  In this paper we have investigated the driven cubic-quintic Duffing equation.
Using the Melnikov analysis technique, we are able to predict the number of
limit cycles around the equilibrium and to develop a theoretical approach to
chaos suppression in damped driven systems. The analytical solution and the
bifurcation of limit cycles of the investigated system have been studied., We
discussed our nonlinear dynamics, namely, the cubic quintic Duffing equation in
the absence of noise and we may discusse it in the presence of noise in the
futur research, investigating for the Hamiltonian of the system (not perturbed)
in both cases is presented.
