Wave localized structures in relaxing media with fluctuations - Danylenko, V.A.; Skurativskyi, S. I. (Reports of the National Academy of Sciences of Ukraine, 2014)
The article deals with the wave solutions of a mathematical model for relaxing media. When the fluctuations of the model parameters are absent, the wave solutions satisfy the nonlinear planar dynamical system, which is studied by means of qualitative analysis methods. The aim of the article is the incorporation of parameters with noise and investigations of the influence of fluctuations on the steady and periodic modes of the dynamical system. In particular, the direction of a displacement of the Andronov-Hopf bifurcation for the steady solutions is estimated with the help of the top Lyapunov exponent, which is derived analytically and numerically. Stochastic limit cycles are considered by means of the sensitivity function. This function is evaluated from a deterministic differential equation by the shooting method and characterizes the dispersion of trajectories in a vicinity of the deterministic limit cycle. It is shown that the trajectories of a stochastic cycle undergo the most dispersion near the saddle fixed point.