Dynamics of traveling waves in fluctuating nonlocal media

The article deals with nonlocal hydrodynamic models for structured media with a fluctuating parameter. We are interested in the structure of traveling wave solutions disturbed by noise. Using the stochastic sensitivity function technique, the confidence ellipses for periodic trajectories obeying the period doubling scenario, hidden and spiral periodic orbits are derived. To identify the peculiarities of confidence ellipses, we consider the variation of eccentricity and area over the period of a periodic trajectory. We show that the dynamics of eccentricity of noisy limit cycle, up to triple period, has the number of minima coinciding with the cycle's multiplicity, whereas this is not in the case of quadruple cycle. The profiles of function for the areas of confidence ellipses characterize the heterogeneous anatomy of stochastic attractors and possess scaling properties for multiple cycles. Considering the eccentricity and area of confidence ellipses for the spiral trajectory existing in the vicinity of Shilnikov homoclinic loop, the intensive oscillations of eccentricity and area are observed when the confidence ellipses are derived for the flow near the one dimensional manifold of Shilnikov's orbit.