John Mallet-Paret

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)

The Global Structure of Traveling Waves in Spatially Discrete Dynamical Systems

We obtain existence of traveling wave solutions for a class of spatially discrete systems, namely, lattice differential equations. Uniqueness of the wave speed c, and uniqueness of the solution with c≠0, are also shown. More generally, the global structure of the set of all traveling wave solutions is shown to be a smooth manifold where c≠0. Convergence results for solutions are obtained at the singular perturbation limit c → 0.

Inertial Manifolds for Reaction Diffusion Equations in Higher Space Dimensions

In this paper we show that the scalar reaction di1rusion equation u/ = v6.u + f(x, u), UER with x E On C Rn (n = 2,3) and with Dirichlet, Neumann, or periodic boundary conditions, has an inertial manifold when (1) the equation is dissipative, and (2) f is of class C3 and for 03 = (O,2n:)3 or Oz = (O,2n:/at> x (0, 2n:/az), where al and az are positive. The proof is based on an (abstract) Invariant Manifold Theorem for flows on a Hilbert space. It is significant that on 0 3 the spectrum of the Laplacian 6. does not have arbitrary large gaps, as required in other theories of inertial manifolds. Our proof is based on a crucial property of the Schroedinger operator 6. + v(x), which is valid only in space dimension n ~ 3. This property says that 6. + v(x) can be well approximated by the constant coefficient problem 6. + f} over large segments of the Hilbert space L2(0) , where v = (vol 0)-1 fn v dx is the average value of v . We call this property the Principle of Spatial Averaging. The proof that the Schroedinger operator satisfies the Principle of Spatial Averaging on the regions Oz and 03 described above follows from a gap theorem for finite families of quadratic forms, which we present in an Appendix to this paper. DIVISION OF ApPLIED MATHEMATICS, BROWN UNIVERSITY, PROVIDENCE, RHODE ISLAND 02912 INSTITUTE FOR MATHEMATICS AND ITS ApPLICATIONS, UNIVERSITY OF MINNESOTA, MINNEAPOLIS, MINNESOTA 55455 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

The Fredholm Alternative for Functional Differential Equations of Mixed Type

We prove a Fredholm alternative theorem for a class of asymptotically hyperbolic linear differential difference equations of mixed type. We also establish the cocycle property and the spectral flow property for such equations, providing an effective means of calculating the Fredholm index. Such systems can arise from equations which describe traveling waves in a spatial lattice.

The Poincare-Bendixson theorem for monotone cyclic feedback systems

AbstractWe prove the Poincare-Bendixson theorem for monotone cyclic feedback systems; that is, systems inRn of the form

Pattern formation and spatial chaos in lattice dynamical systems. II

x_i = f_i (x_i , x_{i - 1} ), i = 1, 2, ..., n (\bmod n).

The Fuller index and global Hopf bifurcation

We apply our results to a variety of models of biological systems.