Existence and roughness of nonuniform $(h,k,\mu ,\nu )$-trichotomy for nonautonomous differential equations

Abstract

The objective of this paper is to explore the existence and roughness of the nonuniform $(h,k,\mu ,\nu )$-trichotomy for nonautonomous differential equations. We first propose a more general notion of trichotomies called the nonuniform $(h,k,\mu ,\nu )$-trichotomy for linear nonautonomous differential equations. Then, we give a complete characterization of the notion of nonuniform $(h,k,\mu ,\nu )$-trichotomy for linear nonautonomous differential equations and prove that any linear nonautonomous differential equation admits a nonuniform $(h,k,\mu ,\nu )$-trichotomy if it has an $(H,K,L)$ Lyapunov exponent with different signs in a finite-dimensional space. Finally, we establish the roughness of nonuniform $(h,k,\mu ,\nu )$-trichotomies in a very concise manner, which implies that the nonuniform $(h,k,\mu ,\nu )$-trichotomy persists under sufficiently small linear perturbations. This study exhibits some new interesting findings in trichotomy that extend the corresponding results for uniform and nonuniform trichotomies.