Dissertations / Theses on the topic 'Bifurcation'
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1
Henderson, Michael E. Keller Herbert Bishop Keller Herbert Bishop. "Complex bifurcation /." Diss., Pasadena, Calif. : California Institute of Technology, 1985. http://resolver.caltech.edu/CaltechETD:etd-03262008-112516.
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Salih, Rizgar Haji. "Hopf bifurcation and centre bifurcation in three dimensional Lotka-Volterra systems." Thesis, University of Plymouth, 2015. http://hdl.handle.net/10026.1/3504.
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This thesis presents a study of the centre bifurcation and chaotic behaviour of three dimensional Lotka-Volterra systems. In two dimensional systems, Christopher (2005) considered a simple computational approach to estimate the cyclicity bifurcating from the centre. We generalized the technique to estimate the cyclicity of the centre in three dimensional systems. A lower bounds is given for the cyclicity of a hopf point in the three dimensional Lotka-Volterra systems via centre bifurcations. Sufficient conditions for the existence of a centre are obtained via the Darboux method using inverse Jacobi multiplier functions. For a given centre, the cyclicity is bounded from below by considering the linear parts of the corresponding Liapunov quantities of the perturbed system. Although the number obtained is not new, the technique is fast and can easily be adapted to other systems. The same technique is applied to estimate the cyclicity of a three dimensional system with a plane of singularities. As a result, eight limit cycles are shown to bifurcate from the centre by considering the quadratic parts of the corresponding Liapunov quantities of the perturbed system. This thesis also examines the chaotic behaviour of three dimensional Lotka-Volterra systems. For studying the chaotic behaviour, a geometric method is used. We construct an example of a three dimensional Lotka-Volterra system with a saddle-focus critical point of Shilnikov type as well as a loop. A construction of the heteroclinic cycle that joins the critical point with two other critical points of type planar saddle and axial saddle is undertaken. Furthermore, the local behaviour of trajectories in a small neighbourhood of the critical points is investigated. The dynamics of the Poincare map around the heteroclinic cycle can exhibit chaos by demonstrating the existence of a horseshoe map. The proof uses a Shilnikov-type structure adapted to the geometry of these systems. For a good understanding of the global dynamics of the system, the behaviour at infinity is also examined. This helps us to draw the global phase portrait of the system. The last part of this thesis is devoted to a study of the zero-Hopf bifurcation of the three dimensional Lotka-Volterra systems. Explicit conditions for the existence of two first integrals for the system and a line of singularity with zero eigenvalue are given. We characteristic the parameters for which a zero-Hopf equilibrium point takes place at any points on the line. We prove that there are three 3-parameter families exhibiting such equilibria. First order of averaging theory is also applied but we show that it gives no information about the possible periodic orbits bifurcating from the zero-Hopf equilibria.
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Binks, Douglas John. "Bifurcation phenomena in nematodynamics." Thesis, University of Oxford, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.306928.
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Taverner, S. "Bifurcation in physical systems." Thesis, University of Oxford, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.375327.
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Impey, M. D. "Bifurcation in Lapwood convection." Thesis, University of Bristol, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.234799.
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Arakawa, Vinicius Augusto Takahashi. "Um estudo de bifurcações de codimensão dois de campos de vetores /." São José do Rio Preto : [s.n.], 2008. http://hdl.handle.net/11449/94243.
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Orientador: Claudio Aguinaldo BuzziBanca: João Carlos da Rocha Medrado
Banca: Luciana de Fátima Martins
Resumo: Nesse trabalho são apresentados alguns resultados importantes sobre bifurcações de codimensão dois de campos de vetores. O resultado principal dessa dissertação e o teorema que d a o diagrama de bifurcação e os retratos de fase da Bifurcação de Bogdanov-Takens. Para a demonstracão são usadas algumas técnicas basicas de Sistemas Dinâmicos e Teoria das Singularidades, tais como Integrais Abelianas, desdobramentos de Sistemas Hamiltonianos, desdobramentos versais, Teorema de Preparação de Malgrange, entre outros. Outra importante bifurcação clássica apresentada e a Bifurca cão do tipo Hopf-Zero, quando a matriz Jacobiana possui um autovalor simples nulo e um par de autovalores imagin arios puros. Foram usadas algumas hipóteses que garantem propriedades de simetria do sistema, dentre elas, assumiuse que o sistema era revers vel. Assim como na Bifurcação de Bogdanov-Takens, foram apresentados o diagrama de bifurcao e os retratos de fase da Bifurcação Hopf-zero bifurcação reversível. As técnicas usadas para esse estudo foram a forma normal de Belitskii e o método do Blow-up polar.
Abstract: In this work is presented some important results about codimension two bifurcations of vector elds. The main result of this work is the theorem that gives the local bifurcation diagram and the phase portraits of the Bogdanov-Takens bifurcation. In order to give the proof, some classic tools in Dynamical System and Singularities Theory are used, such as Abelian Integral, versal deformation, Hamiltonian Systems, Malgrange Preparation Theorem, etc. Another classic bifurcation phenomena, known as the Hopf-Zero bifurcation, when the Jacobian matrix has a simple zero and a pair of purely imaginary eigenvalues, is presented. In here, is added the hypothesis that the system is reversible, which gives some symmetry in the problem. Like in Bogdanov-Takens bifurcation, the bifurcation diagram and the local phase portraits of the reversible Hopf-zero bifurcation were presented. The main techniques used are the Belitskii theory to nd a normal forms and the polar Blow-up method.
