Three coupled oscillators: Mode-locking, global bifurcations and toroidal chaos

DOI10.1016/0167-2789(91)90155-3zbMath0734.58036OpenAlexW2069032592MaRDI QIDQ810971

N. E. Zubov

Publication date: 1991

Published in: Physica D (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/0167-2789(91)90155-3

zbMATH Keywords

coupled oscillatorsglobal bifurcationsmode-lockingtoroidal chaos

Mathematics Subject Classification ID

Local and nonlocal bifurcation theory for dynamical systems (37G99)

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Cites Work

• Bifurcations de points fixes elliptiques. III: Orbites périodiques de ``petites périodes et élimination résonnante des couples de courbes invariantes. (Bifurcations of elliptic fixed points. III: Periodic orbits with ``small periods and resonant elimination of pairs of invariant curves.)
• The dynamics of the Hénon map
• Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnold et de Moser sur le tore de dimension 2
• On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane
• Structural stability on two-dimensional manifolds
• Chaotic attractors on a 3-torus, and torus break-up
• Resonance regions for Mathieu type dynamical systems on a torus
• Stochastic webs
• Homoclinic bifurcation at resonant eigenvalues
• Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
• Attractors on an N-torus: Quasiperiodicity versus chaos
• The geometry of cross sections to flows
• Transition to topological chaos for circle maps
• Universal transition in a dynamical system forced at two incommensurate frequencies
• Existence d'orbites quasi-périodiques dans les attracteurs de Birkhoff. (Existence of quasiperiodic orbits in Birkhoff attractors)
• Global Hopf bifurcation of two-parameter flows
• On a four parameter family of planar vector fields
• Quasi-genericity of bifurcations to high dimensional invariant tori for maps
• Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. (On smooth conjugacy of diffeomorphisms of the circle with rotations)
• Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer-assisted study
• Flow equivalence, hyperbolic systems and a new zeta function for flows
• Occurrence of strange Axiom A attractors near quasi periodic flows on \(T^m\), \(m\geq 3\)
• Bifurcations de tores invariants
• Three-frequency quasiperiodicity, phase locking, and the onset of chaos
• Diffeomorphisms with infinitely many sinks
• Note concerning our paper On the nature of turbulence. Commun. math. Phys. 20, 167-192 (1971)
• Note on differential equations on the torus
• Periodic and Steady-State Mode Interactions Lead to Tori
• Stable and Random Motions in Dynamical Systems
• Resonance Zones in Two-Parameter Families of Circle Homeomorphisms
• Remarks on the perturbation theory for problems of Mathieu type
• Integrability of hamiltonian systems on cantor sets
• Topologies and Rotation Numbers for Families of Monotone Functions on the Circle
• Converse KAM theory for symplectic twist maps
• Rotation Sets of Toral Flows
• The necessity of the Hopf bifurcation for periodically forced oscillators
• Roads to turbulence in dissipative dynamical systems
• On the structure of the family of Cherry fields on the torus
• Periodic orbits for dissipative twist maps
• Limit Cycles in the Josephson Equation
• Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3
• Recurrence and fixed points of surface homeomorphisms
• A simple proof of Denjoy's theorem
• Stabilité ou instabilité des points fixes elliptiques
• Propriétés des attracteurs de Birkhoff
• Rotation Sets for Maps of Tori
• Realizing Rotation Vectors for Torus Homeomorphisms
• Homoclinic and heteroclinic bifurcations of Vector fields
• Resonance regions for families of torus maps
• Construction d'un difféomorphisme minimal d'entropie topologique non nulle
• SOURCES AND SINKS OFA-DIFFEOMORPHISMS OF SURFACES
• The Poincare-Bendixson Theorem for the Klein Bottle
• Unfoldings and bifurcations of quasi-periodic tori
• Analytic Quasi-Periodic Curves of Discontinuous Type on a Torus
• Rotation interval from a time series
• Invariant manifolds
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