Analysis of noise-induced transitions from regular to chaotic oscillations in the Chen system.

The stochastically perturbed Chen system is studied within the parameter region which permits both regular and chaotic oscillations. As noise intensity increases and passes some threshold value, noise-induced hopping between close portions of the stochastic cycle can be observed. Through these transitions, the stochastic cycle is deformed to be a stochastic attractor that looks like chaotic. In this paper for investigation of these transitions, a constructive method based on the stochastic sensitivity function technique with confidence ellipses is suggested and discussed in detail. Analyzing a mutual arrangement of these ellipses, we estimate the threshold noise intensity corresponding to chaotization of the stochastic attractor. Capabilities of this geometric method for detailed analysis of the noise-induced hopping which generates chaos are demonstrated on the stochastic Chen system.

[1]  K. Mallick,et al.  Noise-induced bifurcations, multiscaling and on–off intermittency , 2007, 0710.4066.

[2]  H. Haken,et al.  The influence of noise on the logistic model , 1981 .

[3]  L. Ryashko,et al.  Constructive analysis of noise-induced transitions for coexisting periodic attractors of the Lorenz model. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Stephen P. Ellner,et al.  When can noise induce chaos and why does it matter: a critique , 2005 .

[5]  L. Ryashko,et al.  NOISE-INDUCED BACKWARD BIFURCATIONS OF STOCHASTIC 3D-CYCLES , 2010 .

[6]  Guanrong Chen,et al.  On a Generalized Lorenz Canonical Form of Chaotic Systems , 2002, Int. J. Bifurc. Chaos.

[7]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[8]  Jianbo Gao,et al.  When Can Noise Induce Chaos , 1999 .

[9]  Alexander N Pisarchik,et al.  Control of on-off intermittency by slow parametric modulation. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Lefever,et al.  Sensitivity of a Hopf bifurcation to multiplicative colored noise. , 1986, Physical review letters.

[11]  Ying-Cheng Lai,et al.  Noise-induced unstable dimension variability and transition to chaos in random dynamical systems. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Badii,et al.  Generalized multistability and noise-induced jumps in a nonlinear dynamical system. , 1985, Physical review. A, General physics.

[13]  J. García-Ojalvo,et al.  Effects of noise in excitable systems , 2004 .

[14]  Wen-Wen Tung,et al.  Noise-induced Hopf-bifurcation-type sequence and transition to chaos in the lorenz equations. , 2002, Physical review letters.

[15]  B. Huberman,et al.  Fluctuations and simple chaotic dynamics , 1982 .

[16]  Irina Bashkirtseva,et al.  Analysis of excitability for the FitzHugh-Nagumo model via a stochastic sensitivity function technique. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Irina A. Bashkirtseva,et al.  Confidence tori in the analysis of stochastic 3D-cycles , 2009, Math. Comput. Simul..

[18]  G. Benettin,et al.  Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application , 1980 .

[19]  Ying-Cheng Lai,et al.  Quasipotential approach to critical scaling in noise-induced chaos. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Guanrong Chen,et al.  Analysis of Stochastic Cycles in the Chen System , 2010, Int. J. Bifurc. Chaos.

[21]  Charalampos Skokos,et al.  The Lyapunov Characteristic Exponents and Their Computation , 2008, 0811.0882.

[22]  Guanrong Chen,et al.  Bifurcation Analysis of Chen's equation , 2000, Int. J. Bifurc. Chaos.

[23]  G A Pavliotis,et al.  Noise induced state transitions, intermittency, and universality in the noisy Kuramoto-Sivashinksy equation. , 2010, Physical review letters.

[24]  Jing Hu,et al.  Diffusion, intermittency, and Noise-Sustained Metastable Chaos in the Lorenz Equations: Effects of Noise on Multistability , 2008, Int. J. Bifurc. Chaos.

[25]  Philippe Marcq,et al.  Stability analysis of a noise-induced Hopf bifurcation , 2003, cond-mat/0312360.

[26]  Peter V. E. McClintock,et al.  Changes in the dynamical behavior of nonlinear systems induced by noise , 2000 .

[27]  F. Gassmann,et al.  Noise-induced chaos-order transitions , 1997 .

[28]  Ulrike Feudel,et al.  Multistability, noise, and attractor hopping: the crucial role of chaotic saddles. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  J. Kurths,et al.  Coherence Resonance in a Noise-Driven Excitable System , 1997 .

[30]  Vadim S. Anishchenko,et al.  Effect of noise-induced crisis of attractor on characteristics of Poincaré recurrence , 2011 .

[31]  Grebogi,et al.  Scaling law for characteristic times of noise-induced crises. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[32]  Irina A. Bashkirtseva,et al.  Stochastic sensitivity of 3D-cycles , 2004, Math. Comput. Simul..

[33]  Eric Vanden-Eijnden,et al.  Noise-induced mixed-mode oscillations in a relaxation oscillator near the onset of a limit cycle. , 2008, Chaos.