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Jones, Mark C. W. "The bifurcation and secondary bifurcation of capillary-gravity waves in the presence of symmetry." Thesis, University of Bath, 1986. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.370986.
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Gaunersdorfer, Andrea, Cars H. Hommes, and Florian O. O. Wagener. "Bifurcation routes to volatility clustering." SFB Adaptive Information Systems and Modelling in Economics and Management Science, WU Vienna University of Economics and Business, 2000. http://epub.wu.ac.at/522/1/document.pdf.
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A simple asset pricing model with two types of adaptively learning traders, fundamentalists and technical analysts, is studied. Fractions of these trader types, which are both boundedly rational, change over time according to evolutionary learning, with technical analysts conditioning their forecasting rule upon deviations from a benchmark fundamental. Volatility clustering arises endogenously in this model. Two mechanisms are proposed as an explanation. The first is coexistence of a stable steady state and a stable limit cycle, which arise as a consequence of a so-called Chenciner bifurcation of the system. The second is intermittency and associated bifurcation routes to strange attractors. Both phenomena are persistent and occur generically in nonlinear multi-agent evolutionary systems. (author's abstract)Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"
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Fujihira, Takeo. "Hamiltonian Hopf bifurcation with symmetry." Thesis, Imperial College London, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.444087.
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Duka, E. D. "Bifurcation problems in finite elasticity." Thesis, University of Nottingham, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.384747.
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Melbourne, I. "Bifurcation problems with octahedral symmetry." Thesis, University of Warwick, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.383295.
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Giacomoni, Jacques. "Problèmes non compacts et bifurcation." Paris 9, 1997. https://portail.bu.dauphine.fr/fileviewer/index.php?doc=1997PA090063.
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Les travaux présentés dans cette thèse portent sur certains problèmes non linéaires et non compacts et s'organisent autour de deux axes principaux : bifurcation globale pour certains problèmes semi-linéaires elliptiques non compacts d'une part et existence locale et comportement global des solutions pour certains problèmes paraboliques dégénérés. La première partie est consacrée à l'étude de deux problèmes semi-linéaires elliptiques non compacts. On démontre pour chacun de ces deux problèmes, l'existence de branches globales de solutions dans R x H ou dans R x L#. Les résultats obtenus sont surprenants et se distinguent de ceux obtenus par la théorie de la bifurcation de Rabinowitz qui ne peut s'appliquer que dans le cadre de problèmes compacts. Les deux problèmes abordés ont des types de défaut de compacité différents : le premier est lié au fait que l'on se place dans un ouvert non borné et le second résulte d'une dégénérescence de l'operateur. Les résultats concernant le second problème ont été obtenus en collaboration avec M. J. Esteban. Enfin, pour certaines non linéarités, des résultats d'unicité et de stabilité des solutions sont établis. La seconde partie porte sur l'étude de deux problèmes paraboliques dégénérés. Des résultats d'existence locale, d'explosion en temps fini ainsi que de convergence vers une solution stationnaire quand t tend vers l'infini, sont démontrés. Pour le second problème, on exhibe un paramètre critique d'explosion qui permet d'établir une complète description du comportement global des solutions de ce problème.
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Ravnås, Eirik. "Continuation and Bifurcation software in MATLAB." Thesis, Norwegian University of Science and Technology, Department of Mathematical Sciences, 2008. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-8954.
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This article contains discussions of the algorithms used for the construction of the continuation software implemented in this thesis. The aim of the continuation was to be able to perform continuation of equilibria and periodic solutions originating from a Hopf bifurcation point. Algorithms for detection of simple branch points, folds, and Hopf bifurcation points have also been implemented. Some considerations are made with regard to optimization, and two schemes for mesh adaptation of periodic solutions based on moving mesh equations are suggested.
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Feudel, Fred, Norbert Seehafer, and Olaf Schmidtmann. "Bifurcation phenomena of the magnetofluid equations." Universität Potsdam, 1995. http://opus.kobv.de/ubp/volltexte/2007/1358/.
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We report on bifurcation studies for the incompressible magnetohydrodynamic equations in three space dimensions with periodic boundary conditions and a temporally constant external forcing. Fourier reprsentations of velocity, pressure and magnetic field have been used to transform the original partial differential equations into systems of ordinary differential equations (ODE), to which then special numerical methods for the qualitative analysis of systems of ODE have been applied, supplemented by the simulative calculation of solutions for selected initial conditions. In a part of the calculations, in order to reduce the number of modes to be retained, the concept of approximate inertial manifolds has been applied. For varying (incereasing from zero) strength of the imposed forcing, or varying Reynolds number, respectively, time-asymptotic states, notably stable stationary solutions, have been traced. A primary non-magnetic steady state loses, in a Hopf bifurcation, stability to a periodic state with a non-vanishing magnetic field, showing the appearance of a generic dynamo effect. From now on the magnetic field is present for all values of the forcing. The Hopf bifurcation is followed by furhter, symmetry-breaking, bifurcations, leading finally to chaos. We pay particular attention to kinetic and magnetic helicities. The dynamo effect is observed only if the forcing is chosen such that a mean kinetic helicity is generated; otherwise the magnetic field diffuses away, and the time-asymptotic states are non-magnetic, in accordance with traditional kinematic dynamo theory.
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Du, Yimian. "Bifurcation analysis in chemical reaction network." Thesis, Imperial College London, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.511282.
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Harlim, John. "Codimension three Hopf and cusp bifurcation." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2001. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/MQ58343.pdf.
