THE BEHAVIOR OF SOLUTIONS IN THE VICINITY OF A BOUNDED SOLUTION TO AUTONOMOUS DIFFERENTIAL EQUATIONS

By using the exponential dichotomy,this paper investigates the behavior of solutionsin the vicinity of a bounded solution to the autonomous differential systemdx/dt=f(x).(1)Suppose x=u(t)is a nontrivial bounded solution of system(1).By discussing theequivalent equations of system(1)dθ/dt=1 (p,θ)dp/dt=A(θ)p (p,θ)(2)with respect to the moving orthonormal transformationx=u(θ) s(θ)p,the author proves that if linear system corresponding to(2)admits exponential dichotomy,then the given bounded solution x=u(t)should be periodic.The author also discusses thestadility of the obtained periodic solution.Finally,this paper discusses perturbation of thebounded solution of autonomous system(1).

THE BEHAVIOR OF SOLUTIONS IN THE VICINITY OF A BOUNDED SOLUTION TO AUTONOMOUS DIFFERENTIAL EQUATIONS

By using the exponential dichotomy, this paper investigates the behavior of solutions in the vicinity of a bounded.solution to the autonomous differerntial system: $$\\frac{{dx}}{{dt}} = f(x)\]$$ Suppose $\x = u(t)\]$ is a nontrivial bounded solution of system (1), By discussing the equivalent equations of system(1) $$\\frac{{d\theta }}{{dt}} = 1 + {\overline f _1}(\rho ,\theta )\]$$ $$\\frac{{d\rho }}{{dt}} = A(\theta )\rho + {\overline f _2}(\rho ,\theta )\]$$ with respect to the moving orthonormal transformation $$\x = u(\theta ) + s(\theta )\rho \]$$ the author proves that if linear system corresponding to (2) admits exponential dichotomy, then the given bounded solution $\x = u(t)\]$ should be periodic. The author also discusses the stadility of the obtained periodic solution. Finally, this paper discusses perturbation of the bounded solution of autonomous system (l).