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Добірка наукової літератури з теми "Hyperbolic dynamical systems"

Автор: Grafiati

Опубліковано: 7 грудня 2024

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Статті в журналах з теми "Hyperbolic dynamical systems"

Bandtlow, Oscar F., Wolfram Just, and Julia Slipantschuk. "A numerical study of rigidity of hyperbolic splittings in simple two-dimensional maps." Nonlinearity 37, no. 4 (March 4, 2024): 045007. http://dx.doi.org/10.1088/1361-6544/ad2b58.

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Abstract Chaotic hyperbolic dynamical systems enjoy a surprising degree of rigidity, a fact which is well known in the mathematics community but perhaps less so in theoretical physics circles. Low-dimensional hyperbolic systems are either conjugate to linear automorphisms, that is, dynamically equivalent to the Arnold cat map and its variants, or their hyperbolic structure is not smooth. We illustrate this dichotomy using a family of analytic maps, for which we show by means of numerical simulations that the corresponding hyperbolic structure is not smooth, thereby providing an example for a global mechanism which produces non-smooth phase space structures in an otherwise smooth dynamical system.

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Barinova, Marina K., and Evgenia K. Shustova. "Dynamical properties of direct products of discrete dynamical systems." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 24, no. 1 (March 31, 2022): 21–30. http://dx.doi.org/10.15507/2079-6900.24.202201.21-30.

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A natural way for creating new dynamical systems is to consider direct products of already known systems. The paper studies some dynamical properties of direct products of homeomorphisms and diffeomorphisms. In particular, authors prove that a chain-recurrent set of the direct product of homeomorphisms is a direct product of the chain-recurrent sets. Another result established in the paper is that the direct product of diffeomorphisms holds hyperbolic structure on the direct product of hyperbolic sets. It is known that if a diffeomorphism has a hyperbolic chain-recurrent set, then this mapping is Ω-stable. Therefore, it follows from the results of the paper that the direct product of Ω-stable diffeomorphisms is also Ω-stable. Another question which is raised in the article concerns the existence of an energy function for the direct product of diffeomorphisms which already have such functions (recall that energy function is a smooth Lyapunov function whose set of critical points coincides with the chain-recurrent set of the system). Authors show that in this case the function can be found as a weighted sum of energy functions of initial diffeomorphisms.

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Whittaker, Michael. "Spectral triples for hyperbolic dynamical systems." Journal of Noncommutative Geometry 7, no. 2 (2013): 563–82. http://dx.doi.org/10.4171/jncg/127.

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Gogolev, Andrey, Pedro Ontaneda, and Federico Rodriguez Hertz. "New partially hyperbolic dynamical systems I." Acta Mathematica 215, no. 2 (2015): 363–93. http://dx.doi.org/10.1007/s11511-016-0135-3.

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Lokutsievskii, L. V. "Fractal structure of hyperbolic Lipschitzian dynamical systems." Russian Journal of Mathematical Physics 19, no. 1 (March 2012): 27–43. http://dx.doi.org/10.1134/s1061920812010050.

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CARVALHO, ALEXANDRE N., JOSÉ A. LANGA, and JAMES C. ROBINSON. "Lower semicontinuity of attractors for non-autonomous dynamical systems." Ergodic Theory and Dynamical Systems 29, no. 6 (February 3, 2009): 1765–80. http://dx.doi.org/10.1017/s0143385708000850.

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AbstractThis paper is concerned with the lower semicontinuity of attractors for semilinear non-autonomous differential equations in Banach spaces. We require the unperturbed attractor to be given as the union of unstable manifolds of time-dependent hyperbolic solutions, generalizing previous results valid only for gradient-like systems in which the hyperbolic solutions are equilibria. The tools employed are a study of the continuity of the local unstable manifolds of the hyperbolic solutions and results on the continuity of the exponential dichotomy of the linearization around each of these solutions.

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Blankers, Vance, Tristan Rendfrey, Aaron Shukert, and Patrick Shipman. "Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers." Fractal and Fractional 3, no. 1 (February 20, 2019): 6. http://dx.doi.org/10.3390/fractalfract3010006.

