Jian-e Xu

The global bifurcation characteristics of the forced van der Pol oscillator

Abstract In this paper, the bifurcation characteristics of the forced van der Pol oscillator on a specific parameter plane, including intermediate parameter regions, are investigated. The successive arrangement of the dominant mode-locking regions, where a single subharmonic solution with the rotation number, 1 (2k + 1) , exists, and the transitional zones between them are depicted. The transitional zones are explicitly proposed to be classified into two groups according to the different global characters: (1) the simple transitional zones, in which coexistence of two mode-locked solutions with rotation numbers 1 (2k ± 1) appear; (2) the complex transitional zones, in which the sub-zones with the mode-locked solutions, whose rotation numbers are rational fractions between 1 (2k + 1) and 1 (2k − 1) , and the quasi-periodic solutions exist. The emphasis of this paper is to study the evolution of the global structures in the transitional zones. A complex transitional zone generally evolves from a Farey tree, when the forcing amplitude is small, to a chaotic regime, when forcing amplitude is sufficiently large. It is of great interest that the sub-zone with a rotation number, 1 2k , which has the largest width within a complex transitional zone, can usually intrude into the dominant regions of 1 (2k − 1) before it completely vanishes. Moreover, the features of overlaps of mode-locking sub-zones and the number of coexistence of different attractors are also discussed.

Resonance in a noise-driven excitable neuron model

Abstract The effects of variations in bifurcation parameter on coherence resonance in the noisy FitzHugh–Nagumo (FHN) neuron model are studied. We find that the coherence resonance effect depends monotonically on the firing bifurcation parameter, and this result is interpreted analytically. Then an external control method is presented to modulate coherence resonance in the excitable neuron model, and our scheme is based on the result that a weak periodic perturbation can be used to change the critical firing onset value. This method can be used to either enhance the effect of coherence resonance or delay the occurrence of coherence resonance, and the application of the method to another neuron model is also discussed.

Relation between responsiveness to neurotransmitters and complexity of epileptiform activity in rat hippocampal CA1 neurons

Summary:  Purpose: Our previous works suggested that sensitivity of neurons with chaotic firing patterns to stimuli is significantly greater than that in neurons with periodic firing patterns, which shows that responsiveness of neurons may depend on the complexity of the firing series. This study was performed to determine the relation between responsiveness of the hippocampal CA1 neurons with epileptiform activity (EA) to neurotransmitters and their complexity of firing series.

A chaotic crisis between chaotic saddle and attractor in forced Duffing oscillators

Abstract We investigate crises in forced Duffing oscillators by the generalized cell mapping digraph method to efficiently complete the global analysis of non-linear systems, which includes global transient analysis through digraphic algorithms based on a strictly theoretical correspondence between generalized cell mappings and digraphs. A process of generalized cell mapping method is developed to refine persistent and transient self-cycling sets. The refining procedures of persistent and transient self-cycling sets are respectively given on the basis of their definitions in the cell state space. A chaotic boundary crisis and a chaotic interior crisis are discovered. A chaotic boundary crisis owing to a collision between chaotic attractor and saddle occurs in its basin boundary possessing a fractal structure. In such a case the chaotic attractor together with its basin of attraction is suddenly destroyed as the parameter passes through the critical value, and the chaotic saddle also undergoes an abrupt enlargement in its size. Namely, the chaotic attractor is converted into an incremental portion of the chaotic saddle after the collision. For a chaotic interior crisis, there is a sudden increase in the size of a chaotic attractor as the parameter passes through the critical value. For the chaotic interior crisis, it is demonstrated that the chaotic attractor collides with a chaotic saddle in its basin interior when the crisis occurs. This chaotic saddle is an invariant and non-attracting set. The origin and evolution of the chaotic saddle are investigated as well.