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Welsh, S. C. "Generalised topological degree and bifurcation theory." Thesis, University of Glasgow, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.372419.
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Buchendorfer, Thomas. "Bifurcation properties of dynamic urban models." Thesis, Cranfield University, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.360072.
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Bougherara, Brahim. "Problèmes non-linéaires singuliers et bifurcation." Thesis, Pau, 2014. http://www.theses.fr/2014PAUU3012/document.
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Cette thèse s’inscrit dans le domaine mathématique de l’analyse des équations aux dérivées partielles non linéaires. Précisément, nous nous sommes intéressés à une classe de problèmes elliptiques et paraboliques avec coefficients singuliers. Ce manque de régularité pose un certain nombre de difficultés qui ne permettent pas d’utiliser directement les méthodes classiques de l’analyse non-linéaire fondées entre autres sur des résultats de compacité. Dans les démonstrations des principaux résultats, nous montrons comment pallier ces difficultés. Ceci suppose d’adapter certaines techniques bien connues mais aussi d’introduire de nouvelles méthodes. Dans ce contexte, une étape importante est l’estimation fine du comportement des solutions qui permet d’adapter le principe de comparaison faible, d’utiliser la régularité elliptique et parabolique et d’appliquer dans un nouveau contexte la théorie globale de la bifurcation analytique. La thèse se présente sous forme de deux parties indépendantes. 1- Dans la première partie (chapitre I de la thèse), nous avons étudié un problème quasi-linéaire parabolique fortement singulier faisant intervenir l’opérateur p-Laplacien. On a démontré l’existence locale et la régularité de solutions faibles. Ce résultat repose sur des estimations a priori obtenues via l’utilisation d’inégalités de type log-Sobolev combinées à des inégalités de Gagliardo-Nirenberg. On démontre l’unicité de la solution pour un intervalle de valeurs du paramètre de la singularité en utilisant un principe de comparaison faible fondé sur la monotonie d’un opérateur non linéaire adéquat. 2- Dans la deuxième partie (correspondant aux Chapitres II, III et IV de la thèse), nous sommes intéressés à l’étude de problèmes de bifurcation globale. On a établi pour ces problèmes l’existence de continuas non bornés de solutions qui admettent localement une paramétrisation analytique. Pour établir ces résultats, nous faisons appel à différents outils d’analyse non linéaire. Un outil important est la théorie analytique de la bifurcation globale qui a été introduite par Dancer (voir Chapitre II de la thèse). Pour un problème semi linéaire elliptique avec croissance critique en dimension 2, on montre que les solutions le long de la branche convergent vers une solution singulière (solution non bornée) lorsque la norme des solutions converge vers l’infini. Par ailleurs nous montrons que la branche admet une infinité dénombrable de "points de retournement" correspondant à un changement de l’indice de Morse des solutions qui tend vers l’infini le long de la brancheThis thesis is concerned with the mathematical study of nonlinear partial differential equations. Precisely, we have investigated a class of nonlinear elliptic and parabolic problems with singular coefficients. This lack of regularity involves some difficulties which prevent the straight-orward application of classical methods of nonlinear analysis based on compactness results. In the proofs of the main results, we show how to overcome these difficulties. Precisely we adapt some well-known techniques together with the use of new methods. In this framework, an important step is to estimate accurately the solutions in order to apply the weak comparison principle, to use the regularity theory of parabolic and elliptic equations and to develop in a new context the analytic theory of global bifurcation. The thesis presents two independent parts. 1- In the first part (corresponding to Chapter I), we are interested by a nonlinear and singular parabolic equation involving the p-Laplacian operator. We established for this problem that for any non-negative initial datum chosen in a certain Lebeque space, there exists a local positive weak solution. For that we use some a priori bounds based on logarithmic Sobolev inequalities to get ultracontractivity of the associated semi-group. Additionaly, for a range of values of the singular coefficient, we prove the uniqueness of the solution and further regularity results. 2- In the second part (corresponding to Chapters II, III and IV of the thesis), we are concerned with the study of global bifurcation problems involving singular nonlinearities. We establish the existence of a piecewise analytic global path of solutions to these problems. For that we use crucially the analytic bifurcation theory introduced by Dancer (described in Chapter II of the thesis). In the frame of a class of semilinear elliptic problems involving a critical nonlinearity in two dimensions, we further prove that the piecewise analytic path of solutions admits asymptotically a singular solution (i.e. an unbounded solution), whose Morse index is infinite. As a consequence, this path admits a countable infinitely many “turning points” where the Morse index is increasing
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Caron, Jean-François. "Phénomène de bifurcation en électro-élasticité." Lille 1, 1997. http://www.theses.fr/1997LIL10120.
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En 1930, Signorini découvrit que le problème de traction en élasticité tridimensionnelle non linéaire possède des solutions non triviales et (Stopelli 1958), que si le problème de l'équilibre d'un corps élastique soumis a des charges données, a une solution (unique a un déplacement rigide près), dans le cas linéarise, il peut en présenter plusieurs dans le cas non linéaire. Aujourd'hui ceci est considère comme un phénomène de bifurcation dans l'espace des solutions élastiques appelé instabilité a la linéarisation. Marsden-hughes en 1978 et 1983 ainsi que Chillingworth, Marsden et Wan en 1982 ont complète l'analyse de Signorini et Stopelli en traitant de façon plus générale le problème des solutions et de leur stabilité. Le présent travail étend cette étude et les techniques de (Marsden et collaborateurs) au cas ou les forces sont d'origine mécanique mais aussi électromagnétique, dans le cas d'un diélectrique piézo-électrique qui, lorsqu'il est soumis a des forces mécaniques, engendre un champ de polarisation (effet piézo-électrique) et, quand il est soumis a un champ électrique se déforme (électro-striction). En particulier il établit que, dans certains cas, une instabilité mécanique peut, sous l'effet de forces électrique, devenir une stabilité.
