Cycling behavior involving steady-states and periodic solutions is known to be a generic feature of continuous dynamical systems with symmetry. Using Chua’s circuit equations and Lorenz equations, Dellnitz et al. [1995] showed that “cycling chaos”, in which solution trajectories cycle around symmetrically related chaotic sets, can also be found generically in coupled cell systems of differentia...

We present an asymptotic stability result for a class of discontinuous dynamical systems (DDS) determined by differential equations in Banach space (resp., Cauchy problems on abstract spaces). We demonstrate the applicability of our result in the analysis of several important classes of DDS, including systems determined by functional differential equations and partial differential equations. Co...

An interactive program for visualizing the long term behavior of dynamical systems, e.g., attractors and bifurcation diagrams, is presented. The program allows an easy specification of a set of formulas and constants which describe a dynamical system. This set of equations is then used for displaying the geometrical structure of the long term development of the dynamical system. The user define...

We study the o -equilibrium dynamics of a particle in a general N dimensional random potential whenN !1. We demonstrate the existence of two asymptotic time regimes: i. stationary dynamics, ii. slow aging dynamics with violation of equilibrium theorems. We derive the equations obeyed by the slowly varying part of the two-times correlation and response functions and obtain an analytical solution...

The singularity confinement test is very useful for isolating integrable cases of discrete-time dynamical systems, but it does not provide a sufficient criterion for integrability. Quite recently a new property of the bilinear equations appearing in discrete soliton theory has been noticed: the iterates of such equations are Laurent polynomials in the initial data. A large class of non-integrab...

Stochastic partial differential equations arise as mathematical models of complex multiscale systems under random influences. Invariant manifolds often provide geometric structure for understanding stochastic dynamics. In this paper, a random invariant manifold reduction principle is proved for a class of stochastic partial differential equations. The dynamical behavior is shown to be described...

Keywords: Relative controllability Time delays Distributed delays Fractional derivative Mittag–Leffler function a b s t r a c t This paper is concerned with the controllability of nonlinear fractional dynamical systems with time varying multiple delays and distributed delays in control defined in finite dimensional spaces. Sufficient conditions for controllability results are obtained using the...

In this case-study, we examine the effects of linear control on continuous dynamical systems that exhibit chaotic behavior using the symbolic computer algebra system Mathematica. Stabilizing (or controlling) higher-dimensional chaotic dynamical systems is generally a difficult problem, Musielak and Musielak, [1]. We numerically illustrate that sometimes elementary approaches can yield the desir...