajem. U drugom delu rada su naprevljeni numericki eksperimenti koji potvrdjuju teoijske rezultate.
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Sroczinski, Matthias [Verfasser]. "Global existence and asymptotic decay for quasilinear second-order symmetric hyperbolic systems of partial differential equations occurring in the relativistic dynamics of dissipative fluids / Matthias Sroczinski." Konstanz : KOPS Universität Konstanz, 2019. http://d-nb.info/1184795460/34.
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Mohammadian, Saeed. "Freeway traffic flow dynamics and safety: A behavioural continuum framework." Thesis, Queensland University of Technology, 2021. https://eprints.qut.edu.au/227209/1/Saeed_Mohammadian_Thesis.pdf.
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Congestion and rear-end crashes are two undesirable phenomena of freeway traffic flows, which are interrelated and highly affected by human psychological factors. Since congestion is an everyday problem, and crashes are rare events, congestion management and crash risk prevention strategies are often implemented through separate research directions. However, overwhelming evidence has underscored the inter-relation between rear-end crashes and freeway traffic flow dynamics in recent decades. This dissertation develops novel mathematical models for freeway traffic flow dynamics and safety to integrate them into a unifiable framework. The outcomes of this PhD can enable moving towards faster and safer roads.
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Trinh, Ngoc Tu. "Étude sur le contrôle / régulation automatique des systèmes non-linéaires hyperboliques." Thesis, Lyon, 2017. http://www.theses.fr/2017LYSE1195/document.
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Dans cette étude on s'intéresse à la dynamique d'une classe de systèmes non-linéaires décrits par des équations aux dérivées partielles (EDP) du type hyperbolique. L'objectif de l'étude est de construire des lois de contrôle par feedback dynamique de la sortie afin de stabiliser le système autour d'un point d'équilibre d'une part, et, d'autre part, de réguler la sortie vers le point de consigne. Nous considérons la classe des systèmes gouvernés par des EDP quasi-linéaires du type hyperbolique à deux variables indépendantes (une variable temporelle et une variable spatiale). Pour le bien-posé du système dynamique non seulement l'état initial mais aussi certaines conditions frontières doivent être prescrites en cohérence avec les EDP. Nous supposons que l'observation et le contrôle sont ponctuels. Autrement dit l'action du contrôle intervient dans le système via les conditions frontières et l'observation est effectuée aux points de la frontière. Notre étude est motivée par l'observation que de nombreux processus physiques sont modélisés par ce type d'équations EDP. Nous citons, par exemple, des processus tels que flux trafique en transport, flux de gaz dans un réseau de pipeline, échangeurs thermiques en génie des procédés, équations de télégraphe dans des lignes de transmission, canaux d'irrigation en génie civil etc. Nous commençons l'étude par une EDP non-linéaire scalaire. Dans ce cas-là nous proposons un correcteur intégral stabilisant qui assure la régulation de la sortie avec l'erreur statique nulle. Nous prouvons la stabilisation locale du système non-linéaire par le correcteur intégral en construisant une fonctionnelle de Lyapunov appropriée. La conception des correcteurs proportionnels et intégraux (PI) que nous proposons est étendue dans un cadre de systèmes de deux EDP. Nous prouvons la stabilisation du système en boucle fermée à l'aide d'une nouvelle fonctionnelle de Lyapunov. La synthèse des correcteurs PI stabilisants se poursuit dans un cadre de réseaux formés d'un nombre fini de systèmes à deux EDP : réseau étoilé et réseau série en cascade. Les contrôles et les observations se trouvent localisés aux différents nœuds de connexion. Pour ces configurations nous présentons un ensemble de correcteurs PI stabilisants qui assurent la régulation vers le point de consigne. Les correcteurs PI que nous concevons sont validés par des simulations numériques à partir des modèles non-linéaires EDP. La contribution de la thèse, par rapport à la littérature existante, consiste en l'élaboration de nouvelles fonctionnelles de Lyapunov pour une classe de systèmes stabilisés par correcteur PI. En effet une grande quantité de résultats ont été obtenus sur la stabilisation des systèmes hyperboliques par feedback statique de la sortie. Toutefois il existe encore peu de résultats sur la stabilisation de ces systèmes par feedback dynamique de la