On the preservation of Lyapunov exponents of integrally separated systems of differential equations under small nonlinear perturbations

Abstract

This paper addresses the Lyapunov exponents of non-vanishing solutions to quasi-linear time-varying systems of differential equations. The linear part is not required to be regular but it is assumed to be integrally separated, which ensures that the associated Lyapunov exponents are distinct and stable. The nonlinear perturbations are assumed to be small in a certain sense, though less restrictive than the condition in Barreira and Valls’ paper, J. Differential Equations 258(2015), 339–361. The main result is a Perron-type theorem for upper and lower Lyapunov exponents, offering an alternative to Barreira and Valls’ result. In addition, an analogous result holds for Bohl exponents.

Keywords

• quasi-linear system
• lyapunov exponent
• integral separation
• nonlinear perturbation
• perron-type theorem