Invariant Manifolds and Their Applications

A special session in

Fourth International Conference on Dynamical Systems and Differential Equations

Wilmington, NC, USA, May 24-27, 2002

Organizers:

 

List of Confirmed Invited Speakers:

  • Lora Billings Montclair State University
  • R.E. Lee DeVille, Rensselaer Polytechnic Institute
  • Brian Hunt, University of Maryland, College Park
  • Kresimir Josic, Boston University
  • Igor Kukavica, University of Southern California
  • Weishi Liu, University of Kansas
  • Horacio Rotstein, Boston University
  • Gene Wayne, Boston University
  • Chongchun Zeng, University of Virginia

    • Titles and Abstracts:

  • C. Eugene Wayne: Invariant Manifolds and the Navier-Stokes Equation
    I will explain some recent work on the construction of finite dimensional invariant manifolds in the phase space of the Navier-Stokes equation on ${\bf R}^n$. These manifolds control the long-term behavior of small solutions, give geometric insight into the host of existing results on the asymptotics of such solutions, and allow one to extend those results in a number of ways. Our results also allow us to prove the stability of certain vortex solutions of the Navier-Stokes equation, even at very large Reynold's number. This is joint work with T. Gallay.


  • Lora Billings: Noise induced chaos
    We identify a global mechanism to induce chaos by stochastic perturbations. Two systems in which we find this are the class B laser and the SEIR population dynamics model. The bifurcation to chaos requires two co-existing saddle periodic orbits in a multistable system, which we call a bi-instability. The noise induces a heteroclinic connection between the invariant manifolds of the saddle periodic orbits, therefore inducing a chaotic attractor. To refine the possibility of control, we have also analyzed the stochastic transport between basins. This is joint work with Ira Schwartz and Erik Bollt.


  • Brian Hunt: Bubbling Bifurcations
    For a one-parameter family of dynamical systems with a persistent invariant submanifold, I will characterize how a chaotic attractor in the invariant manifold loses asymptotic stability transverse to the manifold as the parameter is varied. After this bifurcation, the attractor generically remains weakly stable, having a basin of attraction that is "riddled" -- it has positive Lebesgue measure but is not open. Small perturbations of the system can then lead to intermittent behavior called "bubbling" -- trajectories spend most of their time near the (formerly) invariant manifold but occasionally burst far away. I will describe different types of bifurcations that can lead to bubbling and the resulting size and frequency of bursts near the bifurcation. The results are relevant to the synchronization of coupled chaotic systems, where bursting represents temporary loss of synchronization.


  • Weishi Liu: On the viscous shock profiles and viscous wave fan profiles of Riemann solutions
    In this talk, I will first describe briefly the Exchange Lemmas for singularly perturbed systems with a class of turning points. As an application, we consider a system of conservation laws in one space dimension and study the structural stability of Riemann solutions. We show that, in particular, there are Riemann solutions which are generically structrally unstable in terms of viscous shock profile criterion but are generically structurally stable in terms of viscous wave fan profile criterion. This research is closely related and motivated by some of the works of Marchesin, Plohr, and Schecter.


  • R. E. Lee DeVille: Approximating the dynamics of thin elastic media
    In this talk, we describe a method for deriving and justifying a hierarchy of "reduced equations" for the dynamical motion of thin elastic media, i.e., starting with a PDE defined on a three-dimensional domain, we will show that its solutions can be approximated by the solutions of equations defined on a two-dimensional domain, and, furthermore, there is a sequence of approximating equations, each of which affords a successively better approximation. The approach is based on ideas from Hamiltonian mechanics, and is related to Nekhoroshev theory. This is joint work with C. Eugene Wayne.


  • Chongchun Zeng: Approximate normally hyperbolic invariant manifolds
    In this talk, we consider a semiflow in a Banach space where a C^1 submanifold is approximately invariant and normally hyperbolic. Assuming the semiflow is inflowing (overflowing) along the boundry, we prove there exists a unique positively invariant stable (unstable) manifold which has an invariant stable (unstable) foliation.


  • Kresimir Josic: Limits to the detection of nonlinear synchrony
    It has long been recognized that the phenomenon of synchronization of chaotic systems can be naturally described in terms of smooth invariant manifolds. Recent evidence suggests that systems exhibiting complex behavior may be synchronized in a weaker sense. A number of such examples and the analytical methods needed to study them will be discussed. I will address the effect of nonsmoothnes of the synchronization manifold on the detectability of the synchronized state. Moreover, in the case the driving system is not invertible the synchronization set is no longer even a manifold but a far more complicated set. I will discuss how the usual graph transform methods can be extended to this case to gain information about the structure of this set. In conclusion I will discuss how these examples provide clues about about the dynamical nature of weak synchrony and discuss practical methods for the detection of such coherent states.


  • Igor Kukavica: Oscillation properties of the complex Ginzburg-Landau equation
    We present estimates on complexity of solutions of the 1D Complex Ginzburg-Landau equation. We will discuss optimality of bounds and discuss extensions to the 2D case. The methods are based on analyticity and unique continuation properties of solutions to the equation.


  • Horacio Rotstein: The canard phenomenon as a mechanism of localization of oscilllations
    We study a system of relaxation oscillators of Fitzhugh-Nagumo type with global inhibitory feedback. In particular we study the mechanism of localization of oscillations for such a system. A cluster is a set of synchroneous oscillators having the same amplitude. In a localized solution the system is divided into two or more clusters and at least two of them oscillate with a ``big'' difference in their amplitudes.

    We argue that the mechanism of localization is based on the canard phenomenon that occurs in relaxation oscillators. For synchronized oscillations we show that the canard phenomenon can be induced by increasing the value of the global inhibitory feedback parameter. For a two-cluster system we show that localized solutions exist in an interval of values of the global feedback parameter that increases with increasing size differences between the two clusters. We argue that under specific assumptions the existence of localized clusters can be understood as a composition of two phenomena: self-inhibition and inhibition (forcing) exerted on each oscillator by the remaining ones. We analyze the effect of self-inhibition on localization and present the results of our simulations.

    Our work is motivated by experimental and simulation results on the Belousov-Zhabotinsky reaction with global inhibitory feedback. We present some numerical results supporting our claim that the same mechanism is responsible for the localization phenomenon in this system. Abstract

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