sortie. L'étude de la thèse est consacrée sur l'élaboration des fonctionnelles de Lyapunov permettant d'obtenir des correcteurs PI stabilisants. L'approche de Lyapunov direct que nous avons proposée a pour l'avantage de permettre d'étudier la robustesse des lois de feedback de la sortie PI vis-à-vis de la non-linéarité. Une autre contribution de la thèse consiste en la construction des programmes de Malab permettant d'effectuer des simulations numériques pour la validation des correcteurs conçus. Pour la résolution numérique des EDP hyperboliques nous avons discrétisé nos systèmes par le schéma numérique de Preissmann. Nous avons chaque fois un système d'équations algébriques non-linéaires à résoudre de façon récurrente. L'apport des simulations numériques nous permet de mieux comprendre la méthodologie applicative de la théorie du contrôle en dimension infinieIn this study we are interested in the dynamics of a class of nonlinear systems described by partial differential equations (PDE) of the hyperbolic type. The aim of the study is to construct control laws by dynamic feedback of the output in order to stabilize the system around an equilibrium point on the one hand and to regulate the output to the set-point. We consider the class of systems governed by hyperbolic PDEs with two independent variables (one time variable and one spatial variable). For the well-posed dynamic system not only the initial state but also certain boundary conditions must be prescribed in coherence with the PDEs. We assume that observation and control are punctual. In other words, the action of the control intervenes in the system via the boundary conditions and the observation is carried out at the points of the border. Our study is motivated by the observation that many physical processes are modeled by this type of PDE equations. Examples include processes such as traffic flow in transportation, gas flows in a pipeline network, heat exchangers in process engineering, telegraph equations in transmission lines, civil engineering irrigation channels, to cite but a few.We begin the study with a scalar nonlinear PDE. In this case we propose a stabilizing integral controller which ensures the regulation of the output with zero static error. We prove the local stabilization of the nonlinear system by the integral controller by constructing an appropriate Lyapunov functional. The design of the proportional and integral (PI) controllers that we propose is extended in a framework of two PDE systems. We prove the stabilization of the closed-loop system with a new Lyapunov functional. The synthesis of stabilizing PI controllers is carried out in a framework of networks formed by a finite number of two PDE systems: star network and serial network in cascade. Controls and observations are located at the different connection nodes. For these configurations we present a set of stabilizing PI controllers that regulate the output to the set-point. The PI controllers that we design are validated by numerical simulations from the nonlinear PDE models. The contribution of the thesis compared to the existing literature consists in the development of new Lyapunov functionals for the class of systems looped by a PI controller. Indeed, a large number of results have been obtained from the stabilization of hyperbolic systems by static feedback of the output. However, there are still few results with the stabilization of these systems by the output dynamic feedback. The study of the thesis is devoted to the development of the Lyapunov functional to obtain stabilizing PI controllers. The direct Lyapunov approach that we have proposed has the advantage of allowing to study the robustness of the output dynamic feedback laws in the form of PI controllers with respect to the nonlinearity. Another contribution of the thesis consists of the Malab program construction allowing to carry out numerical simulations for the validation of the conceived controllers. For the numerical resolution of hyperbolic PDEs, we have discretized our systems using the Preissmann numerical scheme. Each time moment we have a system of non-linear algebraic equations to be solved in a recurring way. The contribution of numerical simulations allows us to better understand the application methodology of the infinite dimension control theory
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Mercier, Magali. "Étude de différents aspects des EDP hyperboliques : persistance d’onde de choc dans la dynamique des fluides compressibles, modélisation du trafic routier, stabilité des lois de conservation scalaires." Thesis, Lyon 1, 2009. http://www.theses.fr/2009LYO10246/document.