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Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form x + τ y for x , y ∈ R , and τ 2 = 1 but τ ≠ ± 1 , are the natural number system in which to encode geometric properties of the Minkowski space R 1 , 1 . We show that the hyperbolic analog of the Mandelbrot set parameterizes the connectedness of hyperbolic Julia sets. We give a wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia sets.

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Pesin, Ya B. "Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties." Ergodic Theory and Dynamical Systems 12, no. 1 (March 1992): 123–51. http://dx.doi.org/10.1017/s0143385700006635.

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AbstractWe introduce a class of dynamical systems on a Riemannian manifold with singularities having attractors with strong hyperbolic behavior of trajectories. This class includes a number of famous examples such as the Lorenz type attractor, the Lozi attractor and some others which have been of great interest in recent years. We prove the existence of a special invariant measure which is an analog of the Bowen-Ruelle-Sinai measure for classical hyperbolic attractors and study the ergodic properties of the system with respect to this measure. We also describe some topological properties of the system on the attractor. Our results can be considered a dissipative version of the theory of systems with singularities preserving the smooth measure.

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Cong, Nguyen Dinh. "Structural stability of linear random dynamical systems." Ergodic Theory and Dynamical Systems 16, no. 6 (December 1996): 1207–20. http://dx.doi.org/10.1017/s0143385700009998.

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AbstractIn this paper, structural stability of discrete-time linear random dynamical systems is studied. A random dynamical system is called structurally stable with respect to a random norm if it is topologically conjugate to any random dynamical system which is sufficiently close to it in this norm. We prove that a discrete-time linear random dynamical system is structurally stable with respect to its Lyapunov norms if and only if it is hyperbolic.

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REY-BELLET, LUC, and LAI-SANG YOUNG. "Large deviations in non-uniformly hyperbolic dynamical systems." Ergodic Theory and Dynamical Systems 28, no. 2 (April 2008): 587–612. http://dx.doi.org/10.1017/s0143385707000478.

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AbstractWe prove large deviation principles for ergodic averages of dynamical systems admitting Markov tower extensions with exponential return times. Our main technical result from which a number of limit theorems are derived is the analyticity of logarithmic moment generating functions. Among the classes of dynamical systems to which our results apply are piecewise hyperbolic diffeomorphisms, dispersing billiards including Lorentz gases, and strange attractors of rank one including Hénon-type attractors.

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Дисертації з теми "Hyperbolic dynamical systems"

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-produit
This 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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Книги з теми "Hyperbolic dynamical systems"

Anosov, D. V. Dynamical Systems IX: Dynamical Systems with Hyperbolic Behaviour. Berlin, Heidelberg: Springer Berlin Heidelberg, 1995.

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V, Anosov D., ed. Dynamical systems with hyperbolic behavior. Berlin: Springer-Verlag, 1995.

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Wiggins, Stephen. Normally Hyperbolic Invariant Manifolds in Dynamical Systems. New York, NY: Springer New York, 1994. http://dx.doi.org/10.1007/978-1-4612-4312-0.

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Barreira, Luis. Ergodic Theory, Hyperbolic Dynamics and Dimension Theory. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012.

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Avila, Artur. Cocycles over partially hyperbolic maps. Paris: Société mathématique de France, 2013.

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Barreira, Luis. Dynamical Systems: An Introduction. London: Springer London, 2013.

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A, Rand D., and Ferreira Flávio, eds. Fine structures of hyperbolic diffeomorphisms. Berlin: Springer, 2009.

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Gaito, Stephen Thomas. Shadowing of weakly pseudo-hyperbolic pseudo-orbits in discrete dynamical systems. [s.l.]: typescript, 1992.

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W, Bates Peter. Existence and persistence of invariant manifolds for semiflows in Banach space. Providence, R.I: American Mathematical Society, 1998.

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Waddington, Simon. Prime orbit theorems for closed orbits and knots in hyperbolic dynamical systems. [s.l.]: typescript, 1992.

