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Three-dimensional tori and Arnold tongues.

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This study reveals how periodic solutions emerge in coupled delayed logistic maps through complex bifurcations. It details the transition from conventional Arnold tongues to invariant closed curve-Arnold tongues via Neimark-Sacker bifurcations.

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Area of Science:

  • Nonlinear Dynamics
  • Chaos Theory
  • Mathematical Biology

Background:

  • Coupled delayed logistic maps are fundamental models for complex systems.
  • Understanding invariant tori and closed curves is crucial for analyzing dynamical systems.
  • Arnold resonance webs describe complex frequency interactions in nonlinear systems.

Purpose of the Study:

  • To analyze the Arnold resonance web in a coupled delayed logistic map.
  • To investigate quasi-periodic bifurcations and their impact on system dynamics.
  • To characterize the emergence of periodic solutions within invariant structures.

Main Methods:

  • Lyapunov analysis was employed to study the coupled delayed logistic map.
  • The study focused on identifying and analyzing invariant tori (ITs) and invariant closed curves (ICCs).
  • Bifurcation analysis was used to understand transitions between different dynamical regimes.

Main Results:

  • The coupled delayed logistic map exhibits two-dimensional invariant tori (ITs) and numerous one-dimensional invariant closed curves (ICCs).
  • Periodic solutions arise at the intersections of ICC-generating regions, termed ICC-Arnold tongues, where frequency components become rational.
  • A transition from conventional Arnold tongues to ICC-Arnold tongues was observed through Neimark-Sacker bifurcations near quasi-periodic Hopf bifurcation boundaries.

Conclusions:

  • The study elucidates a complex bifurcation structure within the Arnold resonance web of the coupled delayed logistic map.
  • The findings provide insights into the mechanisms driving the emergence of periodic solutions from quasi-periodic dynamics.
  • The identified ICC-Arnold tongues offer a novel perspective on resonance phenomena in delayed dynamical systems.