Linear properties of chaotic states of systems described by equations of nonlinear dynamics: Analogy with quantum theory

Authors

DOI:

https://doi.org/10.33910/2687-153X-2020-1-4-150-157

Keywords:

nonlinear dynamics, strange attractor, probability density, chaos, perturbation theory

Abstract

The paper provides a theoretical exploration of properties of systems described by equations of nonlinear dynamics in a chaotic state. Using the example of a system described by Duffing equations, it is shown that when the state of the system corresponds to a chaotic (strange) attractor, it is possible to determine a function whose meaning corresponds to the probability density. In this case, the resulting equation for the probability density is linear, so that the solution methods developed for linear differential equations, in particular the method of perturbation theory, can be applied to solve the equation in question. This results in a linear dependence of the average values of physical quantities on the parameter that characterizes small perturbations of the system. The numerical experiment confirms this linear relationship.

References

Gonchenko, A. S., Gonchenko, S. V., Kazakov, A. O., Kozlov, A. D. (2017) Matematicheskaya teoriya dinamicheskogo khaosa i ee prilozheniya: Obzor. Chast’ 1. Psevdogiperbolicheskie attraktory [Mathematical theory of dynamical chaos and its applications: Review. Part 1. Pseudohyperbolic attractors. Pseudohyperbolic attractors]. Izvestiya Vysshikh uchebnykh zavedeniy. Prikladnaya nelineynaya dinamika — Izvestiya VUZ. Applied Nonlinear Dynamics, 25 (2), 4–36. (In Russian)

Kamke, E. (1971) Differentialgleichungen: Lösungsmethoden und Lösungen. Vol. 2. Leipzig: Akademische Verlagsgesellschaft, 243 p. (In German)

Kiselev, A. A., Lyapcev, A. V. (1989) Kvantovomekhanicheskaya teoriya vozmushchenij. Diagrammnyj metod [Quantum mechanical perturbation theory. Diagram method]. Leningrad: Leningrad State University Publ., 360 p. (In Russian)

Kondrat’ev, A. S., Lyaptsev, A. V. (2008) Fizika. Zadachi na komp’yutere [Physics. Tasks on the computer]. Moscow: Fizmatlit Publ., 400 p. (In Russian)

Liaptsev, A. V. (2013) Simmetriya regulyarnykh i khaoticheskikh dvizhenij v zadachakh nelinejnoj dinamiki. Uravnenie Duffinga [The symmetry of regular and chaotic motions in nonlinear dynamics problems. Duffing Equation]. Izvestia Rossijskogo gosudarstvennogo pedagogicheskogo universiteta im. A. I. Gertsena — Izvestia: Herzen University Journal of Humanities & Sciences, 157, 24–34. (In Russian)

Liaptsev, A. V. (2014a) Simmetriya regulyarnykh i khaoticheskikh dvizhenij v zadachakh nelinejnoj dinamiki. Rotator v periodicheskom pole [Symmetry of regular and chaotic motions in nonlinear dynamic problems. Rotator in periodic field]. Izvestia Rossijskogo gosudarstvennogo pedagogicheskogo universiteta im. A. I. Gertsena — Izvestia: Herzen University Journal of Humanities & Sciences, 165, 23–35. (In Russian)

Liaptsev, A. V. (2014b) Simmetriya v zadachakh nelinejnoj dinamiki. Proyavlenie svojstv simmetrii v polyarizatsii izlucheniya [Symmetry in problems of nonlinear dynamics. The manifestation of the properties of the symmetry in the polarization of radiation]. Izvestia Rossijskogo gosudarstvennogo pedagogicheskogo universiteta im. A. I. Gertsena — Izvestia: Herzen University Journal of Humanities & Sciences, 168, 16–28. (In Russian)

Liaptsev, A. V. (2015) Proyavlenie svojstv simmetrii v zadachakh nelinejnoj dinamiki. Effekty, analogichnye vyrozhdeniyu v kvantovomekhanicheskikh zadachakh [Manifestation of symmetry properties in problems of nonlinear dynamics. Effects analogous to degeneration in quantum mecanical problems]. Izvestia Rossijskogo gosudarstvennogo pedagogicheskogo universiteta im. A. I. Gertsena — Izvestia: Herzen University Journal of Humanities & Sciences, 173, 64–77. (In Russian)

Liapzev, A. V. (2019) The calculation of the probability density in phase space of a chaotic system on the example of rotator in the harmonic field. Computer Assisted Mathematics, 1, 55–65. (In English)

Loskutov, A. Yu. (2007) Dynamical chaos: Systems of classical mechanics. Physics-Uspekhi, 50 (9), 939–964. DOI: 10.1070/PU2007v050n09ABEH006341 (In English)

Sagdeev, R. Z., Usikov, D. A, Zaslavskii, G. M. (1988) Nonlinear physics: From the pendulum to turbulence and chaos. 2nd ed. Chur: Harwood Academic Publ., 675 p. (In English)

Published

2020-12-24

Issue

Section

Theoretical Physics