Unpredictable solutions of Duffing type equations with Markov coefficients

TL;DR

This paper investigates the existence, uniqueness, and stability of unpredictable solutions in stochastic Duffing equations with Markov chain-driven coefficients, supported by theoretical proofs and numerical examples.

Contribution

It introduces the concept of unpredictability in solutions of stochastic Duffing equations with Markov coefficients and proves their existence, uniqueness, and stability.

Findings

Unpredictable solutions exist for the considered equations.

Solutions are unique and exponentially stable.

Numerical examples support theoretical results.

Abstract

The paper considers a stochastic differential equation of Duffing type with Markov coefficients. The existence of unpredictable solutions is considered. The unpredictability is a property of bounded functions characterized by unbounded sequences of moments of divergence and convergence in Bebutov dynamics. Markov components of the equation coefficients admit the unpredictability property. The components of the equation coefficients are derived from a Markov chain. The existence, uniqueness and exponential stability of an unpredictable solution are proved. The sequences of divergence and convergence of the coefficients and the solution are synchronized. Numerical example that support the theoretical results are provided.