Coexisting solutions and their neighbourhood in the dynamical system describing second-order optical processes

Abstract

Coexisting periodic solutions of a dynamical system describing nonlinear optical processes of the second-order are studied. The analytical results concern both the simplified autonomous model and the extended nonautonomous model, including the pump and damping mechanism. The nonlinearity in the coexisting solutions of the autonomous system is in concealed frequencies depending on the initial conditions. In the solutions of the nonautonomous system the nonlinearity is convoluted in amplitudes. The neighbourhood of periodic solutions is studied numerically, mainly in phase portraits. As a result of disturbance, for example detuning, the periodic solutions are shown to escape to other states, periodic, quasiperiodic (beats) or chaotic. The chaotic behavior is indicated by the Lypunov exponents. We also investigate selected aspects of synchronization (unidirectional or mutual) of two identical systems being in two different coexisting states. The effects of quenching of oscillations are shown. In the autonomous system the quenching is caused by a change in frequency, whereas in the nonautonomous one by a change in amplitude. The quenching seems very promising for design of some advanced signal processing.