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Sammon, Michel P. "Bifurcation analyses of respiratory vagal reflexes." Case Western Reserve University School of Graduate Studies / OhioLINK, 1992. http://rave.ohiolink.edu/etdc/view?acc_num=case1060098227.
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Hachich, Mohamed. "Conditions de bifurcation dans les solides /." Cachan : Laboratoire de mécanique et technologie, 1994. http://catalogue.bnf.fr/ark:/12148/cb35813140w.
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Kwok, Loong-Piu. "Viscous cross-waves: Stability and bifurcation." Diss., The University of Arizona, 1988. http://hdl.handle.net/10150/184441.
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In the first part of this thesis, the nonlinear Schrodinger equation for inviscid cross-waves near onset is found to be modified by viscous linear damping and detuning. The accompanying boundary condition at the wavemaker is also modified by damping from the wavemaker meniscus. The relative contributions of the free-surface, sidewalls, bottom, and wavemaker viscous boundary layers are computed. It is shown that viscous dissipation due to the wavemaker meniscus breaks the symmetry of the neutral curve. In the second part, existence and stability of steady solutions to the nonlinear Schrodinger equation are examined numerically. It is found that at forcing frequency above a critical value, f(c), only one solution exists. However, below f(c), multiple steady solutions, the number of which is determined, are possible. This multiplicity leads to hysteresis for f < f(c), in agreement with observation. A Hopf bifurcation of the steady solutions is found. This bifurcation is compared with the transition from unmodulated to periodically modulated cross-waves observed experimentally.
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Hima, Nikolin. "Bifurcation based mechanisms for elastic metainterfaces." Doctoral thesis, Università degli studi di Trento, 2022. https://hdl.handle.net/11572/362105.
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Towards the design of innovative elastic metainterfaces, rigid-elastic mechanisms based on bifurcation phenomena are developed and analysed under quasi-static and dynamic loading conditions. In particular, a rigid-elastic mechanism is designed by introducing a strut, which displays a bifurcation at either vanishing tensile or compressive load. Analysis under quasi-static loading shows that such bifurcation allows for devising simple mechanisms with low degrees of freedom, which can exhibit a wide range of mechanical behaviours characterized by multiple stable configurations (from monostable to tetrastable states within a narrow range of deformations). Within this context, a bifurcation mechanism is proven to be mechanically equivalent to a unilateral constraint, a finding instrumental in enabling numerical solutions for the treatment of dynamics. Vibration analysis reveals the inherent non-smoothness in the dynamic response of the considered structural system, similarly to the rocking motion of rigid bodies. Due to the presence of the unilateral constraints, dissipation mechanisms may emerge from impacts of each layer, in addition to that occurring when damping sources are continuous in time. A complex behaviour under dynamic settings is observed with the coexistence of multiple stable attractors (dynamic equilibria) as well as quasi-periodic and chaotic regions in the space parameters, the latter possibly comprising stable attractors too. It is also found that the dynamic effects reduce the monostability region of loading conditions, by enlarging that of multistability. The obtained results show the potential in harnessing bifurcation phenomena as design practice, that can lead to mechanisms with highly desirable mechanical properties, without compromising the global stability of the system. Moreover, the multistable and multisource dissipation mechanisms embedded in these concepts, open new possibilities in coupling vibration attenuation and control with energy harvesting, thus paving the way towards more sustainable solutions.
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Pacha, Andújar Juan Ramón. "On the quasiperiodic hamiltonian andronov-hopf bifurcation." Doctoral thesis, Universitat Politècnica de Catalunya, 2002. http://hdl.handle.net/10803/5830.
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Aquest treball es situa dintre del marc dels sistemes dinàmics hamiltonians de tres graus de llibertat. Allà considerem famílies d'òrbites periòdiques amb una transició estable-complex inestable: sigui L el paràmetre que descriu la família i suposarem que per a valors del paràmetre més petits que un cert valor crític, L', els multiplicadors característics de les òrbites periòdiques corresponents hi són sobre el cercle unitat, quan L=L' aquests col·lisionen per parelles conjugades (òrbita ressonant o crítica) i per L > L', abandonen el cercle unitat cap al pla complex (col·lisió de Krein amb signatura oposada). El canvi d'estabilitat subseqüent es descriu a la literatura com "transició estable a complex inestable". Tanmateix, a partir d'estudis numèrics sobre certes aplicacions simplèctiques (n'esmentarem D. Pfenniger, Astron. Astrophys. 150, 97-111, 1985), és coneguda l'aparició (sota condicions d'incommensurabilitat) de fenòmens de bifurcació quasi-periòdica, en particular, el desplegament de famílies de tors 2-dimensionals. A més aquesta bifurcació s'assembla a la (clàssica) bifurcació d'Andronov-Hopf, en el sentit de què hi sorgeixen objectes linealment estables (tors-2D el·líptics) "al voltant" d'objectes inestables de dimensionalitat més baixa (òrbites periòdiques), i recíprocament, n'apareixen tors inestables (hiperbòlics) "al voltant" d'òrbites periòdiques linealment estables.Nostre objectiu és entendre la dinàmica local en un entorn de l'òrbita periòdica ressonant per tal de provar, analíticament, l'existència dels tors invariants bifurcats segons l'esquema descrit dalt. Això el portem a terme mitjançant l'anàlisi següent:
(i) Primer de tot obtenim d'una manera constructiva (això és, donant algorismes) una forma normal ressonant en un entorn de l'òrbita periòdica crítica. Aquesta forma normal la portem fins a qualsevol ordre arbitrari r. Així doncs, mostrem que el hamiltonià inicial es pot posar com la suma de la forma normal (integrable) més una resta no integrable. A partir d'aquí, podem estudiar la dinàmica de la forma normal, prescindint dels altres termes i, amb aquest tractament (formal) del problema, som capaços d'identificar els paràmetres que governen tant l'existència de la bifurcació com la seva tipologia (directa, inversa). Cal, remarcar que el que es fa fins aquí, no és només un procés qualitatiu, ja que a més ens permet derivar parametritzacions molt acurades dels tors no pertorbats.
