Time Dependent Stabilization of a Hamiltonian System

In this paper we consider the unstable chaotic attractor of a Hamiltonian system with Toda lattice potential and stabilize it by an integral form control. In order to obtain stability results, we use a control function in an integral form: $u\left(t\right)={\int }_0^{t}k(t,s)X\left(s\right)ds$ in which all the back story of the process X(t) is taken into consideration. Using the exponential kernel $k(t,s)=e^{-\beta (t-s)}$, we replace the study of integro-differential system of order 4 with an analysis of 5^th order system of ordinary differential equations (without integrals). Numerical solution of the resulting system leads to the asymptotically stabilization of the unstable fixed point. (paper)