Topological conjugacy between two kinds of nonlinear differential equations via generalized exponential dichotomy (Q762997)

For the differential equation \[ \dot x= A(t)x \] in \(\mathbb{R}^n\), the authors consider a generalized exponential dichotomy and the nonlinear equations \[ \dot x= A(t)x+ f(t,x),\tag{1} \] \[ \dot x= A(t)x+ g(t,x),\tag{2} \] \(x\in \mathbb{R}^n,~\;A(t),\;B(t)\text{ are }n\times n\text{ matrices}\). The equations (1), (2) are called topological equivalent if there is a homeomorphism between the solutions of (1) and (2).