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

Keywords

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