(ii) A continuació, calculem acotacions "òptimes" per a la resta. D'aquesta manera, esperem provar que un bon nombre de tors (en sentit de la mesura) es preserven quan s'afegeix la pertorbació.
(iii) Finalment, apliquem mètodes KAM per establir que la majoria (veure comentari dalt) dels tors bifurcats sobreviuen. Aquests mètodes es basen en la construcció d'un esquema de convergència quadràtica capaç de contrarestar l'efecte dels petits divisors que apareixen quan s'aplica teoria de pertorbacions per trobar solucions quasi-periòdiques. En el nostre cas, a més, resulta que alguna de les condicions "típiques" que s'imposen sobre les freqüències (intrínseques i normals) dels tors no pertorbats, no estan ben definides per als tors bifurcats, de manera que ens ha calgut desenvolupar un tractament més específic.
keywords: Bifurcation problems, perturbations, normal forms, small divisors, KAM theory.
Classificació AMS: 37J20, 37J25, 37J40
This work is placed into the context of the three-degree of freedom Hamiltonian systems, where we consider families of periodic orbits undergoing transitions stable-complex unstable. More precisely: Let L be the parameter of the family and assuming that, for values of L smaller than some critical value say, L', the characteristic multipliers of the periodic orbits lie on the unit circle, when L=L' they colllide pairwise (critical or resonant periodic orbit) and, for L > L' leave the unit circle towards the complex plane (Krein collision with opposite signature).
From numerical studies on some concrete symplectic maps (for instance, D. Pfennniger, Astron. Astrophys. 150, 97-111, 1985) it is known the rising (under certain irrationality conditions), of quasi-periodic bifurcation phenomena, in particular, the appearance of unfolded 2D invariant tori families. Moreover, the bifurcation takes place in a way that resembles the classical Andronov-Hopf one, in the sense that either stable invariant objects (elliptic tori) unfold "around" linear unstable periodic orbits, or conversely, unstable invariant structures (hyperbolic tori) appear "surrounding" stable periodic orbits.
Our objective is, thus, to understand the (local) dynamics in a neighbourhood of the critical periodic orbit well enough to prove analytically, the existence of such quasi-periodic solutions together with the bifurcation pattern described above. This is carried out through three steps:
(i) First, we derive, in a constructive way (i. e., giving algorithms), a resonant normal form around the critical periodic orbit up to any arbitrary order r. Whence, we show that the initial raw Hamiltonian can be casted --through a symplectic change--, into an integrable part, the normal form itself, plus a (non-integrable) remainder. From here, one can study the dynamics of the normal form, skipping the remainder off. As a result of this (formal) approach, we are able to indentify the parameters governing both, the presence of the bifurcation and its type (direct, inverse). We remark that this is not a merely qualitative process for, in addition, accurate parametrizations of the bifurcated families of invariant tori are derived in this way.
(ii) Beyond the formal approach, we compute "optimal" bounds for the remainder of the normal form, so one expects to prove the preservation of a higher (in the measure sense) number of invariant tori --than, indeed, with a less sharp estimates--.
(iii) Finally, we apply KAM methods to establish the persistence of (most, in the measure sense) of the bifurcated invariant tori. These methods involve the design of a suitable quadratic convergent scheme, able to overcome the effect of the small divisors appearing in perturbation techniques when one looks for quasi-periodic solutions. In this case though, some of the "typical" conditions that one imposes on the frequencies (intrinsic and normal) of the unperturbed invariant tori do not work, due to the proximity to parabolic tori, so one is bound to sketch specific tricks.
keywords: Bifurcation problems, perturbations, normal forms, small divisors, KAM theory
AMS classification: 37J20, 37J25, 37J40
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Gemici, Omer Caner. "Numerical Bifurcation Analysis Of Cosymmetric Dynamical Systems." Thesis, METU, 2003. http://etd.lib.metu.edu.tr/upload/1260425/index.pdf.
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In this thesis, bifurcation phenomena in dynamical systems with cosymmetry
and Hamiltonian structure were investigated using numerical methods.
Several numerical continuation methods and test functions for detecting bifurcations
were presented. The numerical results for various examples are
given using a numerical bifurcation analysis toolbox.
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Schumacher, Jörg, and Norbert Seehafer. "Bifurcation analysis of the plane sheet pinch." Universität Potsdam, 1999. http://opus.kobv.de/ubp/volltexte/2007/1492/.
