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Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits
S. V. Gonchenko1, L. P. Shil'nikov, D. V. Turaev
1Research Institute for Applied Mathematics and Cybernetics, Nizhniy Novgorod, Russia.
Chaos (Woodbury, N.Y.)
|March 1, 1996
Summary
Systems with homoclinic tangencies exhibit complex dynamics. Their quasiattractors defy simple descriptions and can contain multiple strange attractors, unlike simpler hyperbolic models.
Area of Science:
- Dynamical Systems Theory
- Nonlinear Dynamics
- Chaos Theory
Background:
- Systems with homoclinic tangencies are prevalent in natural applications.
- Quasiattractors in these systems present unique dynamical behaviors.
- These behaviors differ significantly from hyperbolic or Lorenz-like attractors.
Purpose of the Study:
- To represent recent findings on non-trivial dynamical phenomena in systems with homoclinic tangencies.
- To highlight the distinct features of quasiattractors in such systems.
- To explore the complexity and limitations in describing these dynamics.
Main Methods:
- Analysis of dynamical systems exhibiting homoclinic tangencies.
- Characterization of quasiattractor properties.
- Investigation of periodic orbits and Lyapunov exponents.
Main Results:
- Quasiattractors display non-trivial features, contrasting with hyperbolic attractors.
- A finite-parameter description of quasiattractor dynamics and bifurcations is impossible.
- Quasiattractors can host saddle periodic orbits with varying positive Lyapunov exponents.
- In phase spaces of sufficient dimension, quasiattractors may contain infinitely many coexisting strange attractors.
Conclusions:
- The dynamics of systems with homoclinic tangencies are inherently complex.
- Quasiattractors represent a class of attractors with profound theoretical implications.
- These findings challenge traditional understanding of attractor behavior in dynamical systems.