Complex dynamics of perturbed solitary waves in a nonlinear saturable medium: A Melnikov approach

Abstract

Objective:To investigate the nonlinear dynamics of a periodically perturbed second-order ordinary differential equation obtained by using traveling wave variables in the model of pulse propagation in a nonlinear medium with saturation. Method:The Melnikov function of the investigated system along its homoclinic and heteroclinic orbits is constructed. It is established that the necessary condition for the occurrence of Melnikov chaos is always met. By analogy with the well-known Duffing equation, a damping term is added to the system to control chaos. Using the numerical calculation of the Melnikov integrals, conditions are found on the parameters of the new system for which the Melnikov chaos takes place. To verify the results obtained by the Melnikov method, attraction basins of the system are constructed. Results:The results obtained by the Melnikov method go in agreement with the structure of the constructed basin boundaries.