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A numerical bifurcation analysis of the electrically driven plane sheet pinch is presented. The electrical conductivity varies across the sheet such as to allow instability of the quiescent basic state at some critical Hartmann number. The most unstable perturbation is the two-dimensional tearing mode. Restricting the whole problem to two spatial dimensions, this mode is followed up to a time-asymptotic steady state, which proves to be sensitive to three-dimensional perturbations even close to the point where the primary instability sets in. A comprehensive three-dimensional stability analysis of the two-dimensional steady tearing-mode state is performed by varying parameters of the sheet pinch. The instability with respect to three-dimensional perturbations is suppressed by a sufficiently strong magnetic field in the invariant direction of the equilibrium. For a special choice of the system parameters, the unstably perturbed state is followed up in its nonlinear evolution and is found to approach a three-dimensional steady state.
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Park, Jungho. "Bifurcation and stability problems in fluid dynamics." [Bloomington, Ind.] : Indiana University, 2007. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3274924.
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Thesis (Ph.D.)--Indiana University, Dept. of Mathematics, 2007.Source: Dissertation Abstracts International, Volume: 68-07, Section: B, page: 4529. Adviser: Shouhong Wang. Title from dissertation home page (viewed Apr. 22, 2008).
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Kwalik, Kristina Mary. "Bifurcation characteristics in closed-loop polymerization reactors." Diss., Georgia Institute of Technology, 1988. http://hdl.handle.net/1853/11711.
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Anayiotos, Andreas Stavrou. "Fluid dynamics at a compliant bifurcation model." Diss., Georgia Institute of Technology, 1990. http://hdl.handle.net/1853/12939.
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Shen, Wenxian. "Staility and bifurcation of traveling wave solutions." Diss., Georgia Institute of Technology, 1992. http://hdl.handle.net/1853/29354.
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Baghdadi, Nadjib. "Bifurcation and continuation analysis : flexible aircraft dynamics." Thesis, University of Bristol, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.539769.
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Tzanov, Vassil Vassilev. "Bifurcation analysis applied to inclined cable dynamics." Thesis, University of Bristol, 2012. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.606568.
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徐善強 and Sin-keung Chui. "Stability and bifurcation in flow induced vibration." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1997. http://hub.hku.hk/bib/B31235724.
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Davidson, Fordyce A. "Bifurcation in systems of reaction-diffusion equations." Thesis, Heriot-Watt University, 1993. http://hdl.handle.net/10399/1444.
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McGarry, John Kevin. "Application of bifurcation theory to physical problems." Thesis, University of Leeds, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.252925.
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Tomlin, Alison Sarah. "Bifurcation analysis for non-linear chemical kinetics." Thesis, University of Leeds, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.255345.
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Charles, Guy Alexander. "Bifurcation tailoring applied to nonlinear flight dynamics." Thesis, University of Bristol, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.274630.
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Orr, Anthony. "The eversion and bifurcation of elastic cylinders." Thesis, University of Glasgow, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.307194.
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OPREA, PETRESCU IULIANA. "Bifurcation et evolution temporelle de dynamos convectives." Nice, 1994. http://www.theses.fr/1994NICE4787.
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On etudie le modele mathematique de la generation et de l'evolution du champ magnetique dans les etoiles et les planetes dues au mouvement du fluide conducteur electrique situe dans leur noyau. Dans la i#e#r#e partie on considere le probleme de la dynamo cinematique dans une couronne spherique. Pour la simulation numerique de l'equation de l'induction magnetique on developpe un algorithme original base sur la decomposition en harmoniques spheriques generalisees de la vitesse et du champ magnetique, qui offre une alternative nettement plus simple a la decomposition usuelle en termes de composantes toroidales et poloidales. La dynamo convective est abordee pour differents regimes de vitesse (stationnaires, periodiques et chaotiques), obtenus par une analyse de bifurcation du probleme de benard spherique. Dans ce contexte est egalement considere l'effet alpha (i. E. L'amplification par des mouvements fluides a petite echelle d'une semence de champ magnetique a bien plus grande echelle) dans le cas ou le coefficient de transport (coefficient alpha) decroit avec l'energie magnetique totale. Dans la ii#e#m#e partie on etudie l'apparition d'une dynamo pour un ecoulement horizontalement periodique avec symetrie hexagonale, dans un domaine fixe ou en rotation, limite par deux plaques rigides paralleles. En utilisant la theorie des bifurcations en presence des symetries et la theorie des representations des groupes, le probleme se ramene a l'etude numerique du spectre de l'operateur de l'induction, realisee au moyen d'une methode spectrale de type galerkin
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Lari-Lavassani, Ali. "Multiparameter bifurcation with symmetry via singularity theory /." The Ohio State University, 1990. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487683049377079.
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Kumeno, Hironori. "Bifurcation and Synchronization in Parametrically Forced Systems." Thesis, Toulouse, INSA, 2012. http://www.theses.fr/2012ISAT0024/document.