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On étudie dans ce travail des systèmes de lois de conservation hyperboliques. La première partie étudie le temps d'existence des solutions régulières et régulières par morceaux de la dynamique des fluides compressibles. Après avoir présenté l'état de l'art en matière de solutions régulières, on montre une extension d'un théorème de Grassin à des gaz de Van der Waals. On étudie ensuite les solutions ondes de chocs : on poursuit l'approche de T. T. Li pour estimer leur temps d'existence dans le cas isentropique à symétrie sphérique, et l'approche de Whitham afin d'obtenir une équation approchée vérifiée par la surface de discontinuité. Dans une deuxième partie, motivée par la modélisation d'un rond-point en trafic routier, on étudie une extension multi-classe du modèle macroscopique de Lighthill-Whitham-Richards sur une route infinie avec des jonctions. On différencie les véhicules selon leur origine et leur destination et on introduit des conditions aux bords adaptées au niveau des jonctions. On obtient existence et unicité d'une solution au problème de Riemann pour ce modèle. Des simulations numériques attestent que les solutions obtenues existent en temps long. On aborde enfin le problème de Cauchy par la méthode de front tracking. La dernière partie concerne les lois de conservation scalaires. La première question abordée est le contrôle de la variation totale de la solution et la stabilité des solutions faibles entropiques par rapport au flux et à la source. Ce résultat nous permet d'étudier des équations avec flux non-local. Une fois établi leur caractère bien posé, on montre la Gâteaux-différentiabilité du semi-groupe obtenu par rapport aux conditions initialesIn this work, we study hyperbolic systems of balance laws. The first part is devoted to compressible fluid dynamics, and particularly to the lifespan of smooth or piecewise smooth solutions. After presenting the state of art, we show an extension to more general gases of a theorem by Grassin.We also study shock waves solutions: first, we extend T. T. Li's approach to estimate the time of existence in the isentropic spherical case; second, we develop Whitham's ideas to obtain an approximated equation satisfied by the discontinuity surface. In the second part, we set up a new model for a roundabout. This leads us to study a multi-class extension of the macroscopic Lighthill-Whitham-Richards' model. We study the traffic on an infinite road, with some points of junction. We distinguish vehicles according to their origin and destination and add some boundary conditions at the junctions. We obtain existence and uniqueness of a weak entropy solution for the Riemann problem. As a complement, we provide numerical simulations that exhibit solutions with a long time of existence. Finally, the Cauchy problem is tackled by the front tracking method. In the last part, we are interested in scalar hyperbolic balance laws. The first question addressed is the control of the total variation and the stability of entropy solutions with respect to flow and source. With this result, we can study equations with non-local flow, which do not fit into the framework of classical theorems. We show here that these kinds of equations are well posed and we show the Gâteaux-differentiability with respect to initial conditions, which is important to characterize maxima or minima of a given cost functional
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Al, Zohbi Maryam. "Contributions to the existence, uniqueness, and contraction of the solutions to some evolutionary partial differential equations." Thesis, Compiègne, 2021. http://www.theses.fr/2021COMP2646.
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Dans cette thèse, nous nous sommes principalement intéressés à l’étude théorique et numérique de quelques équations qui décrivent la dynamique des densités des dislocations. Les dislocations sont des défauts microscopiques qui se déplacent dans les matériaux sous l’effet des contraintes extérieures. Dans un premier travail, nous démontrons un résultat d’existence globale en temps des solutions discontinues pour un système hyperbolique diagonal qui n’est pas nécessairement strictement hyperbolique, dans un espace unidimensionnel. Ainsi dans un deuxième travail, nous élargissons notre portée en démontrant un résultat similaire pour un système d’équations de type eikonal non-linéaire qui est en fait une généralisation du système hyperbolique déjà étudié. En effet, nous prouvons aussi l’existence et l’unicité d’une solution continue pour le système eikonal. Ensuite, nous nous sommes intéressés à l’analyse numérique de ce système en proposant un schéma aux différences finies, par lequel nous montrons la convergence vers le problème continu et nous consolidons nos résultats avec quelques simulations numériques. Dans une autre direction, nous nous sommes intéressés à la théorie de contraction différentielle pour les équations d’évolutions. Après avoir introduit une nouvelle distance, nous construisons une nouvelle famille des solutions contractantes positives pour l’équation d’évolution p-LaplaceIn this thesis, we are mainly interested in the theoretical and numerical study of certain equations that describe the dynamics of dislocation densities. Dislocations are microscopic defects in materials, which move under the effect of an external stress. As a first work, we prove a global in time existence result of a discontinuous solution to a diagonal hyperbolic system, which is not necessarily strictly hyperbolic, in one space dimension. Then in another work, we broaden our scope by proving a similar result to a non-linear eikonal system, which is in fact a generalization of the hyperbolic system studied first. We also prove the existence and uniqueness of a continuous solution to the eikonal system. After that, we study this system numerically in a third work through proposing a finite difference scheme approximating it, of which we prove the convergence to the continuous problem, strengthening our outcomes with some numerical simulations. On a different direction, we were enthused by the theory of differential contraction to evolutionary equations. By introducing a new distance, we create a new family of contracting positive solutions to the evolutionary p-Laplacian equation
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Mummert, Anna. "Thermodynamic formalism for nonuniformly hyperbolic dynamical systems." 2006. http://etda.libraries.psu.edu/theses/approved/WorldWideIndex/ETD-1432/index.html.
