Shui-Nee Chow

An example of bifurcation to homoclinic orbits

Abstract Consider the equation x − x + x 2 = −λ 1 x + λ 2 ƒ(t) where ƒ(t + 1) = ƒ(t) and λ = (λ 1 , λ 2 ) is small. For λ = 0, there is a homoclinic orbit Γ through zero. For λ ≠ 0 and small, there can be “strange” attractors near Γ. The purpose of this paper is to determine the curves in λ-space of bifurcation to “strange” attractors and to relate this to hyperbolic subharmonic bifurcations.

Integral averaging and bifurcation

One of the simplest topological variations of the phase space of a one-parameter family of differential equations (vector fields, flows) is the creation of periodic orbits from equilibria as the parameter crosses a critical value. The study of such topological variations about an equilibrium was initiated and developed by Poincare perhaps 90 years ago and belongs today to the classical theory of periodic solutions. It was Hopf [23] who presented the bifurcation theorem in 1942 and it is now commonly known as the Hopf bifurcation theorem. Specifically, consider a one-parameter family of ODE (ordinary differential equations)

Invariant manifolds for flows in Banach spaces

On presente une theorie des varietes invariantes lisses basee sur la methode classique de Lyapunov-Penon pour des semi-flots continus dans des espaces de Banach

Pattern formation and spatial chaos in lattice dynamical systems. II

For part I see ibid., vol.42, no.10, pp.746-51 (1995). We survey a class of continuous-time lattice dynamical systems, with an idealized nonlinearity. We introduce a class of equilibria called mosaic solutions, which are composed of the elements 1, -1, and 0, placed at each lattice point. A stability criterion for such solutions is given. The spatial entropy h of the set of all such stable solutions is defined, and we study how this quantity varies with parameters. Systems are qualitatively distinguished according to whether h=0 (termed pattern formation), or h>0 (termed spatial chaos). Numerical techniques for calculating h are described. >

Homoclinic bifurcation at resonant eigenvalues

We consider a bifurcation of homoclinic orbits, which is an analogue of period doubling in the limit of infinite period. This bifurcation can occur in generic two parameter vector fields when a homoclinic orbit is attached to a stationary point with resonant eigenvalues. The resonance condition requires the eigenvalues with positive/negative real part closest to zero to be real, simple, and equidistant to zero. Under an additional global twist condition, an exponentially flat bifurcation of double homoclinic orbits from the primary homoclinic branch is established rigorously. Moreover, associated period doublings of periodic orbits with almost infinite period are detected. If the global twist condition is violated, a resonant side switching occurs. This corresponds to an exponentially flat bifurcation of periodic saddle-node orbits from the homoclinic branch.

The Fuller index and global Hopf bifurcation

Abstract Using an index for periodic solutions of an autonomous equation defined by Fuller, we prove Alexander and Yorkes global Hopf bifurcation theorem. As the Fuller index can be defined for retarded functional differential equations, the global bifurcation theorem can also be proved in this case. These results imply the existence of periodic solutions for delay equations with several rationally related delays, for example, x (t) = −α[ax(t − 1) + bx(t − 2)]g(x(t)) , with a and b non-negative and α greater than some computable quantity ξ(a, b) calculated from the linearized equation.

Lattice Dynamical Systems

The following notes are based on my lectures given at CIME Session on ”Dynamical Systems” from June 19 to June 26, 2000 in Cetraro (Cosenza). In Section 1, we study spatially discrete nonlinear diffusion equations and discuss various phenomenon; in Section 2, we introduce the idea of spatial chaos by studying Nagumo equation; in Section 3, 4 and 5, we present a general theory for pattern formation and spatial chaos in lattice dynamics systems; finally in Section 6 and 7, we study the special case of synchronisation phenomena of lattice systems.

Smooth Invariant Foliations in Infinite Dimensional Spaces

One of the most useful properties of dynamical systems is the existence of invariant manifolds and their invariant foliations near an equilibrium or a periodic orbit. These manifolds and foliations serve as a convenient setting to describe the qualitative behavior of the local flows, and in many cases they are useful tools for technical estimates which facilitate the study of the local bifurcation diagram (see [6]). Many other important concepts in dynamical systems are closely related to the invariant manifolds and foliations. In finite dimensional space, the relations among invariant manifolds, invariant foliations, l-lemma, linearization, and homoclinic bifurcation have been studied in [ll]. It is well known that if each leaf is used as a coordinate, the original system is completely decoupled and the linearization follows easily (for example, see [27, 221). As a motivation, let us consider a linear system in Rmfn

Synchronization in lattices of coupled oscillators

Abstract We consider coupled nonlinear oscillators with external periodic forces and the Dirichlet boundary conditions. We prove that synchronization occurs provided that the coupling is dissipative and the coupling coefficients are sufficiently large. The synchronization here is of an obvious type — the size of an attractor is comparable to the difference of the subsystems.

DYNAMICS OF LATTICE DIFFERENTIAL EQUATIONS

In this paper recent work on the dynamics of lattice differential equations is surveyed. In particular, results on propagation failure and lattice induced anisotropy for traveling wave or plane wav...