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Dans cette thèse, nous étudions un système à temps discret de dimension N, dont les paramètres varient périodiquement. Le système de dimension N est construit à partir de n sous-systèmes de dimension un couplés symétriquement. Dans un premier temps, nous donnons les propriétés générales du système de dimension N. Dans un second temps, nous étudions le cas particulier où le sous-système de dimension un est défini à l’aide d’une transformation logistique. Nous nous intéressons plus particulièrement à la structure des bifurcations lorsque N=1 ou 2. Des zones échangeurs centrées sur des points cuspidaux sont obtenues dans le cas de courbes de bifurcation de type fold (noeud-col).Ensuite, nous nous intéressons au comportement de circuits de type Chua couplés lorsqu’un paramètre varie lui aussi périodiquement, la période étant celle d’une des variables d’état interne au système. A partir de l’étude des bifurcations du système, la non existence de cycles d’ordre impair et la coexistence de plusieurs attracteurs est mise en évidence. D’autre part, on peut mettre en évidence la coexistence de différents attracteurs pour lesquels les états de synchronisation sont distincts. Le cas continu est comparé avec le cas discret. Des phénomènes tout à fait similaires sont obtenus. Il est important de noter que l’étude d’un système à temps discret est plus facile et plus rapide que celle d’un système à temps continu. L’étude du premier système permet donc d’avoir des informations sur ce qui peut se produire dans le cas continu. Pour terminer, nous analysons le comportement d’un autre système couplé à temps continu, basé lui aussi sur le circuit de Chua, mais pour lequel la commutation qui contrôle la variation du paramètre s’effectue différemment du premier système. Ce type de commutation génère une augmentation du nombre d’attracteursIn this thesis, we propose a N-dimensional coupled discrete-time system whose parameters are forced into periodic varying, the N-dimensional system being constructed of n same one-dimensional subsystems with mutually influencing coupling and also coupled continuous-time system including periodically parameter varying which correspond to the periodic varying in the discrete-time system.Firstly, we introduce the N-dimensional coupled parametrically forced discrete-time system and its general properties. Then, when logistic maps is used as the one-dimensional subsystem constructing the system, bifurcations in the one or two-dimensional parametrically forced logistic map are investigated. Crossroad area centered at fold cusp points regarding several order cycles are confirmed.Next, we investigated behaviors of the coupled Chua's circuit whose parameter is forced into periodic varying associated with the period of an internal state value. From the investigation of bifurcations in the system, non-existence of odd order cycles and coexistence of different attractors are observed. From the investigation of synchronizations coexisting of many attractors whose synchronizations states are different are observed. Observed phenomena in the system is compared with the parametrically forced discrete-time system. Similar phenomena are confirmed between the parametrically forced discrete-time system and the parametrically forced Chua's circuit. It is worth noting that this facilitates to analyze parametrically forced continuous-time systems, because to analyze discrete-time systems is easier than continuous-time systems. Finally, we investigated behaviors of another coupled continuous-time system in which Chua's circuit is used, while, the motion of the switch controlling the parametric varying is different from the above system. Coexisting of many attractors whose synchronizations states are different are observed. Comparing with theabove system, the number of coexisting stable state is increased by the effect of the different switching motion
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Chui, Sin-keung. "Stability and bifurcation in flow induced vibration /." Hong Kong : University of Hong Kong, 1997. http://sunzi.lib.hku.hk/hkuto/record.jsp?B1904155X.
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Meissen, Emily Philomena, and Emily Philomena Meissen. "Invading a Structured Population: A Bifurcation Approach." Diss., The University of Arizona, 2017. http://hdl.handle.net/10150/625610.
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Matrix population models are discrete in both time and state-space, where a matrix with density-dependent entries is used to project a population vector of a stage-structured population from one time to the next. Such models are useful for modeling populations with discrete categorizations (e.g. developmental cycles, communities of multiple species, differing sizes, etc.). We present a general matrix model of two interacting populations where one (the resident) has a stable cycle, and we analyze when the other population (the invader) can successfully invade. Specifically, we study the local bifurcations of coexistence cycles as the resident cycle destabilizes, where a cycle of length 1 corresponds to an equilibrium.
We make no assumptions on the types of interactions between the populations or on the population structure of the resident; we consider when the invader's projection matrix is primitive or imprimitive and 2x2. The simplest biological scenarios for such structures are an iteroparous invader and a two-stage semelparous invader. When the invader has a primitive projection matrix, coexistence cycles (of the same period as the resident cycle) bifurcate from the resident-cycle. When the invader has an imprimitive two-stage projection matrix, two types of coexistence cycles bifurcate from the resident-cycle: cycles of the same period and cycles of double the period. In both the primitive and imprimitive cases, we provide diagnostic quantities to determine the direction of bifurcation and the stability of the bifurcating cycles. Because we only perform a local stability analysis, the only successful invasion provided by our results is through stable coexistence cycles. As we show in some simple examples, however, the invader may persist when the coexistence cycles are unstable through competitive exclusion where the branch of bifurcating cycles connects to a branch of invader attractors and creates a multi-attractor scenario known as a strong Allee effect.
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Gomez, Maria Gabriela Miranda. "Symmetries in bifurcation theory : the appropriate context." Thesis, University of Warwick, 1992. http://wrap.warwick.ac.uk/110618/.
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Many phenomena in nature can be modeled by differential equations depending on parameters that are being varied continuously. We say that a given solution undergoes a bifurcation with respect to a given parameter if the qualitative behaviour of the system changes arbitrarily close to this solution when the parameter is varied across a critical value. Bifurcation problems can achieve a very high level of complexity because nature is complex. Several assumptions can be made in order to introduce considerable simplifications without going too far from reality. In this thesis we are mainly concerned in setting the problem in a symmetric context and showing that this is a realistic assumption that makes analysis much simpler. We want to emphasize that a lot of behaviour can be much easier to understand and predict when the appropriate symmetry context has been set. The message in part I of this thesis is that the full set of symmetries is not always obvious. We give examples of phenomena that are modeled by partial differential equations on rectangular domains and show that these problems have more than rectangular symmetry. Such hidden symmetries are found by embedding our problem into a larger one satisfying periodic boundary conditions and then consider all the symmetries that satisfy the original boundary conditions. In part II we study the behaviour of an electric circuit which can be modeled by a 3-dimensional system of ordinary differential equations. We begin by analysing this system under a symmetry assumption. Then in order to be more realistic we break the symmetry with a small perturbation. Most of the results for the asymmetric system are obtained by numerical and experimental search since a rigorous analysis became much harder. We observe a smooth change in qualitative behaviour by increasing the symmetry breaking perturbation. There is no dramatic change and we conclude that the original symmetry assumption was convenient and not misleading.
