Diffusion, intermittency, and Noise-Sustained Metastable Chaos in the Lorenz Equations: Effects of Noise on Multistability

Multistability is an interesting phenomenon of nonlinear dynamical systems. To gain insights into the effects of noise on multistability, we consider the parameter region of the Lorenz equations that admits two stable fixed point attractors, two unstable periodic solutions, and a metastable chaotic "attractor". Depending on the values of the parameters, we observe and characterize three interesting dynamical behaviors: (i) noise induces oscillatory motions with a well-defined period, a phenomenon similar to stochastic resonance but without a weak periodic forcing; (ii) noise annihilates the two stable fixed point solutions, leaving the originally transient metastable chaos the only observable; and (iii) noise induces hopping between one of the fixed point solutions and the metastable chaos, a three-state intermittency phenomenon.

[1]  Jianbo Gao,et al.  Pathological tremors as diffusional processes , 2002, Biological Cybernetics.

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

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

[4]  Gao,et al.  Direct dynamical test for deterministic chaos and optimal embedding of a chaotic time series. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[5]  Jianbo Gao,et al.  Direct Dynamical Test for Deterministic Chaos , 1994 .

[6]  Huw G. Davies,et al.  A period–doubling bifurcation with slow parametric variation and additive noise , 2001, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[7]  V. Araújo.  Random Dynamical Systems , 2006, math/0608162.

[8]  Kurths,et al.  Period-doubling bifurcations in the presence of colored noise. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  Frank Moss,et al.  Noise in human muscle spindles , 1996, Nature.

[10]  Jianbo Gao,et al.  Recognizing randomness in a time series , 1997 .

[11]  Binoy Krishna Goswami.  Self-similarity in the bifurcation structure involving period tripling, and a suggested generalization to period n-tupling , 1998 .

[12]  A N Pisarchik,et al.  Experimental demonstration of attractor annihilation in a multistable fiber laser. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Alexander N. Pisarchik,et al.  Using periodic modulation to control coexisting attractors induced by delayed feedback , 2003 .

[14]  Jianbo Gao,et al.  NOISE-INDUCED CHAOS , 1999 .

[15]  Richard L. Kautz,et al.  Chaos and thermal noise in the rf‐biased Josephson junction , 1985 .

[16]  Kung Yao,et al.  Perceptual dominance time distributions in multistable visual perception , 2004, Biological Cybernetics.

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

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

[19]  Geetha Basappa,et al.  Observation of chaotic dynamics in dilute sheared aqueous solutions of CTAT. , 2000, Physical review letters.

[20]  James P. Crutchfield,et al.  Geometry from a Time Series , 1980 .

[21]  Karl Heinz Hoffmann,et al.  The Hopf bifurcation of twodimensional systems under the influence of one external noise source , 1982 .

[22]  A N Pisarchik,et al.  Experimental observation of two-state on-off intermittency. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  A. K. Sood,et al.  Chaotic dynamics in shear-thickening surfactant solutions , 2001 .

[24]  Steven H. Strogatz,et al.  Nonlinear Dynamics and Chaos , 2024 .

[25]  Arecchi,et al.  Dynamic behavior and onset of low-dimensional chaos in a modulated homogeneously broadened single-mode laser: Experiments and theory. , 1986, Physical review. A, General physics.

[26]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

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

[28]  J. Hasty,et al.  Noise-based switches and amplifiers for gene expression. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[29]  Murthy,et al.  Evidence for chaos in an experimental time series from serrated plastic flow. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

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

[31]  F. Kneubühl,et al.  First experimental realization of a distributed feedback gas laser , 1979 .

[32]  Tom Mullin,et al.  The effect of noise on pitchfork and Hopf bifurcations , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[33]  Riccardo Mannella,et al.  Integration Of Stochastic Differential Equations On A Computer , 2002 .

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

[35]  Jianbo Gao,et al.  Multiscale Analysis of Complex Time Series , 2007 .

[36]  B. Kendall.  Nonlinear Dynamics and Chaos , 2001 .

[37]  J B Gao,et al.  Distinguishing chaos from noise by scale-dependent Lyapunov exponent. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[39]  Tomasz Kapitaniak,et al.  Controlling chaos by chaos in geophysical systems , 1995 .

[40]  Gao,et al.  Noise-induced chaos in an optically injected semiconductor laser model , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[41]  R Jaimes-Reátegui,et al.  Synchronization of chaotic systems with coexisting attractors. , 2006, Physical review letters.

[42]  F. Takens.  Detecting strange attractors in turbulence , 1981 .

[43]  Glorieux,et al.  Chaos in a CO2 laser with modulated parameters: Experiments and numerical simulations. , 1987, Physical review. A, General physics.

[44]  Jianbo Gao,et al.  Multiscale Analysis of Complex Time Series: Integration of Chaos and Random Fractal Theory, and Beyond , 2007 .

[45]  Jianbo Gao,et al.  Inertia and memory in ambiguous visual perception , 2006, Cognitive Processing.

[46]  Jianbo Gao,et al.  Power-law sensitivity to initial conditions in a time series with applications to epileptic seizure detection , 2005 .

[47]  R. Holzner,et al.  Observation of order and chaos in a nuclear spin–flip laser , 1985 .

[48]  Michael C. Mackey,et al.  Solution multistability in first-order nonlinear differential delay equations. , 1993, Chaos.

[49]  Jianbo Gao,et al.  Local exponential divergence plot and optimal embedding of a chaotic time-series , 1993 .

[50]  Foss,et al.  Multistability and delayed recurrent loops. , 1996, Physical review letters.

[51]  Goswami,et al.  Annihilation of one of the coexisting attractors in a bistable system , 2000, Physical review letters.

[52]  Massimo Riani,et al.  Visual Perception of Stochastic Resonance , 1997 .

[53]  Hernán G. Solari,et al.  Influence of coexisting attractors on the dynamics of a laser system , 1987 .

[54]  Celso Grebogi,et al.  Multistability, Basin Boundary Structure, and Chaotic Behavior in a Suspension Bridge Model , 2004, Int. J. Bifurc. Chaos.

[55]  Alexander N. Pisarchik,et al.  Efficiency of the control of coexisting attractors by harmonic modulation applied in different ways , 2005 .

[56]  Alexander N. Pisarchik,et al.  Control of multistability in a directly modulated diode laser , 2002 .

[57]  J. Maurer,et al.  Effect of the Prandtl number on the onset of turbulence in liquid 4He , 1980 .

[58]  V. Roychowdhury,et al.  Assessment of long-range correlation in time series: how to avoid pitfalls. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.