  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.
