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Torus breakdown in noninvertible maps.

V Maistrenko1, Yu Maistrenko, E Mosekilde

  • 1Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev 252601, Ukraine.maistren@nas.gov.ua

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 6, 2003
PubMed
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We identified a new criterion for destroying two-dimensional tori in noninvertible maps. This destruction occurs via cusp points forming on invariant curves, leading to complex chaotic sets.

Area of Science:

  • Dynamical Systems
  • Chaos Theory
  • Nonlinear Dynamics

Background:

  • Resonance tori are fundamental structures in dynamical systems.
  • Understanding torus destruction is key to characterizing chaotic behavior.
  • Noninvertible maps present unique challenges in analyzing invariant structures.

Purpose of the Study:

  • To propose a novel criterion for the destruction of two-dimensional tori.
  • To elucidate the mechanism of torus destruction through cusp point formation.
  • To investigate the transition from ordered torus structures to chaotic sets.

Main Methods:

  • Analysis of invariant curves and critical curves in noninvertible maps.
  • Identification of cusp points arising from specific tangent-eigenvalue conditions.

Related Experiment Videos

  • Examination of parameter-dependent transformations of invariant manifolds.

    • Main Results:

    • A criterion for torus destruction based on cusp point formation is proposed.
    • Cusp points emerge when the torus tangent aligns with a vanishing eigenvalue eigendirection.
    • Parameter changes lead to manifold self-intersections (loops) and torus breakdown into chaotic sets.

    Conclusions:

    • The formation of infinite cusp points provides a specific mechanism for torus destruction in noninvertible maps.
    • This mechanism highlights the unique dynamics of noninvertible systems.
    • The study details a pathway from regular torus structures to complex chaotic behavior.