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Arakawa, Vinicius Augusto Takahashi [UNESP]. "Um estudo de bifurcações de codimensão dois de campos de vetores." Universidade Estadual Paulista (UNESP), 2008. http://hdl.handle.net/11449/94243.
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Nesse trabalho são apresentados alguns resultados importantes sobre bifurcações de codimensão dois de campos de vetores. O resultado principal dessa dissertação e o teorema que d a o diagrama de bifurcação e os retratos de fase da Bifurcação de Bogdanov-Takens. Para a demonstracão são usadas algumas técnicas basicas de Sistemas Dinâmicos e Teoria das Singularidades, tais como Integrais Abelianas, desdobramentos de Sistemas Hamiltonianos, desdobramentos versais, Teorema de Preparação de Malgrange, entre outros. Outra importante bifurcação clássica apresentada e a Bifurca cão do tipo Hopf-Zero, quando a matriz Jacobiana possui um autovalor simples nulo e um par de autovalores imagin arios puros. Foram usadas algumas hipóteses que garantem propriedades de simetria do sistema, dentre elas, assumiuse que o sistema era revers vel. Assim como na Bifurcação de Bogdanov-Takens, foram apresentados o diagrama de bifurcao e os retratos de fase da Bifurcação Hopf-zero bifurcação reversível. As técnicas usadas para esse estudo foram a forma normal de Belitskii e o método do Blow-up polar.
In this work is presented some important results about codimension two bifurcations of vector elds. The main result of this work is the theorem that gives the local bifurcation diagram and the phase portraits of the Bogdanov-Takens bifurcation. In order to give the proof, some classic tools in Dynamical System and Singularities Theory are used, such as Abelian Integral, versal deformation, Hamiltonian Systems, Malgrange Preparation Theorem, etc. Another classic bifurcation phenomena, known as the Hopf-Zero bifurcation, when the Jacobian matrix has a simple zero and a pair of purely imaginary eigenvalues, is presented. In here, is added the hypothesis that the system is reversible, which gives some symmetry in the problem. Like in Bogdanov-Takens bifurcation, the bifurcation diagram and the local phase portraits of the reversible Hopf-zero bifurcation were presented. The main techniques used are the Belitskii theory to nd a normal forms and the polar Blow-up method.
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Swat, Maciej J. "Bifurcation analysis of regulatory modules in cell biology." [S.l.] : [s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=981033113.
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Korobeinikov, Andrei. "Stability and bifurcation of deterministic infectious disease models." Thesis, University of Auckland, 2001. http://wwwlib.umi.com/dissertations/fullcit/3015611.
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Autonomous deterministic epidemiological models are known to be asymptotically stable. Asymptotic stability of these models contradicts observations. In this thesis we consider some factors which were suggested as able to destabilise the system. We consider discrete-time and continuous-time autonomous epidemiological models. We try to keep our models as simple as possible and investigate the impact of different factors on the system behaviour. Global methods of dynamical systems theory, especially the theory of bifurcations and the direct Lyapunov method are the main tools of our analysis. Lyapunov functions for a range of classical epidemiological models are introduced. The direct Lyapunov method allows us to establish their boundedness and asymptotic stability. It also helps investigate the impact of such factors as susceptibles' mortality, horizontal and vertical transmission and immunity failure on the global behaviour of the system. The Lyapunov functions appear to be useful for more complicated epidemiological models as well. The impact of mass vaccination on the system is also considered. The discrete-time model introduced here enables us to solve a practical problem-to estimate the rate of immunity failure for pertussis in New Zealand. It has been suggested by a number of authors that a non-linear dependence of disease transmission on the numbers of infectives and susceptibles can reverse the stability of the system. However it is shown in this thesis that under biologically plausible constraints the non-linear transmission is unable to destabilise the system. The main constraint is a condition that disease transmission must be a concave function with respect to the number of infectives. This result is valid for both the discrete-time and the continuous-time models. We also consider the impact of mortality associated with a disease. This factor has never before been considered systematically. We indicate mechanisms through which the disease-induced mortality can affect the system and show that the disease-induced mortality is a destabilising factor and is able to reverse the system stability. However the critical level of mortality which is necessary to reverse the system stability exceeds the mortality expectation for the majority of human infections. Nevertheless the disease-induced mortality is an important factor for understanding animal diseases. It appears that in the case of autonomous systems there is no single factor able to cause the recurrent outbreaks of epidemics of such magnitudes as have been observed. It is most likely that in reality they are caused by a combination of factors.Subscription resource available via Digital Dissertations
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Smith, Robert Frederick. "Geometric models of the stenosed human carotid bifurcation." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp01/MQ32510.pdf.
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Venkatagiri, Shankar C. "The peak-crossing bifurcation in lattice dynamical systems." Diss., Georgia Institute of Technology, 1996. http://hdl.handle.net/1853/29340.
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