This paper investigates border-collision bifurcations in piecewise-linear planar maps that are non-invertible in one region. Maps of this type arise as normal forms for grazing–sliding bifurcations in three-dimensional Filippovtype systems. A possible strategy is presented for classifying fixed and period-2 points, that are involved in such bifurcations. This allows one to determine a region of...

We construct a Cantor set ̂ of limit-periodic Jacobi operators having the spectrum on the Julia set J of the quadratic map z ι-> z + E for large negative real numbers E. The density of states of each of these operators is equal to the unique equilibrium measure μ on J. The Jacobi operators in $ are defined over the von Neumann-Kakutani system, a group translation on the compact topological group...

We construct a determining form for the surface quasi-geostrophic (SQG) equation with subcritical dissipation. In particular, we show that the global attractor for this equation can be embedded in the long-time dynamics of an ordinary differential equation (ODE) called a determining form. Indeed, there is a one-to-one correspondence between the trajectories in the global attractor of the SQG eq...

A simple and transparent example of a nonautonomous flow system with a hyperbolic strange attractor is suggested. The system is constructed on the basis of two coupled van der Pol oscillators, the characteristic frequencies differ twice, and the parameters controlling generation in both oscillators undergo a slow periodic counterphase variation in time. In terms of stroboscopic Poincaré section...

We consider an autonomous system of partial differential equations for a onedimensional distributed medium with periodic boundary conditions. Dynamics in time consists of alternating birth and death of patterns with spatial phases transformed from one stage of activity to another by the doubly expanding circle map. So, the attractor in the Poincaré section is uniformly hyperbolic, a kind of Sma...

Wolfram divided the 256 elementary cellular automata rules informally into four classes using dynamical concepts like periodicity, stability, and chaos. Rule 14, which is Bernoulli στ -shift rule and is a member of Wolfram’s class II, is said to be simple as periodic before. Therefore, it is worthwhile studying dynamical behaviors of rule 14, whether it possesses chaotic attractors or not. In t...

The forced and weakly damped Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. Starting from L and mean-zero initial data we prove that the solution decomposes into two parts; a linear one which decays to zero as time goes to infinity and a nonlinear one which always belongs to a smoother space. As a corollary we prove that all solutions are attracted by a ball i...

Abstract. The properties of the mean first passage time in a system characterized by multiple periodic attractors are studied. Using a transformation from a high dimensional space to 1D, the problem is reduced to a stochastic process along the path from the fixed point attractor to a saddle point located between two neighboring attractors. It is found that the time to switch between attractors ...

Abstract We use entropy theory as a new tool to study sectional hyperbolic flows in any dimension. show that for C 1 flows, every set Λ is expansive, and the topological varies continuously with flow. Furthermore, if Lyapunov stable, then it has positive entropy; addition, chain recurrent class, contains periodic orbit. As corollary, we prove generic Lorenz-like class an attractor.

We investigate the prevalence of Li-Yorke pairs for C and C multimodal maps f with non-flat critical points. We show that every measurable scrambled set has zero Lebesgue measure and that all strongly wandering sets have zero Lebesgue measure, as does the set of pairs of asymptotic (but not asymptotically periodic) points. If f is topologically mixing and has no Cantor attractor, then typical (...