Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Experiment Videos

Basic structures of the Shilnikov homoclinic bifurcation scenario.

Rene O Medrano-T1, Murilo S Baptista, Iberê L Caldas

  • 1Instituto de Física, Universidade de São Paulo, C. P. 66318, CEP 05315-970 São Paulo, SP, Brazil. medrano@if.usp.br

Chaos (Woodbury, N.Y.)
|October 29, 2005
PubMed
Summary

Researchers discovered small-scale structures of homoclinic bifurcation curves in the Chua circuit, revealing a complete bifurcation scenario. This scenario is generic for systems with Shilnikov homoclinic orbits, advancing the understanding of complex dynamical systems.

Related Concept Videos

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Neuronal silence as a predictive biomarker and target for epileptic seizures suppression.

Scientific reports·2026
Same author

All-optical chaos synchronization between nonidentical optomechanical cavities.

Physical review. E·2026
Same author

Geometric perspective of linear stability of q-states in finite Kuramoto networks on circulant graphs.

Physical review. E·2025
Same author

Dependence of isochronous bifurcations on the driving-mode phase shift in two-harmonic standard maps.

Physical review. E·2025
Same author

Collapse of multi-headed chimera states in biologically based neuronal networks.

Chaos (Woodbury, N.Y.)·2025
Same author

Dynamic behavior analysis of a fractal tumor-immune model with drug resistance.

Physical review. E·2025

Area of Science:

  • Dynamical Systems and Chaos Theory
  • Nonlinear Dynamics
  • Circuit Theory

Background:

  • The Chua circuit is a canonical nonlinear electronic circuit exhibiting complex chaotic dynamics.
  • Homoclinic bifurcations, particularly Shilnikov type, are crucial for understanding chaos onset in many systems.
  • Characterizing bifurcation structures in parameter space is essential for predicting system behavior.

Purpose of the Study:

  • To numerically identify and characterize small-scale structures of homoclinic bifurcation curves in the Chua circuit's parameter space.
  • To theoretically demonstrate the generic nature of these structures for systems with Shilnikov homoclinic orbits.
  • To classify the complexity of homoclinic orbits and extend the study to higher-order orbits.

Main Methods:

Related Experiment Videos

  • Numerical computation of homoclinic bifurcation curves.
  • Analysis of the geometrical properties and distribution of these curves.
  • Theoretical investigation of the genericity of observed structures.
  • Classification of homoclinic orbits based on their order (number of returning loops).

    • Main Results:

    • Discovery of numerically small-scale basic structures of homoclinic bifurcation curves.
    • Establishment of a complete homoclinic bifurcation scenario for the Chua circuit.
    • Theoretical confirmation that these structures and the scenario are generic for systems with Shilnikov homoclinic orbits.
    • Classification of primary and subsidiary homoclinic orbits by order and identification of curve accumulations.

    Conclusions:

    • The identified homoclinic bifurcation structures and scenario are fundamental and generic for a broad class of dynamical systems.
    • The study confirms and extends previous predictions regarding homoclinic bifurcation curves, including high-order primary orbits.
    • New insights into the complex organization of bifurcation curves, including accumulations of subsidiary orbits, were revealed.