[Paper Review] Peculiarities of wave fields in nonlocal media

V.A. Danylenko, Sergii Skurativskyi|arXiv (Cornell University)|Mar 2, 2015

Nonlinear Waves and Solitons4 references3 citations

TL;DR

This paper investigates wave field dynamics in nonlocal, structured, non-equilibrium media using hydrodynamic models with temporal and spatial nonlocality. By applying symmetry reduction and qualitative/numerical analysis to autonomous dynamical systems derived from these models, the authors identify a rich spectrum of wave solutions—including periodic, quasiperiodic, chaotic, soliton-like, and homoclinic trajectories—along with bifurcations driven by nonlinearity and nonlocality, demonstrating the models' capacity to describe complex spatio-temporal wave regimes.

ABSTRACT

The article summarizes the studies of wave fields in structured non-equilibrium media describing by means of nonlocal hydrodynamic models. Due to the symmetry properties of models, we derived the invariant wave solutions satisfying autonomous dynamical systems. Using the methods of numerical and qualitative analysis, we have shown that these systems possess periodic, multiperiodic, quasiperiodic, chaotic, and soliton-like solutions. Bifurcation phenomena caused by the varying of nonlinearity and nonlocality degree are investigated as well.

Motivation & Objective

• To understand the behavior of wave fields in structured, non-equilibrium media where intrinsic microstructure and relaxation effects cannot be neglected.
• To develop and analyze nonlocal hydrodynamic models that incorporate both temporal and spatial nonlocality in the equations of state.
• To classify invariant wave solutions and their bifurcations in hierarchical nonlocal models using qualitative and numerical methods.
• To investigate the emergence of complex dynamics such as chaos, quasiperiodicity, and homoclinic structures in these systems.
• To establish the relevance of these models for describing dissipative structures and noise-induced dynamics in non-equilibrium systems.

Proposed method

• Derives dimensionless nonlocal hydrodynamic models from physical principles, incorporating relaxation processes and structural interactions via nonlocal equations of state.
• Applies symmetry reduction to transform the original PDEs into autonomous dynamical systems describing invariant wave solutions.
• Employs qualitative analysis and numerical integration to study the phase space behavior of the reduced systems.
• Uses Poincaré sections, bifurcation diagrams, and attractor visualization to identify periodic, quasiperiodic, chaotic, and homoclinic regimes.
• Analyzes bifurcations via parameter variation (e.g., nonlinearity, nonlocality, relaxation time), identifying Feigenbaum scenarios and intermittency.
• Investigates hidden attractors and hysteretic phenomena through numerical continuation and initial condition sensitivity.

Experimental results

Research questions

• RQ1 What types of wave solutions (periodic, chaotic, soliton-like) emerge in nonlocal hydrodynamic models of structured non-equilibrium media?
• RQ2 How do bifurcations in nonlinearity and nonlocality parameters affect the transition to chaos and the emergence of complex dynamics?
• RQ3 What role do homoclinic orbits of Shilnikov type play in the formation of chaotic attractors in these systems?
• RQ4 How do quasiperiodic regimes and torus bifurcations manifest in spatially nonlocal models?
• RQ5 In what ways do hidden attractors and hysteretic phenomena influence the predictability and stability of wave regimes?

Key findings

• The models support a wide range of wave solutions, including periodic, multiperiodic, quasiperiodic, chaotic, soliton-like, and homoclinic trajectories.
• Chaotic dynamics follow the Feigenbaum scenario, with period-doubling cascades leading to chaos, and intermittency observed in certain parameter regimes.
• Homoclinic structures of Shilnikov type exist and undergo bifurcations as nonlocality and nonlinearity parameters vary.
• Quasiperiodic regimes emerge via torus bifurcations, with Poincaré sections showing closed curves that evolve into chaotic attractors upon frequency locking.
• Bifurcation diagrams reveal increasing chaotic attractor regions with higher nonlinearity and nonlocality, including windows of periodicity within chaos.
• Hidden attractors and hysteresis are identified, indicating complex basin structures and non-trivial initial condition dependence in the system's dynamics.