Topological equivalence of nonhyperbolic nonautonomous systems

Abstract

Backes, Dragičević and Palmer proved in [1, J. Differential Equations 2021] a C0 linearization result for nonautonomous coupled systems, whose linear part is nonhyperbolic but the second subsystem in their equations remains unchanged. In this paper, we establish a topological equivalence sending the solutions of the semilinear evolution equationx′=A(t)x+f(t,x,y),y′=r(t,x,y), onto those of its nonhyperbolic linearized partx′=A(t)x,y′=0. We require that one of the linear subsystem x′=A(t)x admits a nonuniform strong exponential dichotomy, but the other is a critical case, i.e., its linear part is a null operator. Then the difficulty arises: how to implement the linearization of the second subsystem? We will overcome the difficulty by introducing an implicit function, which enable us to construct equivalent functions to achieve the topological conjugacy between nonhyperbolic systems.