108 research outputs found
Dense heteroclinic tangencies near a Bykov cycle
This article presents a mechanism for the coexistence of hyperbolic and
non-hyperbolic dynamics arising in a neighbourhood of a Bykov cycle where
trajectories turn in opposite directions near the two nodes --- we say that the
nodes have different chirality. We show that in the set of vector fields
defined on a three-dimensional manifold, there is a class where tangencies of
the invariant manifolds of two hyperbolic saddle-foci occur densely. The class
is defined by the presence of the Bykov cycle, and by a condition on the
parameters that determine the linear part of the vector field at the
equilibria. This has important consequences: the global dynamics is
persistently dominated by heteroclinic tangencies and by Newhouse phenomena,
coexisting with hyperbolic dynamics arising from transversality. The
coexistence gives rise to linked suspensions of Cantor sets, with hyperbolic
and non-hyperbolic dynamics, in contrast with the case where the nodes have the
same chirality.
We illustrate our theory with an explicit example where tangencies arise in
the unfolding of a symmetric vector field on the three-dimensional sphere
Applications of dynamical systems with symmetry
This thesis examines the application of symmetric dynamical systems theory to
two areas in applied mathematics: weakly coupled oscillators with symmetry, and
bifurcations in flame front equations.
After a general introduction in the first chapter, chapter 2 develops a theoretical
framework for the study of identical oscillators with arbitrary symmetry group under an
assumption of weak coupling. It focusses on networks with 'all to all' Sn coupling. The
structure imposed by the symmetry on the phase space for weakly coupled oscillators
with Sn, Zn or Dn symmetries is discussed, and the interaction of internal symmetries
and network symmetries is shown to cause decoupling under certain conditions.
Chapter 3 discusses what this implies for generic dynamical behaviour of coupled
oscillator systems, and concentrates on application to small numbers of oscillators (three
or four). We find strong restrictions on bifurcations, and structurally stable heteroclinic
cycles.
Following this, chapter 4 reports on experimental results from electronic oscillator
systems and relates it to results in chapter 3. In a forced oscillator system, breakdown
of regular motion is observed to occur through break up of tori followed by a symmetric
bifurcation of chaotic attractors to fully symmetric chaos.
Chapter 5 discusses reduction of a system of identical coupled oscillators to phase
equations in a weakly coupled limit, considering them as weakly dissipative Hamiltonian
oscillators with very weakly coupling. This provides a derivation of example phase
equations discussed in chapter 2. Applications are shown for two van der Pol-Duffing
oscillators in the case of a twin-well potential.
Finally, we turn our attention to the Kuramoto-Sivashinsky equation. Chapter 6
starts by discussing flame front equations in general, and non-linear models in particular.
The Kuramoto-Sivashinsky equation on a rectangular domain with simple
boundary conditions is found to be an example of a large class of systems whose linear
behaviour gives rise to arbitrarily high order mode interactions.
Chapter 7 presents computation of some of these mode interactions using competerised
Liapunov-Schmidt reduction onto the kernel of the linearisation, and investigates
the bifurcation diagrams in two parameters
Nearly inviscid Faraday waves in containers with broken symmetry
In the weakly inviscid regime parametrically driven surface gravity-capillary waves generate oscillatory viscous boundary layers along the container walls and the free surface. Through nonlinear rectification these generate Reynolds stresses which drive a streaming flow in the nominally inviscid bulk; this flow in turn advects the waves responsible for the boundary layers. The resulting system is described by amplitude equations coupled to a Navier-Stokes-like equation for the bulk streaming flow, with boundary conditions obtained by matching to the boundary layers, and represents a novel type of pattern-forming system. The coupling to the streaming flow is responsible for various types of drift instabilities of standing waves, and in appropriate regimes can lead to the presence of relaxations oscillations. These are present because in the nearly inviscid regime the streaming flow decays much more slowly than the waves. Two model systems, obtained by projection of the Navier-Stokes-like equation onto the slowest mode of the domain, are examined to clarify the origin of this behavior. In the first the domain is an elliptically distorted cylinder while in the second it is an almost square rectangle. In both cases the forced symmetry breaking results in a nonlinear competition between two nearly degenerate oscillatory modes. This interaction destabilizes standing waves at small amplitudes and amplifies the role played by the streaming flow. In both systems the coupling to the streaming flow triggered by these instabilities leads to slow drifts along slow manifolds of fixed points or periodic orbits of the fast system, and generates behavior that resembles bursting in excitable systems. The results are compared to experiments
Dynamics of nearly inviscid Faraday waves in almost circular containers
Parametrically driven surface gravity-capillary waves in an elliptically distorted circular cylinder are studied. In the nearly inviscid regime, the waves couple to a streaming flow driven in oscillatory viscous boundary layers. In a cylindrical container, the streaming flow couples to the spatial phase of the waves, but in a distorted cylinder, it couples to their amplitudes as well. This coupling may destabilize pure standing oscillations, and lead to complex time-dependent dynamics at onset. Among the new dynamical behavior that results are relaxation oscillations involving abrupt transitions between standing and quasiperiodic oscillations, and exhibiting ‘canards’
