Peculiarities of wave dynamics in media with oscillating inclusions

• The mathematical models for media with oscillating inclusions accompanied by the cubic and nonlocal equations of state are considered.
• Travelling wave solutions satisfying the dynamical systems are studied by means of qualitative analysis methods.
• The local model is shown to admit periodic and solitary wave solutions having both continuous and non-analytic profiles.
• The nonlocal model possesses periodic, multi-periodic, quasi-periodic, hidden and chaotic solutions which undergo bifurcations when the parameter of nonlocality is varied.

The article is concerned with mathematical models for media with oscillating inclusions. These models consist of mutually connected equations, one of which is the wave equation for carrying medium and others are equations of motion for partial oscillators. To close these models, we use cubic and nonlocal equations of state for the carrying medium. Travelling wave solutions to these models are studied in detail. Using qualitative analysis methods, the phase space is shown to contain periodic, homo- and heteroclinic trajectories. Moreover, in the case of nonlocal models we observe the creation of quasiperiodic and chaotic regimes. Bifurcations of localized regimes are studied via the Poincaré section technique.