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Geometry of resonance tongues. (English) Zbl 1120.37034

ChĂ©niot, Denis (ed.) et al., Singularity theory. Proceedings of the 2005 Marseille singularity school and conference, CIRM, Marseille, France, January 24–February 25, 2005. Dedicated to Jean-Paul Brasselet on his 60th birthday. Singapore: World Scientific (ISBN 978-981-270-410-8/hbk). 327-356 (2007).
Summary: Resonance tongues arise in bifurcations of discrete or continuous dynamical systems undergoing bifurcations of a fixed-point or an equilibrium satisfying certain resonance conditions. They occur in several different contexts, depending, for example, on whether the dynamics is dissipative, conservative, or reversible. Generally, resonance tongues are domains in parameter space, with periodic dynamics of a specified type (regarding period of rotation number, stability, etc.). In each case, the tongue boundaries are part of the bifurcation set. We mention here several standard ways that resonance tongues appear.
For the entire collection see [Zbl 1111.14001].

MSC:

37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
58K35 Catastrophe theory
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