Xiaole Yue

Global Bifurcation Analysis of a Duffing–Van der Pol Oscillator with Parametric Excitation

In this paper, a composite cell state space is constructed by a multistage division of the continuous phase space. Based on point mapping method, global properties of dynamical systems can be analyzed more accurately and efficiently, and any small regions can be refined for clearly depicting some special basin boundaries in this cell state space. Global bifurcation of a Duffing–Van der Pol oscillator subjected to harmonic parametrical excitation is investigated. Attractors, basins of attraction, basin boundaries, saddles and invariant manifolds have been obtained. As the amplitude of excitation increases, it can be observed that the boundary crisis occurs twice. Then a basin boundary with Wada property appears in the state space and undergoes metamorphosis in the chaotic boundary crisis. At last, two attractors merge into a chaotic one when they simultaneously collide with the chaotic saddle embedded in the fractal boundary. All these results show the effectiveness of the proposed method in global analysis.

ROLES OF CHAOTIC SADDLE AND BASIN OF ATTRACTION IN BIFURCATION AND CRISIS ANALYSIS

This paper is devoted to the dynamical behavior of a parametrically driven double-well Duffing (PDWD) system. Despite the invariant property of symmetry, this simple model exhibits a large diversit...

The response analysis of fractional-order stochastic system via generalized cell mapping method

This paper is concerned with the response of a fractional-order stochastic system. The short memory principle is introduced to ensure that the response of the system is a Markov process. The generalized cell mapping method is applied to display the global dynamics of the noise-free system, such as attractors, basins of attraction, basin boundary, saddle, and invariant manifolds. The stochastic generalized cell mapping method is employed to obtain the evolutionary process of probability density functions of the response. The fractional-order ϕ6 oscillator and the fractional-order smooth and discontinuous oscillator are taken as examples to give the implementations of our strategies. Studies have shown that the evolutionary direction of the probability density function of the fractional-order stochastic system is consistent with the unstable manifold. The effectiveness of the method is confirmed using Monte Carlo results.

Generation and Evolution of Chaos in Double-Well Duffing Oscillator under Parametrical Excitation

The generation and evolution of chaotic motion in double-well Duffing oscillator under harmonic parametrical excitation are investigated. Firstly, the complex dynamical behaviors are studied by applying multibifurcation diagram and Poincare sections. Secondly, by means of Melnikov’s approach, the threshold value of parameter for generation of chaotic behavior in Smale horseshoe sense is calculated. By the numerical simulation, it is obvious that as exceeds this threshold value, the behavior of Duffing oscillator is still steady-state periodic but the transient motion is chaotic; until the top Lyapunov exponent turns to positive, the motion of system turns to permanent chaos. Therefore, in order to gain an insight into the evolution of chaotic behavior after passing the threshold value, the transient motion, basin of attraction, and basin boundary are also investigated.

The stochastic dynamical behaviors of the gene regulatory circuit in Bacillus subtilis

In recent years, the gene regulatory circuit in biological systems has attracted a considerable interest. In this paper, we consider the dynamical behaviors of the gene regulatory circuit in Bacillus subtilis, including deterministic system and stochastic system effected by colored noise. First of all, the global dynamical behaviors of the deterministic system are exhibited by numerical method. Moreover, we give an effective method to explore the stochastic response and bifurcation by means of the stochastic generalized cell mapping method. To satisfy the Markov nature, we transform the colored-noise system into the equivalent white-noise system. And the stochastic generalized cell mapping method can be used to obtain the dynamical behaviors of the stochastic system. We found that the system parameters and noise can induce the occurrence of the stochastic P-bifurcation in the stochastic system, which means that the conversion between competent state and vegetative state in Bacillus subtilis is possible. In addition, the effectiveness of the stochastic generalized cell mapping method is verified by Monte Carlo simulation.In recent years, the gene regulatory circuit in biological systems has attracted a considerable interest. In this paper, we consider the dynamical behaviors of the gene regulatory circuit in Bacillus subtilis, including deterministic system and stochastic system effected by colored noise. First of all, the global dynamical behaviors of the deterministic system are exhibited by numerical method. Moreover, we give an effective method to explore the stochastic response and bifurcation by means of the stochastic generalized cell mapping method. To satisfy the Markov nature, we transform the colored-noise system into the equivalent white-noise system. And the stochastic generalized cell mapping method can be used to obtain the dynamical behaviors of the stochastic system. We found that the system parameters and noise can induce the occurrence of the stochastic P-bifurcation in the stochastic system, which means that the conversion between competent state and vegetative state in Bacillus subtilis is possible. I...

Suppression of pattern complexity by discrete-space state feedback with equal intervals

The suppression of the pattern complexity of spatiotemporal chaos in Coupled Acousto-optic Bistable Map Lattice system (CABMLs) is investigated. The pattern complexity of spatiotemporal dynamics is exhibited quantitatively by employing the average Kolmogorov–Sinai (KS) entropy density according to the Lyapunov exponents of lattices. By designing a global control method with periodic state feedback, we provide the stability analysis for suppressing spatiotemporal chaos to a stable state. Then, a pattern control strategy using discrete-space state feedback with equal intervals is proposed. Numerical simulations demonstrate that the pattern complexity of spatiotemporal chaos is decreased to a low level by the proposed method. Finally, the dependence of the control effect on the control interval is discussed in detail by considering the control efficiency and control cost and the optimal control interval is obtained.