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Частини книг з теми "Hyperbolic dynamical systems"

Barreira, Luis, and Claudia Valls. "Hyperbolic Dynamics I." In Dynamical Systems, 87–112. London: Springer London, 2013. http://dx.doi.org/10.1007/978-1-4471-4835-7_5.

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Barreira, Luis, and Claudia Valls. "Hyperbolic Dynamics II." In Dynamical Systems, 113–51. London: Springer London, 2013. http://dx.doi.org/10.1007/978-1-4471-4835-7_6.

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Araújo, Vitor, and Marcelo Viana. "Hyperbolic Dynamical Systems." In Mathematics of Complexity and Dynamical Systems, 740–54. New York, NY: Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-1806-1_45.

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Araújo, Vitor, and Marcelo Viana. "Hyperbolic Dynamical Systems." In Encyclopedia of Complexity and Systems Science, 4723–37. New York, NY: Springer New York, 2009. http://dx.doi.org/10.1007/978-0-387-30440-3_279.

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Kuznetsov, Sergey P. "Dynamical Systems and Hyperbolicity." In Hyperbolic Chaos, 3–33. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-23666-2_1.

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Barreira, Luís, and Claudia Valls. "Hyperbolic Dynamics." In Dynamical Systems by Example, 135–67. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-15915-3_10.

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Barreira, Luís, and Claudia Valls. "Hyperbolic Dynamics." In Dynamical Systems by Example, 35–47. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-15915-3_4.

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Shub, Michael. "Hyperbolic Sets." In Global Stability of Dynamical Systems, 20–32. New York, NY: Springer New York, 1987. http://dx.doi.org/10.1007/978-1-4757-1947-5_4.

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Palmer, Ken. "Hyperbolic Sets of Diffeomorphisms." In Shadowing in Dynamical Systems, 21–55. Boston, MA: Springer US, 2000. http://dx.doi.org/10.1007/978-1-4757-3210-8_2.

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Ellis, David B., and Michael G. Branton. "Non-self-similar attractors of hyperbolic iterated function systems." In Dynamical Systems, 158–71. Berlin, Heidelberg: Springer Berlin Heidelberg, 1988. http://dx.doi.org/10.1007/bfb0082829.

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Тези доповідей конференцій з теми "Hyperbolic dynamical systems"

Miranda-Reyes, C., G. Fernandez-Anaya, and J. J. Flores-Godoy. "Preservation of hyperbolic equilibrium points and synchronization in dynamical systems." In 2008 5th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE). IEEE, 2008. http://dx.doi.org/10.1109/iceee.2008.4723367.

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Zhao, Yan, and Huaguang Zhang. "Anticontrol of Chaos for Discrete-time Dynamical Systems via Fuzzy Hyperbolic Models." In 2008 IEEE International Conference on Networking, Sensing and Control (ICNSC). IEEE, 2008. http://dx.doi.org/10.1109/icnsc.2008.4525190.

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Jedrzejewski, F. "Entropy and Lyapunov Exponents Relationships in Stochastic Dynamical Systems." In ASME 2003 Pressure Vessels and Piping Conference. ASMEDC, 2003. http://dx.doi.org/10.1115/pvp2003-1822.

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Stochastic differential equations and classical techniques related to the Fokker-Planck equation are standard bases for the analysis of nonlinear systems perturbed by noise, such as seismic wave propagation in random media and response of structures to turbulent wind. In this paper, a complementary approach based on entropy production is proposed to analyse the stochastic stability of dynamical systems. For a large class of stochastic dynamical systems, it is shown that the entropy information production is equal to the negative sum of Lyapunov exponents as the noise strength tends to zero. This result is correlated to the topological entropy property, which is in some cases such as the hyperbolic case, equal the sum of Lyapunov exponents. Several examples are given to illustrate the proposed procedure.

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Díaz, Jesús Ildefonso, and Alicia Arjona. "Stabilization of a hyperbolic/elliptic system modelling the viscoelastic-gravitational deformation in a multilayered Earth." In The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications (Madrid, Spain). American Institute of Mathematical Sciences, 2015. http://dx.doi.org/10.3934/proc.2015.0066.

