In this paper we study the dynamic bifurcation of the SwiftHohenberg equation on a periodic cell Ω = [−L,L]. It is shown that the equations bifurcates from the trivial solution to an attractor Aλ when the control parameter λ crosses the critical value. In the odd periodic case, Aλ is homeomorphic to S 1 and consists of eight singular points and their connecting orbits. In the periodic case, Aλ ...

In this paper we generalize analytic studies the problems related to suppression of chaos and non–feedback controlling chaotic motion. We develop an analytic method of the investigation of qualitative changes in chaotic dynamical systems under certain external periodic perturbations. It is proven that for polymodal maps one can stabilize chosen in advance periodic orbits. As an example, the qua...

There is at present a doubly discrete classification for strange attractors of low dimension, d(L)<3. A branched manifold describes the stretching and squeezing processes that generate the strange attractor, and a basis set of orbits describes the complete set of unstable periodic orbits in the attractor. To this we add a third discrete classification level. Strange attractors are organized by ...

The long-term unprcdictability associated with chaos may be undesirable in certain settings. Ott, Grebogi, and Yorke (Phys. Rev. Lett. 1990.64, 1196) have recently proposed a method by which any of the infinite number of unstable periodic orbits embedded within a strange attractor can be, in principle, stabilized through small, controlled perturbations of a system constraint. This method is app...

We review the major ideas involved in the control of chaos by considering higher dimensional dynamics. We present the Ott-Grebogi-Yorke (OGY) method of controlling chaos to achieve time periodic motion by utilizing only small feedback control. The time periodic motion results from the stabilization of unstable periodic orbits embedded in the chaotic attractor. We demonstrate that the OGY method...

Abstract The Rayleigh–Bénard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. physical problem be equivalent one periodic and certain symmetries. This enables Gronwall estimate on enstrophy. That then used bound the L 2 norm of temperature gradient attractor, which, turn, find bounding region for enstr...

We continue to study a simple integro-differential equation: the Quasi-Steady equation (QS) of flame front dynamics. This second order quasi-linear parabolic equation with a non-local term is dynamically similar to the Kuramoto-Sivashinsky (KS) equation. In [FGS03], where it was introduced, its well-posedness and proximity for finite time intervals to the KS equation in Sobolev spaces of period...

A discrete-time version of the replicator equation for two-strategy games is studied. The stationary properties differ from those of continuous time for sufficiently large values of the parameters, where periodic and chaotic behavior replace the usual fixed-point population solutions. We observe the familiar period-doubling and chaotic-band-splitting attractor cascades of unimodal maps but in s...

We study the transition from stochasticity to determinism in the three-strategy pairapproximated prisoner’s dilemma game. We show that the stochastic solution converges to the deterministic limit cycle attractor as the number of participating players increases. Importantly though, between the stochastic and periodic solutions, we reveal a broad range of population sizes for which the system exh...

A chaotic model of spontaneous (without external stimulus) neuron firing has been analyzed by mapping the irregular spiking time-series into telegraph signals. In this model the fundamental frequency of chaotic Rössler attractor provides (with a period doubling) the strong periodic component of the generated irregular signal. The exponentially decaying broad-band part of the spectrum of the Rös...