In this paper we consider a new type of bifurcation to chaotic motion. This bifurcation is characterized by the simultaneous appearance of a pair of unstable fixed points or periodic orbits, a, fractal (i.e., nondifferentiable) basin boundary, and a chaotic attractor (also commonly called a strange attractor). Just prior to this type of bifurcation, transient behavior with a chaotic character c...

Robust Exponential stability of continuous-time attractor neural networks with delays is discussed. A new sufficient condition ensuring existence and uniqueness of periodic solution for a general class of interval dynamical systems are obtained. Discretetime analogue of the continuous-time systems with periodic input is formulated and we study their dynamical characteristics. The robust exponen...

We have computed all the orbit-forcing implications, up to period 8, in horseshoe-type flows or maps that generate strange attractors. The results are presented in a horseshoe implication diagram. We describe how this diagram was computed, and show how it can be used to construct a minimal (basis) set of periodic orbits which force the existence of all the periodic orbits associated with a stra...

We consider intermittency in one parameter families of unimodal maps, induced by saddle node and boundary crisis bifurcations. In these bifurcations a periodic orbit or a periodic interval, respectively, disappears to give rise to chaotic bursts. We prove asymptotic formulae for the frequency with which orbits visit the region previously occupied by the attractor. For this, we extend results of...

It has been widely experienced that tracking (targeting) a periodic orbit embedded within a chaotic attractor often encounters an essential issue of numerical sensitivity. In this paper, we develop an effective digital tracker for continuous-time chaotic orbit tracking, which is insensitive to numerical errors. The design is based on some advanced digital redesign techniques equipped with a pre...

For the Kuramoto Sivashinsky equation with L-periodic boundary conditions we show that the radius of space analyticity on the global attractor is lowersemicontinuous function at the stationary solutions, and thereby deduce the existence of a neighborhood in the global attractor of the set of all stationary solutions in which the radius of analyticity is independent of the bifurcation parameter ...

We investigate the bifurcation mechanism for the loss of transverse stability of the chaotic attractor in an invariant subspace in an asymmetric dynamical system. It is found that a direct transition to global riddling occurs through a transcritical contact bifurcation between a periodic saddle embedded in the chaotic attractor on the invariant subspace and a repeller on its basin boundary. Thi...

We revisit the globally coupled map lattice. We show that in the so called turbulent regime various periodic cluster attractor states are formed, even though the coupling between the maps are very small relative to the nonlinearity in the element maps. Most outstanding is a maximally symmetric three cluster attractor in period-3 motion, due to the foliation of the period-3 window of the element...

which is chaotic on a particular parameter set, (σ, ρ, β) = (10, 28, 8/3)1 in the real parameter space of the general system (1); but for other parameter sets, system (1) may not be chaotic. The chaotic Lorenz system (2) has received sustained research interest in the literature since its discovery. For example, in [Tucker, 1999; Stewart, 2000], a rigorous proof confirms the existence of the ch...

Powerful computational techniques have been developed to study chaotic attractors that are generated by stretching and folding processes. These include relative rotation rates for determining the organization of unstable periodic orbits and simplex distortion procedures for estimating the topological entropy of these orbits. These methods are useful for attractors contained in a genus-one torus...