Confidence tori in the analysis of stochastic 3D-cycles

We present a new computer approach to the spatial analysis of stochastically forced 3D-cycles in nonlinear dynamic systems. This approach is based on a stochastic sensitivity analysis and uses the construction of confidence tori. A confidence torus as a simple 3D-model of the stochastic cycle adequately describes its main probabilistic features. We suggest an effective algorithm for construction of the confidence tori using a discrete set of confidence ellipses. The ability of these tori to visualize thin effects observed for the period-doubling bifurcations zone in the stochastic Roessler model are shown. For this zone, the geometrical growth of stochastic sensitivity of the forced cycles under transition to chaos is presented.

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

[2]  R. Val Ery Roy.  Asymptotic Analysis of First-passage Problems , 1994 .

[3]  H. Leung,et al.  STOCHASTIC HOPF BIFURCATION IN A BIASED VAN DER POL MODEL , 1998 .

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

[5]  R. Lefever,et al.  Sensitivity of a Hopf Bifurcation to External Multiplicative Noise , 1984 .

[6]  F. Baras,et al.  Stochastic analysis of limit cycle behavior , 1997 .

[7]  C. Pearce,et al.  Stochastic Resonance: From Suprathreshold Stochastic Resonance to Stochastic Signal Quantization , 2008 .

[8]  K. Schenk-Hoppé,et al.  Bifurcation scenarios of the noisy duffing-van der pol oscillator , 1996 .

[9]  James P. Crutchfield,et al.  Scaling for External Noise at the Onset of Chaos , 1981 .

[10]  Lev B. Ryashko,et al.  The Analysis of the Stochastically Forced Periodic attractors for Chua's Circuit , 2004, Int. J. Bifurc. Chaos.

[11]  Stochastic analysis of subcritical amplification of magnetic energy in a turbulent dynamo , 2004, astro-ph/0402006.

[12]  R. Costantino,et al.  Can noise induce chaos , 2003 .

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

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

[15]  L. Pontryagin,et al.  Noise in nonlinear dynamical systems: Appendix: On the statistical treatment of dynamical systems , 1989 .

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

[17]  N. Berglund,et al.  Geometric singular perturbation theory for stochastic differential equations , 2002 .

[18]  M. Mangel.  Small Fluctuations in Systems with Multiple Limit Cycles , 1980 .

[19]  Annette Zippelius,et al.  The effect of external noise in the Lorenz model of the Bénard problem , 1981 .

[20]  M.W.Kalinowski.  Nonlinear Waves , 2016, 1611.10114.

[21]  James P. Crutchfield,et al.  Fluctuations and the onset of chaos , 1980 .

[22]  Irina Bashkirtseva,et al.  Stochastic analysis of a non-normal dynamical system mimicking a laminar-to-turbulent subcritical transition. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  O. Rössler.  An equation for continuous chaos , 1976 .

[24]  Moss,et al.  Postponement of Hopf bifurcations by multiplicative colored noise. , 1987, Physical review. A, General physics.

[25]  Klaus Schulten,et al.  Effect of noise and perturbations on limit cycle systems , 1991 .

[26]  N. Namachchivaya.  Hopf bifurcation in the presence of both parametric and external stochastic excitations , 1988 .

[27]  Bernard J. Matkowsky,et al.  A direct approach to the exit problem , 1990 .

[28]  Earl H. Dowell,et al.  Parametric Random Vibration , 1985 .

[29]  Vadim N. Smelyanskiy,et al.  Topological features of large fluctuations to the interior of a limit cycle , 1997 .

[30]  James A. Yorke,et al.  A scaling law: How an attractor's volume depends on noise level , 1985 .

[31]  Gabriele Bleckert,et al.  The Stochastic Brusselator: Parametric Noise Destroys Hoft Bifurcation , 1999 .

[32]  I. Tsuda,et al.  Noise-induced order , 1983 .

[33]  R. Lefever,et al.  Noise in nonlinear dynamical systems: Noise-induced transitions , 1989 .

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

[35]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[36]  G. Rowlands,et al.  Nonlinear Waves, Solitons and Chaos , 1990 .

[37]  Gregoire Nicolis,et al.  Stochastic resonance , 2007, Scholarpedia.

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

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

[40]  Irina Bashkirtseva,et al.  Sensitivity and chaos control for the forced nonlinear oscillations , 2005 .

[41]  N. Berglund,et al.  Noise-Induced Phenomena in Slow-Fast Dynamical Systems: A Sample-Paths Approach , 2005 .

[42]  Irina Bashkirtseva,et al.  Sensitivity analysis of the stochastically and periodically forced Brusselator , 2000 .

[43]  R. Fox,et al.  On the amplification of molecular fluctuations for nonstationary systems: hydrodynamic fluctuations for the Lorenz model , 1993 .

[44]  T. T. Soong,et al.  Random Vibration of Mechanical and Structural Systems , 1992 .

[45]  R. G. Medhurst,et al.  Topics in the Theory of Random Noise , 1969 .