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Organized structures of two bidirectionally coupled logistic maps.

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Summary
This summary is machine-generated.

Organized structures emerge in coupled logistic maps, exhibiting chaos through period-bubbling and quasiperiodic routes. These findings reveal novel Arnold tongues and self-similar shrimp structures, offering insights into complex system dynamics.

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Area of Science:

  • Nonlinear Dynamics
  • Chaos Theory
  • Complex Systems

Background:

  • Coupled logistic maps are fundamental models for studying nonlinear dynamics.
  • Standard logistic maps typically exhibit chaos via period-doubling bifurcations.
  • Harvesting and coupling introduce novel dynamical behaviors.

Purpose of the Study:

  • To investigate the organized structures in linearly coupled logistic maps with different harvesting.
  • To explore the routes to chaos, including period-bubbling and quasiperiodic routes.
  • To characterize the emergent Arnold tongues and self-similar structures.

Main Methods:

  • Analysis of two linearly coupled logistic maps with varying harvesting parameters.
  • Numerical simulations to identify bifurcations and route to chaos.
  • Characterization of periodic windows, Arnold tongues, and fractal basin boundaries.

Main Results:

  • Coupled systems exhibit chaos via period-bubbling and quasiperiodic routes, unlike the period-doubling route in simple maps.
  • Infinite families of periodic Arnold tongues and self-similar shrimp structures with period-adding sequences were discovered.
  • Fibonacci-like sequences leading to the Golden Mean and period 3-times self-similarity scaling in shrimp structures were observed.
  • Quasiperiodicity is essential for Arnold tongues, leading to shrimps in chaotic regions. Shrimp existence implies period-bubbling, but not vice-versa.
  • Bifurcation-induced hysteresis results in coexisting attractors with fractal basin boundaries.

Conclusions:

  • Coupled logistic maps display complex dynamics beyond simple period-doubling.
  • The study reveals novel fractal structures and routes to chaos, enriching our understanding of nonlinear systems.
  • Findings highlight the importance of coupling and harvesting in generating complex dynamical behaviors and emergent patterns.