We analyze the double Hopf bifurcations which occur in two geophysical fluid dynamics models: (1) a two-layer quasigeostrophic potential vorticity model with forcing and (2) a mathematical model of the differentially heated rotating annulus experiment. The bifurcations occur at the transition between axisymmetric steady solutions and non-axisymmetric travelling waves. For both models, the results indicate that, close to the transition, there are regions in parameter space where there are multiple stable waves. Hysteresis of these waves is predicted. For each model, center manifold reduction and normal form theory are used to deduce the local behaviour of the full system of partial differential equations from a low-dimensional system of ordinary differential equations. In each case, it is not possible to compute the relevant eigenvalues and eigenfunctions analytically. Therefore, the linear part of the equations is discretized and the eigenvalues and eigenfunctions are approximated from the resulting matrix eigenvalue problem. However, the projection onto the center manifold and reduction to normal form can be done analytically. Thus, a combination of analytical and numerical methods are used to obtain numerical approximations of the normal form coefficients, from which the dynamics are deduced. The first model differs from those previously studied with bifurcation analysis since it supports a steady solution which varies nonlinearly with latitude. The results indicate that the forcing does not qualitatively change the behaviour. However, the form of the bifurcating solution is affected. The second model uses the Navier-Stokes equations in the Boussinesq approximation, in cylindrical geometry. In addition to the double Hopf bifurcation analysis, a detailed axisymmetric to non-axisymmetric transition curve is produced from the computed eigenvalues. A quantitative comparison with experimental data finds that the computed transition curve, critical wave numbers and drift rates of the bifurcating waves are reasonably accurate. This indicates that the analysis, as well as the approximations which are made, are valid. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis contains some results on the behavior of a delay differential equation (DDE) with two delays, at a Hopf bifurcation, for the nonzero equilibrium, using the growth rate, r, as bifurcation parameter. This DDE is a model for population growth, incorporating a maturation delay, and a second delay in the harvesting term. Considering a Taylor expansion of the non-dimensionalized model, we find a region of stability for the nonzero equilibrium, after which we find a pair of ODEs which help define the flow on the center manifold. We then find an expression for the first Lypapunov coefficient, which changes sign, so we also find the second Lyapunov coefficient, allowing us to predict multi-stability in the model. Numerical simulations provide examples of the behavior expected. For a similar model with one delay (PMC model), we prove the Hopf bifurcation at the nonzero equilibrium is always supercritical.

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In this paper we explore bifurcations, in particular the Hopf bifurcation. We study this especially in connection with the Brusselator, which is a model of certain chemical reaction-diffusion systems. After a thorough exploration of what a bifurcation is and what classifications there are, we give graphic representations of an occurring Hopf bifurcation in the Brusselator. When an additional forcing term is added, behavior changes dramatically. This includes the introduction of a horseshoe in the time map as well as a strange attractor in the system.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

We examine discrete models with hexagonal symmetry to compare the sequence of transitions with the alpha-inc-beta phase transition of quartz. We examine a model by Parlinski which employs interactions of nearest and next-nearest neighbor atoms. We numerically determine the configurations which lead to minimum energy for a range of parameters. We then use Golubitsky's results on systems with hexagonal symmetry to derive the bifurcation diagram for Parlinski's model. Finally, we study a large class of modifications to Parlinski's model and show that all such modifications have the same bifurcation picture as the original model. / Ph. D.

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.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Submitted by DIANA YOVANI RODRÍGUEZ VILLENA null (dayaniss_23@hotmail.com) on 2017-10-09T19:16:47Z No. of bitstreams: 1 Dissertação Diana.pdf: 1051753 bytes, checksum: df5c2679c43a774ec3d6809c69271fd4 (MD5) / Approved for entry into archive by Monique Sasaki (sayumi_sasaki@hotmail.com) on 2017-10-09T19:43:34Z (GMT) No. of bitstreams: 1 rodriguezvillena_dy_me_sjrp.pdf: 1051753 bytes, checksum: df5c2679c43a774ec3d6809c69271fd4 (MD5) / Made available in DSpace on 2017-10-09T19:43:34Z (GMT). No. of bitstreams: 1 rodriguezvillena_dy_me_sjrp.pdf: 1051753 bytes, checksum: df5c2679c43a774ec3d6809c69271fd4 (MD5) Previous issue date: 2017-08-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho, estudamos a teoria de bifurcação e algumas das suas aplicações. Apresentamos alguns resultados básicos e definimos o conceito de ponto de bifurcação. Logo, estudamos a teoria do grau topológico. Em seguida, enunciamos dois teoremas importantes que são os teoremas de Krasnoselski e de Rabinowitz. Finalmente apresentamos um exemplo e duas aplicações do teorema de Rabinowitz nas quais os valores característicos com que lidamos são simples, no exemplo se consegue provar que a segunda alternativa do teorema ocorre, a primeira aplicação é um problema de autovalores não lineares de Sturm-Liouville para uma E.D.O de segunda ordem na qual se prova que a primeira alternativa do teorema de Rabinowitz é válida e a segunda aplicação é um problema de autovalores para uma equação diferencial parcial quase-linear a qual se prova que também ocorre a primeira alternativa do teorema. / In this work, we study bifurcation theory and its applications. We present some basic results and define the concept of bifurcation point. Then we study the theory of topological degree. Next we state two important theorems that are Krasnoselski's theorem and Rabinowitz's theorem. Finally we present an example and two applications of Rabinowitz theorem in which the characteristic values we deal with are simple, in an example we can prove that the second item of theorem occurs and the first application is a nonlinear Sturm-Liouville eigenvalue problem for a second order ordinary differential equation were we prove that the first alternative of Rabinowitz's theorem holds and the second application is an eigenvalue problem for a quasilinear elliptic partial differential equation where we prove that the first alternative of the theorem also holds.