Symmetry breaking in dynamical systems
Symmetry breaking bifurcations and dynamical systems have obtained a lot of attention over the last years. This has several reasons: real world applications give rise to systems with symmetry, steady state solutions and periodic orbits may have interesting patterns, symmetry changes the notion of structural stability and introduces degeneracies into the systems as well as geometric simplifications. Therefore symmetric systems are attractive to those who study specific applications as well as to those who are interested in a the abstract theory of dynamical systems. Dynamical systems fall into two classes, those with continuous time and those with discrete time. In this paper we study only the continuous case, although the discrete case is as interesting as the continuous one. Many global results were obtained for the discrete case. Our emphasis are heteroclinic cycles and some mechanisms to create them. We do not pursue the question of stability. Of course many studies have been made to give conditions which imply the existence and stability of such cycles. In contrast to systems without symmetry heteroclinic cycles can be structurally stable in the symmetric case. Sometimes the various solutions on the cycle get mapped onto each other by group elements. Then this cycle will reduce to a homoclinic orbit if we project the equation onto the orbit space. Therefore techniques to study homoclinic bifurcations become available. In recent years some efforts have been made to understand the behaviour of dynamical systems near points where the symmetry of the system was perturbed by outside influences. This can lead to very complicated dynamical behaviour, as was pointed out by several authors. We will discuss some of the technical difficulties which arise in these problems. Then we will review some recent results on a geometric approach to this problem near steady state bifurcation points
Parametric Forcing of Confined and Stratified Flows
abstract: A continuously and stably stratified fluid contained in a square cavity subjected to harmonic body forcing is studied numerically by solving the Navier-Stokes equations under the Boussinesq approximation. Complex dynamics are observed near the onset of instability of the basic state, which is a flow configuration that is always an exact analytical solution of the governing equations. The instability of the basic state to perturbations is first studied with linear stability analysis (Floquet analysis), revealing a multitude of intersecting synchronous and subharmonic resonance tongues in parameter space. A modal reduction method for determining the locus of basic state instability is also shown, greatly simplifying the computational overhead normally required by a Floquet study. Then, a study of the nonlinear governing equations determines the criticality of the basic state's instability, and ultimately characterizes the dynamics of the lowest order spatial mode by the three discovered codimension-two bifurcation points within the resonance tongue. The rich dynamics include a homoclinic doubling cascade that resembles the logistic map and a multitude of gluing bifurcations.
The numerical techniques and methodologies are first demonstrated on a homogeneous fluid contained within a three-dimensional lid-driven cavity. The edge state technique and linear stability analysis through Arnoldi iteration are used to resolve the complex dynamics of the canonical shear-driven benchmark problem. The techniques here lead to a dynamical description of an instability mechanism, and the work serves as a basis for the remainder of the dissertation.Dissertation/ThesisSupplemental Materials Description Filezip file containing 10 mp4 formatted video animations, as well as a text readme and the previously submitted Supplemental Materials Description FileDoctoral Dissertation Mathematics 201
Vector fields with heteroclinic networks
Dissertação de Doutoramento em Matemática apresentada à Faculdade de Ciências da Universidade do PortoO trabalho desenvolvido ao longo desta tese tem como ponto de partida uma famÃlia de equações diferenciais apresentada e estudada por Field (Ver M.J Field, 1996, Lectures on Bifurcations, Dynamics and Symmetry, Pitman Research Notes in Mathematics Series 356, Longman).Field conjectura, com base no seu estudo analÃtico e numérico, a existência, para certos valores dos parâmetros, de uma rede heteroclÃnica envolvendo os equilÃbrios e as trajectórias periódicas na dinâmica do sistema. No caso de as variedades invariantes de dimensão 2 dos equilÃbrios e das trajectórias periódicas se intersectarem transversalmente, Field conjectura também a existência de dinâmica da ferradura da rede heteroclÃnica.Nesta tese provamos as conjecturas de Field. O trabalho aqui desenvolvido indica a existência de uma rede heteroclÃnica de Shilnikov e prova a existência de dinâmica da ferradura na vizinhança de uma tal rede heteroclÃnica.Usamos a simetria do sistema para definir a rede heteroclÃnica quociente. Isto sugeriu-nos uma abordagem para estudar a dinâmica na vizinhança da rede heteroclÃnica de Shilnikov. O estudo da dinâmica é efectuado com recurso a uma codificação da dinâmica ao longo da rede heteroclÃnica e a uma codificação local na vizinhança dos ciclos heteroclÃnicos na rede quociente.ConstruÃmos exemplos simples contendo ciclos heteroclÃnicos de Shilnikov que são topologicamente equivalentes a ciclos heteroclÃnicos quocientes no exemplo de Field. Um facto importante acerca destes exemplos é que, apesar de possuÃrem dinâmica complexa, pela forma como são construÃdos, são mais fáceis de manipular analiticamente. Por exemplo, provamos analiticamente a intersecção transversal das variedades invariantes de dimensão 2.Os exemplos que construÃmos ajudam a compreender o comportamento complexo no exemplo de Field. Provamos a existência de dinâmica da ferradura na vizinhança de ciclos heteroclÃnicos envolvendo duas selas com autovalores complexos. Isto prova a existência ..
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