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Whittaker, Michael Fredrick. "Poincaré duality and spectral triples for hyperbolic dynamical systems." Thesis, 2010. http://hdl.handle.net/1828/2897.
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We study aspects of noncommutative geometry on hyperbolic dynamical systems known as Smale spaces. In particular, there are two C*-algebras, defined on the stable and unstable groupoids arising from the hyperbolic dynamics. These give rise to two additional crossed product C*-algebras known as the stable and unstable Ruelle algebras. We show that the Ruelle algebras exhibit noncommutative Poincaré duality. As a consequence we obtain isomorphisms between the K-theory and K-homology groups of the stable and unstable Ruelle algebras. A second result defines spectral triples on these C*-algebras and we show that the spectral dimension recovers the topological entropy of the Smale space itself. Finally we define a natural Fredholm module on the Ruelle algebras in the special case that the Smale space is a shift of finite type. Using unitary operators arising from the Pimsner-Voiculescu sequence we compute the index pairing with our Fredholm module for specific examples.
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Carrasco, Correa Pablo Daniel. "Compact Dynamical Foliations." Thesis, 2011. http://hdl.handle.net/1807/27574.
Full textAbstract:
According to the work of Dennis Sullivan, there exists a smooth flow on the 5-sphere all of whose orbits are periodic although there is no uniform bound on their periods. The question addressed in this thesis is whether such an example can occur in the partially hyperbolic context. That is, does there exist a partially hyperbolic diffeomorphism of a compact manifold such that all the leaves of its center foliation are compact although there is no uniform bound for their volumes. We will show that the answer to the previous question under the very mild hypothesis of dynamical coherence is no.
The thesis is organized as follows. In the first chapter we give the necessary background and results in partially hyperbolic dynamics needed for the rest of the work, studying in particular the geometry of the center foliation. Chapter two is devoted to a general discussion of compact foliations. We give proof or sketches of all the relevant results used. Chapter three is the core of the thesis, where we establish the non existence of Sullivan's type of examples in the partially hyperbolic domain, and generalize to diffeomorphisms whose center foliation has arbitrary dimension. The last chapter is devoted to applications of the results of chapter three, where in particular it is proved that if the center foliation of a dynamically coherent partially hyperbolic diffeomorphism is compact and without holonomy, then it is plaque expansive.
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Talitskaya, Anna. "Partially hyperbolic phenomena in dynamical systems with discrete and continuous time." 2004. http://etda.libraries.psu.edu/theses/approved/WorldWideIndex/ETD-533/index.html.
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"Some new results on hyperbolic gauss curvature flows." Thesis, 2011. http://library.cuhk.edu.hk/record=b6075160.
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Wo, Weifeng.Thesis (Ph.D.)--Chinese University of Hong Kong, 2011.
Includes bibliographical references (leaves 99-102).
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Abstract also in Chinese.
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Wieler, Susana. "Smale spaces with totally disconnected local stable sets." Thesis, 2012. http://hdl.handle.net/1828/3905.
