The impact of colored noise on the equilibria of nonlinear dynamic systems

I. A. Bashkirtseva

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 28 (2018) no. 2, pp. 133-142

Résumé

The influence of colored noise on the equilibrium regimes of nonlinear dynamical systems is investigated. To study the response of the system to small perturbations, we use an asymptotic approach that develops the stochastic sensitivity function technique. The stochastic sensitivity of equilibrium in a general multidimensional dynamical system is defined by some matrix. For this stochastic sensitivity matrix, we obtain a matrix algebraic equation. An exact solution of this equation is given for an important class of nonlinear oscillators with perturbations in the form of colored noises. This theory is applied to the parametric study of the response of the electronic generator with hard excitation to colored noises with various correlation times. The dependence of the dispersion of random states on the characteristic correlation time is investigated. It is shown that this dependence can be nonmonotonic and have maxima corresponding to the resonances. The paper discusses the probabilistic mechanism of the stochastic generation of large-amplitude oscillations caused by color noise.

Détail

Keywords: colored noise, correlation time, stochastic sensitivity, electronic generator, stochastic excitability.

Bibliographie

[1] Hänggi P., Jung P., Colored noise in dynamical systems, Advances in Chemical Physics, 89, eds. I. Prigogine, S.A. Rice, John Wiley Sons, Inc., Hoboken, NJ, USA | DOI

[2] Anishchenko V. S., Astakhov V. V., Vadivasova T. E., Neiman A. B., Strelkova G. I., Shimanskiy–Geier L., Nonlinear effects in chaotic and stochastic systems, Institute of Computer Sciences, Izhevsk, 2003, 535 pp.

[3] Short R., Mandel L., Roy R., “Correlation functions of a dye laser: Comparison between theory and experiment”, Physical Review Letters, 49:9 (1982), 647–650 | DOI

[4] Marano S., Edwards B., Ferrari G., Faeh D., “Fitting earthquake spectra: colored noise and incomplete data”, Bulletin of the Seismological Society of America, 107:1 (2017), 276–291 | DOI

[5] Sarkar P., “The linear response of a glycolytic oscillator, driven by a multiplicative colored noise”, Journal of Statistical Mechanics: Theory and Experiment, 2016:12 (2016), 123202 | DOI | MR

[6] Spanio T., Hidalgo J., Muñoz M. A., “Impact of environmental colored noise in single-species population dynamics”, Physical Review E, 96:4 (2017), 042301 | DOI

[7] Dong H., He L., Lu H., Li J., “A microbial growth kinetics model driven by hybrid stochastic colored noises in the water environment”, Stochastic Environmental Research and Risk Assessment, 31:8 (2016), 2047–2056 | DOI

[8] Guo Q., Sun Z., Xu W., “The properties of the anti-tumor model with coupling non-Gaussian noise and Gaussian colored noise”, Physica A: Statistical Mechanics and its Applications, 449 (2016), 43–52 | DOI | MR

[9] H'alisz L., Jung P., Hänggi P., Talkner P., Schimansky–Geier L., “Colored noise driven systems with inertia”, Zeitschrift für Physik B Condensed Matter, 77:3 (1989), 471–483 | DOI

[10] Anishchenko V. S., Neiman A. B., “Dynamic chaos and colored noise”, Pis'ma v Zhurnal Tekhnicheskoi Fiziki, 16:7 (1990), 21–25 (in Russian) | Zbl

[11] Jung P., Neiman A., Afghan M. K.N., Nadkarni S., Ullah G., “Thermal activation by power-limited coloured noise”, New Journal of Physics, 7 (2005), 17 | DOI

[12] Zakharova A. S., Vadivasova T. E., Anishchenko V. S., “Influence of noise on chaotic self-sustained oscillations in the regime of spiral attractor”, Izvestiya vuzov. Prikladnaya nelineinaya dinamika, 14:5 (2006), 44–61 (in Russian)

[13] Hänggi P., Jung P., Zerbe C., Moss F., Can colored noise improve stochastic resonance?, Journal of Statistical Physics, 70:1–2 (1993), 25–47 | DOI | MR | Zbl

[14] Xu Y., Gu R., Zhang H., Xu W., Duan J., “Stochastic bifurcations in a bistable Duffing–Van der Pol oscillator with colored noise”, Physical Review E, 83:5 (2011), 056215 | DOI | MR

[15] Lei Y., Hua M., Du L., “Onset of colored-noise-induced chaos in the generalized Duffing system”, Nonlinear Dynamics, 89:2 (2017), 1371–1383 | DOI | MR

[16] Venttsel' A.D., Freidlin M. I., Fluctuations in dynamic systems under the influence of small random disturbances, Nauka, M., 1979, 424 pp.

[17] Bashkirtseva I., Ryashko L., “Analysis of excitability for the FitzHugh–Nagumo model via a stochastic sensitivity function technique”, Physical Review E, 83:6 (2011), 061109 | DOI | MR

[18] Bashkirtseva I., Ryashko L., “Stochastic sensitivity of regular and multi-band chaotic attractors in discrete systems with parametric noise”, Physics Letters A, 381:37 (2017), 3203–3210 | DOI | MR | Zbl

[19] Ryashko L., Slepukhina E., “Noise-induced torus bursting in the stochastic Hindmarsh-Rose neuron model”, Physical Review E, 96:3 (2017), 032212 | DOI | MR

[20] Bashkirtseva I., Ryashko L., “Stochastic sensitivity and variability of glycolytic oscillations in the randomly forced Sel'kov model”, The European Physical Journal B, 90:1 (2017) | DOI | MR | Zbl

[21] Bashkirtseva I., Ryashko L., “Noise-induced shifts in the population model with a weak Allee effect”, Physica A: Statistical Mechanics and its Applications, 491 (2018), 28–36 | DOI | MR

[22] Andronov A. A., Vitt A. A., Khaikin S. E., Theory of oscillations, Nauka, M., 1981, 568 pp.