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This study investigates the origins of homoclinic chaos in the Rössler model, revealing how Shilnikov saddle-foci bifurcations shape chaotic dynamics and attractors using computational methods.

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Area of Science:

  • Nonlinear Dynamics
  • Chaos Theory
  • Mathematical Modeling

Background:

  • The Rössler model, proposed in 1976, is a foundational 3D system for studying chaotic behavior.
  • Homoclinic chaos and Shilnikov bifurcations are key phenomena in understanding complex dynamical systems.

Purpose of the Study:

  • To investigate the origin of homoclinic chaos in the Rössler model.
  • To analyze the role of Shilnikov saddle-foci bifurcations in global bifurcation unfolding.
  • To understand the transformations of chaotic attractors within the model.

Main Methods:

  • Application of computational methods, including interval maps.
  • Utilizing a symbolic approach tailored to the Rössler model.
  • Scrutiny of homoclinic bifurcations and detection of chaotic regions.

Main Results:

  • Detailed analysis of convoluted bifurcations of Shilnikov saddle-foci.
  • Identification of how these bifurcations influence the global dynamics.
  • Mapping of regions exhibiting stable and chaotic dynamics in the parameter space.

Conclusions:

  • The synergy of Shilnikov saddle-foci bifurcations is crucial for the global bifurcation unfolding in the Rössler model.
  • Computational methods effectively reveal the mechanisms driving homoclinic chaos.
  • The study provides insights into the complex interplay between bifurcations and chaotic attractors.