Small perturbations may change the sign of Lyapunov exponents for linear SDEs

TL;DR

This paper investigates how small, exponentially decaying perturbations can alter the sign of Lyapunov exponents in linear stochastic differential equations, revealing that stability properties are sensitive to such perturbations.

Contribution

It demonstrates that perturbations can change the sign of Lyapunov exponents in linear SDEs, providing new insights into their stability behavior under small disturbances.

Findings

Positive Lyapunov exponents can become negative under perturbations.

Negative Lyapunov exponents can turn positive with perturbations.

Different perturbation positions can change the sign of Lyapunov exponents.

Abstract

In this paper, we study the existence of $n$-dimensional linear stochastic differential equations (SDEs) such that the sign of Lyapunov exponents is changed under an exponentially decaying perturbation. First, we show that the equation with all positive Lyapunov exponents will have $n-1$ linearly independent solutions with negative Lyapunov exponents under the perturbation. Meanwhile, we prove that the equation with all negative Lyapunov exponents will also have solutions with positive Lyapunov exponents under another similar perturbation. Finally, we also show that other three kinds of perturbations which appear at different positions of the equation will change the sign of Lyapunov exponents.