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Morse Predecomposition of an Invariant Set.

Michał Lipiński1,2, Konstantin Mischaikow3, Marian Mrozek4

  • 1Intitute of Science and Technology Austria (ISTA), Am Campus 1, 3400 Klosterneuburg, Austria.

Qualitative Theory of Dynamical Systems
|November 18, 2024
PubMed
Summary

We introduce Morse predecomposition, a generalization of Morse decomposition for dynamical systems. This new framework captures recurrent dynamics, offering a finer analysis of complex systems beyond gradient behavior.

Keywords:
Chain recurrenceCombinatorial dynamicsIsolated invariant setMorse decompositionMultivector fieldRecurrence

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Area of Science:

  • Dynamical Systems Theory
  • Topology and Geometry in Dynamics

Background:

  • Morse decomposition is a standard tool for analyzing the gradient dynamics of isolated invariant sets.
  • Existing methods struggle to fully capture the recurrent dynamics within these sets, particularly chain recurrent sets.
  • A need exists for a more comprehensive framework to understand the full dynamics of isolated invariant sets.

Purpose of the Study:

  • To introduce and define the concept of Morse predecomposition for isolated invariant sets.
  • To generalize Morse decomposition by incorporating recurrent dynamics beyond purely gradient behavior.
  • To provide a framework capable of finer analysis for chain recurrent sets.

Main Methods:

  • Developed a new framework, Morse predecomposition, by relaxing the poset structure of Morse decomposition.
  • Introduced the concept of 'links' to represent connections between set elements, replacing strict ordering.
  • Utilized both combinatorial and classical dynamical systems settings for the generalization.

Main Results:

  • Morse predecomposition successfully extends analysis to include recurrent components of dynamical systems.
  • Proved that standard Morse decomposition is a special case of Morse predecomposition.
  • Demonstrated that Morse predecomposition can be condensed to recover a Morse decomposition.

Conclusions:

  • Morse predecomposition offers a more refined and comprehensive analysis of isolated invariant sets in dynamical systems.
  • This generalization enhances the study of recurrent orbits and complex dynamics.
  • The framework provides a valuable tool for understanding systems not fully described by gradient dynamics alone.