Bifurcations of quasi-periodic dynamics: torus breakdown

DOI10.1007/s00033-013-0363-8zbMath1315.34046OpenAlexW2169959775MaRDI QIDQ2253940

Taoufik Bakri, Ferdinand Verhulst

Publication date: 13 February 2015

Published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s00033-013-0363-8

zbMATH Keywords

strange attractorquasi-periodictorus bifurcationtorus breakdown

Mathematics Subject Classification ID

Periodic solutions to ordinary differential equations (34C25) Bifurcation theory for ordinary differential equations (34C23) Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. (34A25) Nonlinear oscillations and coupled oscillators for ordinary differential equations (34C15) Averaging method for ordinary differential equations (34C29) Complex behavior and chaotic systems of ordinary differential equations (34C28) Invariant manifolds for ordinary differential equations (34C45) Multifrequency systems of ordinary differential equations (34C46)

Related Items

Dissecting a resonance wedge on heteroclinic bifurcations, Torus bifurcations in a mechanical system, Persistence of degenerate hyperbolic lower-dimensional invariant tori in Hamiltonian systems with Bruno’s conditions, Chenciner bubbles and torus break-up in a periodically forced delay differential equation, Response solutions and quasi-periodic degenerate bifurcations for quasi-periodically forced systems, Periodic forcing of a heteroclinic network, Resonant tori, transport barriers, and chaos in a vector field with a Neimark-Sacker bifurcation, Computer assisted proofs of two-dimensional attracting invariant tori for ODEs, Construction of response solutions for two classes of quasi-periodically forced four-dimensional nonlinear systems with degenerate equilibrium point under small perturbations, Coexistence and Dynamical Connections between Hyperchaos and Chaos in the 4D Rössler System: A Computer-Assisted Proof, Torus-Breakdown Near a Heteroclinic Attractor: A Case Study, Canonical models for torus canards in elliptic bursters, A Degenerate 2:3 Resonant Hopf--Hopf Bifurcation as Organizing Center of the Dynamics: Numerical Semiglobal Results

Uses Software

• MATCONT
• AUTO
• Torcont

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Resonance and fractal geometry
• Three coupled oscillators: Mode-locking, global bifurcations and toroidal chaos
• Computing Arnol'd tongue scenarios
• Parametric excitation in non-linear dynamics
• Resonance regions for Mathieu type dynamical systems on a torus
• Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer-assisted study
• Growing 1D and quasi-2D unstable manifolds of maps
• Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems
• On the nature of turbulence
• Note concerning our paper On the nature of turbulence. Commun. math. Phys. 20, 167-192 (1971)
• Numerical Periodic Normalization for Codim 2 Bifurcations of Limit Cycles: Computational Formulas, Numerical Implementation, and Examples
• Fourier methods for quasi-periodic oscillations
• New features of the software M<scp>at</scp>C<scp>ont</scp>for bifurcation analysis of dynamical systems
• Differentiable dynamical systems and the problem of turbulence
• Invariant two-dimensional tori, their breakdown and stochasticity
• Towards global models near homoclinic tangencies of dissipative diffeomorphisms
• Secondary Bifurcations of Two Non-Linearly Coupled Oscillators
• ON SOME MATHEMATICAL TOPICS IN CLASSICAL SYNCHRONIZATION.: A TUTORIAL
• MATCONT
• Continuation of Quasi-periodic Invariant Tori
• Averaging methods in nonlinear dynamical systems
• Elements of applied bifurcation theory