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A Smale space is a chaotic dynamical system with canonical coordinates of contracting and expanding directions. The basic sets for Smale’s Axiom A systems are a key class of examples. R.F. Williams considered the special case where the basic set had a totally disconnected contracting set and a Euclidean expanding one. He provided a construction using inverse limits of such examples and also proved that (under appropriate hyptotheses) all such basic sets arose from this construction. We will be working in the metric setting of Smale spaces, but the goal is to extend Williams’ results by removing all hypotheses on the unstable sets. We give criteria on a stationary inverse limit which ensures the result is a Smale space. We also prove that any irreducible Smale space with totally disconnected local stable sets is obtained through this construction.Graduate
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Bohnet, Doris. "Partially hyperbolic diffeomorphisms with a compact center foliation with finite holonomy." Phd thesis, 2011. http://tel.archives-ouvertes.fr/tel-00782664.
Full textAbstract:
On démontre que les difféomorphismes partiellement hyperboliques admettant un feuilletage central invariant compact sont dynamiquement cohérent et en plus ils ont la propriété de pistage. En supposant une direction instable uni-dimensionnelle et orientée on peut prouver que le difféomorphisme projette en un automorphisme hyperbolique de tore dans l'espace des feuilles centrales.
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Wieler, Susana. "Symbolic and geometric representations of unimodular Pisot substitutions." Thesis, 2007. http://hdl.handle.net/1828/131.
Full textAbstract:
We review the construction of three Smale spaces associated to a unimodular Pisot substitution on d letters: a subshift of finite type (SFT), a substitution tiling space, and a hyperbolic toral automorphism on the Euclidean d-torus. By considering an SFT whose elements are biinfinite, rather than infinite, paths in the graph associated to the substitution, we modify a well-known map to obtain a factor map between our SFT and the hyperbolic toral automorphism on the d-torus given by the incidence matrix of the substitution. We prove that if the tiling substitution forces its border, then this factor map is the composition of an s-resolving factor map from the SFT to a one-dimensional substitution tiling space and a u-resolving factor map from the tiling space to the d-torus.
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Garg, Naveen Kumar. "Novel Upwind and Central Schemes for Various Hyperbolic Systems." Thesis, 2017. http://etd.iisc.ac.in/handle/2005/3564.
Full textAbstract:
The class of hyperbolic conservation laws model the phenomena of non-linear wave propagation, including the presence and propagation of discontinuities and expansion waves. Such nonlinear systems can generate discontinuities in the so-lution even for smooth initial conditions. Presence of discontinuities results in break down of a solution in the classical sense and to show existence, weak for-mulation of a problem is required. Moreover, closed form solutions are di cult to obtain and in some cases such solutions are even unavailable. Thus, numerical algorithms play an important role in solving such systems. There are several dis-cretization techniques to solve hyperbolic systems numerically and Finite Volume Method (FVM) is one of such important frameworks. Numerical algorithms based on FVM are broadly classi ed into two categories, central discretization methods and upwind discretization methods. Various upwind and central discretization methods developed so far di er widely in terms of robustness, accuracy and ef-ciency and an ideal scheme with all these characteristics is yet to emerge. In this thesis, novel upwind and central schemes are formulated for various hyper-bolic systems, with the aim of maintaining right balance between accuracy and robustness.
This thesis is divided into two parts. First part consists of the formulation of upwind methods to simulate genuine weakly hyperbolic (GWH) systems. Such systems do not possess full set of linearly independent (LI) eigenvectors and some of the examples include pressureless gas dynamics system, modi ed Burgers' sys-tem and further modi ed Burgers' system. The main challenge while formulating an upwind solver for GWH systems, using the concept of Flux Di erence Splitting (FDS), is to recover full set of LI eigenvectors, which is done through addition of generalized eigenvectors using the theory of Jordan Canonical Forms. Once the defective set of LI eigenvectors are completed, a novel (FDS-J) solver is for-mulated in such a manner that it is independent of generalized eigenvectors, as they are not unique. FDS-J solver is capable of capturing various shocks such as
-shocks, 0-shocks and 00-shocks accurately. In this thesis, the FDS-J schemes are proposed for those GWH systems each of which have one particular repeated eigenvalue with arithmetic multiplicity (AM) greater than one. Moreover, each
ux Jacobian matrix corresponding to such systems is similar to a unique Jordan matrix.
