Abstract:
In this paper, bifurcations near optimal escape for Thompson's escape equation are numerically studied by means of Generalized Cell Mapping Digraph (GCMD) method. We find a chaotic saddle embedded in a Wada fractal basin boundary. The chaotic saddle is an unstable (nonattracting) chaotic invariant set. The Wada fractal basin boundary has the Wada property that any point that is on the boundary of that basin is also simultaneously on the boundary of at least two other basins. The chaotic saddle in the Wada basin boundary plays an extremely important role in the bifurcations governing the escape. We demonstrate that the chaotic saddle in the Wada basin boundary leads to a local saddle-node fold bifurcation with globally indeterminate outcome. In such a case, the attractor (node) and the saddle of the saddle-node fold are merged into the chaotic saddle and the chaotic saddle also undergoes an abrupt enlargement in its size as a parameter passes through the bifurcation value, simultaneously the Wada basin boundary is also converted into the fractal basin boundary of two remaining attractors, in particular, the chaotic saddle after the saddle-node fold bifurcation is in the fractal basin boundary, this implies that the saddle-node fold bifurcation has indeterminate outcome, namely, after the system drifts through the bifurcation, which of the two remaining attractors the orbit goes to is indeterminate in that it is sensitively dependent on arbitrarily small effects such as how the parameter is changed and/or noise and/or computer roundoff, obviously, this presents an extreme form of indeterminacy in a dynamical system. We also investigate the origin and evolution of the chaotic saddle in the Wada basin boundary and demonstrate that the chaotic saddle in the Wada basin boundary is created by the collision between two chaotic saddles in different fractal basin boundaries. We demonstrate that a final escape bifurcation is the boundary crisis caused by the collision between a chaotic attractor and a chaotic saddle, and this implies that Grebogi's definition of the boundary crisis by the collision with a periodic saddle is generalized.

Abstract:
In this paper, according to Hsu's idea that posets and digraphs are introduced intogeneralized cell mapping, a generalized cell mapping digraph method is presented by using thetheory of generalized cell mapping discretizing the continuous state space into the cell state spaceand the theories of set and digraph to achieve the task of global analysis of nonlinear dynamicalsystems. In the cell state space, we make the correspondence between generalized cell mappingdynamical systems and digraphs. The demonstrations of the two theorems of existence of self-cycling set and persistent self-cycling set are given. State cells are classified, and self-cycling sets,persistent self-cycling sets and transient self-cycling sets are defined. The persistent self-cycling setsrepresent the attractors of the systems, while the transient self-cycling sets are usually associatedwith the unstable fixed points and periodic solutions. Digraphs are introduced into generalizedcell mapping systems by defining binary relations in the cell state space, thus, the rich theoriesand the very powerful algorithms in the field of graphs and digraphs are adopted for the purposeof determing the global evolution properties of the systems. After all the self-cycling sets arecondensed by using digraph condensation method, the number of the state cells involved can beefficiently decreased in the global transient analysis, and a topological sorting of the global transientstate cells can be efficiently achieved by digraph algorithms, simultaneously, after transient cellsare classified to transient cell sets according to the number of the domiciles that they have, domainsof attraction and boundary regions can also be determined. Based on the different treatments,the global properties can be divided into qualitative (topological) and quantitative properties. Inthe whole analysis of the qualitative properties, only Boolean operations are used. The Booleanoperations are absolutely accurate, reliable, and time-saving. It is believed that the generalized cellmapping digraph method offers us a new way to examine the complicated behavior of nonlineardynamical systems.

Abstract:
Biological experiments of mammalian brain have shown that real neural systems exhibit a range of phenomena such as oscillations, phase-locking and even chaos. The chaotic behaviors simulate the information processing mechanisms of the real neural systems at a higher level. In this paper the bifurcation and chaos of the high order correlation networks will be studied.In some previous discussions about the high order correlation neural networks, we learned that the high order correlation networks expected to ...

Abstract:
A chaotic synchronized system of two coupled skew tent maps is discussed in this paper. The locally and globally riddled basins of the chaotic synchronized attractor are studied. It is found that there is a novel phenomenon in the local－global riddling bifurcation of the attractive basin of the chaotic synchronized attractor in some specific coupling intervals. The coupling parameter corresponding to the locally riddled basin has a single value which is embedded in the coupling parameter interval corresponding to the globally riddled basin, just like a breakpoint. Also, there is no relation between this phenomenon and the form of the chaotic synchronized attractor. This phenomenon is found analytically. We also try to explain it in a physical sense. It may be that the chaotic synchronized attractor is in the critical state, as it is infinitely close to the boundary of its attractive basin. We conjecture that this isolated critical value phenomenon will be common in a system with a chaotic attractor in the critical state, in spite of the system being discrete or differential.

Abstract:
A study of Hodgkin－Huxley (HH) neuron under external sinusoidal excited stimulus is presented in this paper. As is well known, the stimulus frequency is to be considered as a bifurcate parameter, and numerous phenomena, such as synchronization, period, and chaos appear alternatively with the changing of the stimulus frequency. For the stimulus frequency less than 2f_B (f_B being the base frequency in this paper), the simulation results demonstrate that the single HH neuron could completely convey the sinusoidal signal in anti-phase into interspike interval (ISI) sequences. We also report, perhaps for the first time, another kind of phenomenon, the beat phenomenon, which exists in the phase dynamics of the ISI sequences of the HH neuron stimulated by a sinusoidal current. It is shown furthermore that intermittent transition results in the general route to chaos.