A filtration model with the generalized Darcy’s law making allowance for nonlocal and nonlinear effects has been developed. The expression for the law was derived within the relaxation formalizm of nonequilibrium thermodynamics. The developed model is applied to analyze the influence of relaxation effects on the phase velocity of small wave-like perturbations. The character of nonlinear traveling waves is determined. The properties of polynomial and self-similar solutions are analyzed.

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A one-dimensional mathematical model for a complex medium with van der Pol oscillators has been studied. Using the Bogolyubov-Mitropolsky method, the wave solutions for a weakly nonlinear model are derived, with their amplitudes being described by a three-dimensional dynamical system analyzed in more details by numerical and qualitative methods. In particular, periodic, multiperiodic, and chaotic trajectories are found in the phase space of the dynamical system. Bifurcations of those regimes were considered using the Poincaré section technique. Exact solutions are derived in the case where the three-dimensional system for amplitudes is reduced to the two-dimensional one. K e y w o r d s: nonlinear waves, van der Pol oscillator, chaotic attractor.

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