On the governing equations in relaxing media models and self-similar quasiperiodic solutions

Peculiarities of wave dynamics in media with oscillating inclusions

“…As it is shown from the numerical experiments, in the phase space of system (14) there are trajectories with basins of attraction which are separated from other attractors. To catch this regime, we must choose corresponding initial data and parameters.…”

“…which correlate in time and space. Applying the internal variables concept and the irreversible thermodynamics technique [12,14], the features of intrinsic medium's structure, namely relaxation and nonlocal interactions between structural elements, can be described in the dynamical equation of state [11,13] ( ) σ τσ τ σ θ σ σ η…”

“…in dynamical system (14) and then numerically integrate the system with the help of the Dormand-Prince scheme [15,16]. Starting from the initial data y 1 ¼0.1, and θ θ = > 0.1536 cr , one can observe the creation of complicated trajectories.…”

“…possesses the fixed points O and ± The behaviour of system(14) in the vicinity of fixed points is recognized via the studying of corresponding linearized systems characterized by the matrix J…”

“…The trajectory's phase portrait (a) of dynamical system(14), its profile (b) of y 2 -component, and the Poincaré section (c) projected onto the ( ) y y ; 1 3 -plane. The bifurcation diagram (d) at increasing θ.…”