Wave localized structures in relaxing media with fluctuations

1Danylenko, VA
1Skurativskyi, SI
1Department of Geodynamics explosion of S. I. Subbotin Institute of Geophysics of the NAS of Ukraine, Kyiv
Dopov. Nac. akad. nauk Ukr. 2014, 12:91-98
https://doi.org/10.15407/dopovidi2014.12.091
Section: Geosciences
Language: Ukrainian
Abstract: 

The article deals with the wave solutions of a mathematical model for relaxing media. When the fluctuations of the model parameters are absent, the wave solutions satisfy the nonlinear planar dynamical system, which is studied by means of qualitative analysis methods. The aim of the article is the incorporation of parameters with noise and investigations of the influence of fluctuations on the steady and periodic modes of the dynamical system. In particular, the direction of a displacement of the Andronov-Hopf bifurcation for the steady solutions is estimated with the help of the top Lyapunov exponent, which is derived analytically and numerically. Stochastic limit cycles are considered by means of the sensitivity function. This function is evaluated from a deterministic differential equation by the shooting method and characterizes the dispersion of trajectories in a vicinity of the deterministic limit cycle. It is shown that the trajectories of a stochastic cycle undergo the most dispersion near the saddle fixed point.

Keywords: fluctuations, relaxing media, wave localized structures
References: 

1. Sadovskiy M. A. Vest. AN USSR, 1986, No 8: 3–11 (in Russian).
2. Danylenko V. A., Danevych T. B., Makarenko O. S., Skurativskyi S. I., Vladimirov V. A. Self-organization in nonlocal non-equilibrium media,  Kyiv: Subbotin Inst. of Geophys. NAS of Ukraine, 2011.
3. Danylenko V. A., Sorokina V. V., Vladimirov V. A. J. Phys. A: Math. Gen.  1993, 26: 7125–7135. https://doi.org/10.1088/0305-4470/26/23/047
4. Vladimirov V. A., Kutafina E. V., Zorychta B. J. Phys. A: Math. Theor.,  2012, 45: 085210. https://doi.org/10.1088/1751-8113/45/8/085210
5. Advances in applied self-organizing systems (Ed. M. Prokopenko), London: Springer, 2008.
6. Kharchenko D., Kharchenko V., Lysenko I. Cent. Eur. J. Phys., 2011, 9, No 3: 698–709.
7. Lifshits E. M., Pitaevskiy L. P. Physical kinetics, Moscow: Nauka, 1979 (in Russian).
8. Arnold L. Random dynamical system., Series: Springer Monographs in Mathematics, New York; Berlin: Springer, 1998. https://doi.org/10.1007/978-3-662-12878-7
9. Khas’minskii R. Z. Toer. veroiatn. i ee primenenie, 1967, 12, Iss. 1: 167–172 (in Russian).
10. Kloeden P. E., Platen E., Schurz H. Numerical solution of SDE through computer experiments, Berlin: Springer, 2003.
11. Wedig W. Lyapunov exponents and invariant measures of equilibria and limit cycles. In: Lyapunov Exponents. Lect. Notes Math., No 1486, Berlin: Springer, 1991: P. 309–321. https://doi.org/10.1007/bfb0086678
12. Wedig W. Pitchfork and Hopf bifurcations in stochastic systems – effective methods to calculate Lyapunov exponents. In: Probabilistic methods in applied physics (Eds. P. Kree, W. Wedig). Lecture Notes in Physics, No 451, Berlin: Springer, 1995: 120–148.
13. Bashkirtseva I. A., Ryashko L. Math. and Comp. in Simulation., 2004, 66, No 1: 55–67. https://doi.org/10.1016/j.matcom.2004.02.021
14. Sidorets V. N., Pentegov I. V. Deterministic chaos in nonlinear circuits with electric arc, Kiev: Mezhdunar. assots. “Svarka”, 2013 (in Russian).
15. Kholodniok M., Klich A., Kybichek M., Marek M. Methods of analysis of nonlinear dynamic models, Moscow: Mir, 1991 (in Russian).