After the successful treatment of genuine weakly hyperbolic systems, this strategy is further applied to those weakly hyperbolic subsystems which result on employ-ing various convection-pressure splittings to the Euler ux function. For example, Toro-Vazquez (TV) splitting and Zha-Bilgen (ZB) type splitting approaches to split the Euler ux function yield genuine weakly hyperbolic convective parts and strict hyperbolic pressure parts. Moreover, the ux Jacobian of each convective part is similar to a Jordan matrix with at least two lower order Jordan blocks. Based on the lines of FDS-J scheme, we develop two numerical schemes for Eu-ler equations using TV splitting and ZB type splitting. Both the new ZBS-FDS and TVS-FDS schemes are tested on various 1-D shock tube problems and out of two, contact capturing ZBS-FDS scheme is extended to 2-dimensional Euler system where it is tested successfully on various test cases including many shock instability problems.
Second part of the thesis is associated with the development of simple, robust and accurate central solvers for systems of hyperbolic conservation laws. The idea of splitting schemes together with the notion of FDS is not easily extendable to systems such as shallow water equations. Thus, a novel central solver Convection Isolated Discontinuity Recognizing Algorithm (CIDRA) is formulated for shallow water equations. As the name suggests, the convective ux is isolated from the total ux in such a way that other ux, in present case other ux represents celerity part, must possess non-zero eigenvalue contribution. FVM framework is applied to each part separately and ux equivalence principle is used to x the coe cient of numerical di usion. CIDRA for SWE is computed on various 1-D and 2-D benchmark problems and extended to Euler systems e ortlessly. As a further improvement, a scalar di usion based algorithm CIDRA-1 is designed for
v
Euler systems. The scalar di usion coe cient depends on that particular part of the Rankine-Hugoniot (R-H) condition which involves total energy of the system as a direct contribution. This algorithm is applied to a variety of shock tube test cases including a class of low density ow problems and also to various 2-D test problems successfully.
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Garg, Naveen Kumar. "Novel Upwind and Central Schemes for Various Hyperbolic Systems." Thesis, 2017. http://etd.iisc.ernet.in/2005/3564.
Full textAbstract:
The class of hyperbolic conservation laws model the phenomena of non-linear wave propagation, including the presence and propagation of discontinuities and expansion waves. Such nonlinear systems can generate discontinuities in the so-lution even for smooth initial conditions. Presence of discontinuities results in break down of a solution in the classical sense and to show existence, weak for-mulation of a problem is required. Moreover, closed form solutions are di cult to obtain and in some cases such solutions are even unavailable. Thus, numerical algorithms play an important role in solving such systems. There are several dis-cretization techniques to solve hyperbolic systems numerically and Finite Volume Method (FVM) is one of such important frameworks. Numerical algorithms based on FVM are broadly classi ed into two categories, central discretization methods and upwind discretization methods. Various upwind and central discretization methods developed so far di er widely in terms of robustness, accuracy and ef-ciency and an ideal scheme with all these characteristics is yet to emerge. In this thesis, novel upwind and central schemes are formulated for various hyper-bolic systems, with the aim of maintaining right balance between accuracy and robustness.
This thesis is divided into two parts. First part consists of the formulation of upwind methods to simulate genuine weakly hyperbolic (GWH) systems. Such systems do not possess full set of linearly independent (LI) eigenvectors and some of the examples include pressureless gas dynamics system, modi ed Burgers' sys-tem and further modi ed Burgers' system. The main challenge while formulating an upwind solver for GWH systems, using the concept of Flux Di erence Splitting (FDS), is to recover full set of LI eigenvectors, which is done through addition of generalized eigenvectors using the theory of Jordan Canonical Forms. Once the defective set of LI eigenvectors are completed, a novel (FDS-J) solver is for-mulated in such a manner that it is independent of generalized eigenvectors, as they are not unique. FDS-J solver is capable of capturing various shocks such as
-shocks, 0-shocks and 00-shocks accurately. In this thesis, the FDS-J schemes are proposed for those GWH systems each of which have one particular repeated eigenvalue with arithmetic multiplicity (AM) greater than one. Moreover, each
ux Jacobian matrix corresponding to such systems is similar to a unique Jordan matrix.
