Robust and Nonrobust Dynamical Systems: Classification of Attractor Types

Abstract

We consider a class of autonomous continuous-time dynamical systems with phase space dimension N ≥ 3. Besides robust systems similar to Andronov–Pontryagin systems on the plane, there appears a class of robust systems with nontrivial hyperbolicity, i.e., systems with chaotic dynamics. Chaotic attractors of robust hyperbolic systems are, in the rigorous mathematical sense, strange attractors. They usually represent some mathematical idealization and are not as a rule observed in experiments. In most cases systems with irregular dynamics are nonrobust. Mathematicians have proven that robust hyperbolic systems are not everywhere dense on the set of dynamical systems with N ≥ 3. Structural instability (nonrobustness) is associated with the emergence of nonrobust double-asymptotic trajectories, such as separatrix loops, homoclinic curves, and heteroclinic curves, which are formed when manifolds of saddle cycles and another saddle sets intersect non-transversally.KeywordsUnstable ManifoldChaotic AttractorStrange AttractorMaximal Lyapunov ExponentLorenz AttractorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.