The behavior of solutions in the vicinity of a bounded solution to autonomous differential equations (Q1082515)

By using the exponential dichotomy, this paper investigates the behavior of solutions in the vicinity of a bounded solution to the autonomous differential system (1) \(dx/dt=f(x).\) Suppose \(x=u(t)\) is a nontrivial bounded solution of system (1). By discussing the equivalent equations of system (1) \[ (2)\quad d\theta /dt=1+\bar f_ 1(\rho,\theta),\quad d\rho /dt=A(\theta)\rho +\bar f_ 2(\rho,\theta) \] with respect to the moving orthonormal transformation \(x=u(\theta)+s(\theta)\rho,\) the author proves that if the linear system corresponding to (2) admits exponential dichotomy, then the given bounded solution \(x=u(t)\) should be periodic. The author also discusses the stability of the obtained periodic solution. Finally, this paper discusses perturbation of the bounded solution of autonomous system (1).