After the successful treatment of genuine weakly hyperbolic systems, this strategy is further applied to those weakly hyperbolic subsystems which result on employ-ing various convection-pressure splittings to the Euler ux function. For example, Toro-Vazquez (TV) splitting and Zha-Bilgen (ZB) type splitting approaches to split the Euler ux function yield genuine weakly hyperbolic convective parts and strict hyperbolic pressure parts. Moreover, the ux Jacobian of each convective part is similar to a Jordan matrix with at least two lower order Jordan blocks. Based on the lines of FDS-J scheme, we develop two numerical schemes for Eu-ler equations using TV splitting and ZB type splitting. Both the new ZBS-FDS and TVS-FDS schemes are tested on various 1-D shock tube problems and out of two, contact capturing ZBS-FDS scheme is extended to 2-dimensional Euler system where it is tested successfully on various test cases including many shock instability problems.
Second part of the thesis is associated with the development of simple, robust and accurate central solvers for systems of hyperbolic conservation laws. The idea of splitting schemes together with the notion of FDS is not easily extendable to systems such as shallow water equations. Thus, a novel central solver Convection Isolated Discontinuity Recognizing Algorithm (CIDRA) is formulated for shallow water equations. As the name suggests, the convective ux is isolated from the total ux in such a way that other ux, in present case other ux represents celerity part, must possess non-zero eigenvalue contribution. FVM framework is applied to each part separately and ux equivalence principle is used to x the coe cient of numerical di usion. CIDRA for SWE is computed on various 1-D and 2-D benchmark problems and extended to Euler systems e ortlessly. As a further improvement, a scalar di usion based algorithm CIDRA-1 is designed for
v
Euler systems. The scalar di usion coe cient depends on that particular part of the Rankine-Hugoniot (R-H) condition which involves total energy of the system as a direct contribution. This algorithm is applied to a variety of shock tube test cases including a class of low density ow problems and also to various 2-D test problems successfully.
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Kaushik, K. N. "A Low Dissipative Relaxation Scheme For Hyperbolic Consevation Laws." Thesis, 2005. https://etd.iisc.ac.in/handle/2005/1661.
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Kaushik, K. N. "A Low Dissipative Relaxation Scheme For Hyperbolic Consevation Laws." Thesis, 2005. http://etd.iisc.ernet.in/handle/2005/1661.
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Hante, Falk Michael [Verfasser]. "Hybrid dynamics comprising modes governed by partial differential equations : modeling, analysis and control for semilinear hyperbolic systems in one space dimension / vorgelgt von Falk Michael Hante." 2010. http://d-nb.info/1006656782/34.
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El-Khatib, Mayar. "Highway Development Decision-Making Under Uncertainty: Analysis, Critique and Advancement." Thesis, 2010. http://hdl.handle.net/10012/5741.
Full textAbstract:
While decision-making under uncertainty is a major universal problem, its implications in the field of transportation systems are especially enormous; where the benefits of right decisions are tremendous, the consequences of wrong ones are potentially disastrous.
In the realm of highway systems, decisions related to the highway configuration (number of lanes, right of way, etc.) need to incorporate both the traffic demand and land price uncertainties. In the literature, these uncertainties have generally been modeled using the Geometric Brownian Motion (GBM) process, which has been used extensively in modeling many other real life phenomena. But few scholars, including those who used the GBM in highway configuration decisions, have offered any rigorous justification for the use of this model.
This thesis attempts to offer a detailed analysis of various aspects of transportation systems in relation to decision-making. It reveals some general insights as well as a new concept that extends the notion of opportunity cost to situations where wrong decisions could be made. Claiming deficiency of the GBM model, it also introduces a new formulation that utilizes a large and flexible parametric family of jump models (i.e., Lévy processes). To validate this claim, data related to traffic demand and land prices were collected and analyzed to reveal that their distributions, heavy-tailed and asymmetric, do not match well with the GBM model. As a remedy, this research used the Merton, Kou, and negative inverse Gaussian Lévy processes as possible alternatives.
Though the results show indifference in relation to final decisions among the models, mathematically, they improve the precision of uncertainty models and the decision-making process. This furthers the quest for optimality in highway projects and beyond.