Simultaneity of centres in Zq-equivariant systems

We study the simultaneous existence of centres for two families of planar Zq-equivariant systems. First, we give a short review about Zq-equivariant systems. Next, we present the necessary and sufficient conditions for the simultaneous existence of centres for a Z2-equivariant cubic system and for a Z2- equivariant quintic system

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

Mathematical frameworks for oscillatory network dynamics in neuroscience

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathemat- ical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical frame- work for further successful applications of mathematics to understanding network dynamics in neuroscience

Bifurcations of attractors in 3D diffeomorphisms : a study in experimental mathematics

The research presented in this PhD thesis within the framework of nonlinear deterministic dynamical systems depending on parameters. The work is divided into four Chapters, where the first is a general introduction to the other three. Chapter two deals with the investigation of a time-periodic three-dimensional system of ordinary differential equations depending on three parameters, the Lorenz-84 model with seasonal forcing. The model is a variation on an autonomous system proposed in 1984 by the meteorologist E. Lorenz to describe general atmospheric circulation at mid latitude of the northern hemisphere. ... Zie: Summary

A new instability framework in 2-component reaction-diffusion systems

This paper concerns pattern formation in 2-component reaction-diffusion systems with linear diffusion terms and a local interaction. We propose a new instability framework with 0-mode Hopf instability, $m$ and $m + 1$ mode Turing instabilities in 2-component reaction-diffusion systems. The normal form for the codimension 3 bifurcation is derived via the center manifold reduction, which is one of the main results in the present paper. We also show numerical results on bifurcation of some reaction-diffusion systems and on a chaotic behavior of the normal form

Lectures on Mechanics

Publisher's description: The use of geometric methods in classical mechanics has proven fruitful, with wide applications in physics and engineering. In this book, Professor Marsden concentrates on these geometric aspects, especially on symmetry techniques. The main points he covers are: the stability of relative equilibria, which is analyzed using the block diagonalization technique; geometric phases, studied using the reduction and reconstruction technique; and bifurcation of relative equilibria and chaos in mechanical systems. A unifying theme for these points is provided by reduction theory, the associated mechanical connection and techniques from dynamical systems. These methods can be applied to many control and stabilization situations, and this is illustrated using rigid bodies with internal rotors, and the use of geometric phases in mechanical systems. To illustrate the above ideas and the power of geometric arguments, the author studies a variety of specific systems, including the double spherical pendulum and the classical rotating water molecule

Symmetries in bifurcation theory : the appropriate context

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