CHAOTIC INTERSPIKE INTERVALS WITH MULTIPEAKED HISTOGRAM IN NEURONS

In our paper, we report the findings that the interspike intervals of the injured dorsal root ganglion neurons have multipeaked histograms and the interspike intervals are chaotic. First, the symbolic analysis method presented in the paper and the nonlinear forecasting method are used to study the dynamic differences between the interspike intervals with a multipeaked histogram generated by chaos and those induced by noise in some neuron models. Then based on the dynamic differences, the interspike intervals recorded in our experiment are analyzed by the nonlinear forecasting and the symbolic analysis method. We obtain that the dynamics of our experimental interspike intervals are greatly different from those of the interspike intervals with a multipeaked histogram induced by noise, while they are similar to the dynamics of this kind of interspike intervals generated by chaos. The results show that the experimental interspike intervals with a multipeaked histogram are chaotic.

Stochastic Synchronization and Aperiodic Stochastic Resonance of a Unidirectionally Coupled Single-mode Optical System

The variational method is applied to a unidirectionally coupled single-mode optical system for the general characteristics of aperiodic stochastic resonance (ASR) and stochastic synchronization in the linear response background. It is shown that ASR in the low-frequency domain exists in all stable cases by means of coherence function, but the stochastic synchronization in the weak noise level exists only when the drive subsystem is bistable or multistable. However, the curve of correlation coefficient vs. the noise intensity exhibits ASR-like behavior only when the drive subsystem is monostable. The relaxation property of the drive subsystem is found to be responsible for these different characteristics.

A simple method for the computation of the conditional Lyapunov exponents

Abstract An handy method of calculating the conditional Lyapunov exponents is put forward. Lyapunov exponents of differential dynamical systems and the conditional Lyapunov exponents can be acquired easily with the method. The method has been successfully used in kinds of synchronization, such as continuous driving synchronization, impulsive (sporadic) driving synchronization, intermittently driving synchronization. The conditional Lyapunov exponents obtained with our method can give the largest and the best time interval for impulsive synchronization that can hardly be settled in other ways.

Synchronization in Two Uncoupled Chaotic Neurons

Using the membrane potential of a chaotic neuron as stimulation signal to synchronize two uncoupled Hindmarsh-Rose (HR) neurons under different initial conditions is discussed. Modulating the corresponding parameters of two uncoupled identical HR neurons, full synchronization is realized when the largest condition lyapunov exponent (LCLE) becomes negative at the threshold of stimulation strength. Computing the interspiks interval (ISI) sequence shows synchronized chaotic response of modulated neurons is different from stimulation signal. Modulating input currents of two uncoupled HR neurons with parameters mismatch, phase synchronization is obtained when the LCLEs of two systems change to both negative, and synchronized response of two systems is in phase synchronization with stimulation signal.

A special bifurcation of riddled basin in two coupled skew tent maps

Abstract A chaotic synchronized system of coupled skew tent maps is discussed. The locally and globally riddled basins of the chaotic synchronized attractor are studied. It is found that the coupling parameter values corresponding to the locally riddled basin are isolated points embedded in the coupling parameter intervals of the globally riddled basin. This kind of bifurcation is novel and not like the local–global riddling bifurcation in found other references. We believe this bifurcation will be generic in systems, whose chaotic attractors are close to the attractive basins’ boundaries infinitely, despite discrete or differential systems.

Dynamical Characteristics of the Fractional-Order FitzHugh-Nagumo Model Neuron

Through the research on the fractional-order FitzHugh-Nagumo model, it is found that the Hopf bifurcation point in such a model, where the state of the model neuron changes from the quiescence into periodic spiking, is different from that of the corresponding integer-order model when the externally applied current is considered to be the bifurcation parameter. Moreover, we demonstrate that the range of periodic spiking of the fractional-order model neuron is clearly smaller than that of the corresponding integer-order model neuron, that is, the range of periodic spiking of the former is just embedded in that of the latter. In addition, we show that the firing frequency of the fractional-order model neuron is evidently larger than that of the integer-order counterpart. The Adomian decomposition method is used to calculate fractional-order differential equations numerically due to its rapid convergence and high accuracy.