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Lesniewski, Piotr, and Andrzej Bartoszewicz. "Hyperbolic tangent based switching reaching law for discrete time sliding mode control of dynamical systems." In 2015 International Workshop on Recent Advances in Sliding Modes (RASM 2015). IEEE, 2015. http://dx.doi.org/10.1109/rasm.2015.7154589.

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Lowrie, Robert, and Jim Morel. "Discontinuous Galerkin for stiff hyperbolic systems." In 14th Computational Fluid Dynamics Conference. Reston, Virigina: American Institute of Aeronautics and Astronautics, 1999. http://dx.doi.org/10.2514/6.1999-3307.

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Dietiker, Jean-Francois, Klaus Hoffmann, and James Forsythe. "Assessment of computational boundary conditions for hyperbolic systems." In 14th Computational Fluid Dynamics Conference. Reston, Virigina: American Institute of Aeronautics and Astronautics, 1999. http://dx.doi.org/10.2514/6.1999-3350.

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Ganesan, Vaahini, Tuhin K. Das, Jeffrey L. Kauffman, and Nazanin Rahnavard. "Including Vibration Characteristics Within Compressive Sensing Formulations for Structural Monitoring of Beams." In ASME 2017 Dynamic Systems and Control Conference. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/dscc2017-5213.

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Vibration-based monitoring of mechanical structures often involves continuous monitoring that result in high data volume and instrumentation with a large array of sensors. Previously, we have shown that Compressive Sensing (CS)-based vibration monitoring can significantly reduce both volume of data and number of sensors in temporal and spatial domains respectively. In this work, further analysis of CS-based detection and localization of structural changes is presented. Incorporating damping and noise handling in the CS algorithm improved its performance for frequency recovery. CS-based reconstruction of deflection shape of beams with fixed boundary conditions is addressed. Formulation of suitable bases with improved conditioning is explored. Restricting hyperbolic terms to lower frequencies in the basis functions improves reconstruction. An alternative is to generate an augmented basis that combines harmonic and hyperbolic terms. Incorporating known boundary conditions into the CS problem is studied.

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Burns, John A., and Eugene M. Cliff. "Control of hyperbolic PDE systems with actuator dynamics." In 2014 IEEE 53rd Annual Conference on Decision and Control (CDC). IEEE, 2014. http://dx.doi.org/10.1109/cdc.2014.7039829.

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Luo, Albert C. J., and Chuanping Liu. "On Symmetric Periodic Motions With Different Excitation Periods in a Discontinuous System With a Hyperbolic Boundary." In ASME 2020 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/imece2020-23310.

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Abstract In this paper, symmetric periodic motions with different excitation periods in a discontinuous dynamic system with a hyperbolic boundary are presented analytically. The switchability conditions of flows at the hyperbolic boundaries are developed. Periodic motions with specific mapping structures are predicted analytically, and numerical simulations of periodic motions are carried out. The corresponding G-functions are presented for illustration of motion switchability at the hyperbolic boundaries.

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Bond, W., Maria Seale, and Jeffrey Hensley. A dynamic hyperbolic surface model for responsive data mining. Engineer Research and Development Center (U.S.), April 2022. http://dx.doi.org/10.21079/11681/43886.

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Data management systems impose structure on data via a static representation schema or data structure. Information from the data is extracted by executing queries based on predefined operators. This paradigm restricts the searchability of the data to concepts and relationships that are known or assumed to exist among the objects. While this is an effective and efficient means of retrieving simple information, we propose that such a structure severely limits the ability to derive breakthrough knowledge that exists in data under the guise of “unknown unknowns.” A dynamic system will alleviate this dependence, allowing theoretically infinite projections of the data to reveal discoverable relationships that are hidden by traditional use case-driven, static query systems. In this paper, we propose a framework for a data-responsive query algebra based on a dynamic hyperbolic surface model. Such a model could provide more intuitive access to analytics and insights from massive, aggregated datasets than existing methods. This model will significantly alter the means of addressing the underlying data by representing it as an arrangement on a dynamic, hyperbolic plane. Consequently, querying the data can be viewed as a process similar to quantum annealing, in terms of characterizing data representation as an energy minimization problem with numerous minima.

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