96 research outputs found
Dynamics of nearly inviscid Faraday waves in almost circular containers
Parametrically driven surface gravity-capillary waves in an elliptically distorted circular cylinder are studied. In the nearly inviscid regime, the waves couple to a streaming flow driven in oscillatory viscous boundary layers. In a cylindrical container, the streaming flow couples to the spatial phase of the waves, but in a distorted cylinder, it couples to their amplitudes as well. This coupling may destabilize pure standing oscillations, and lead to complex time-dependent dynamics at onset. Among the new dynamical behavior that results are relaxation oscillations involving abrupt transitions between standing and quasiperiodic oscillations, and exhibiting ‘canards’
Lyapunov-Schmidt Reduction for Unfolding Heteroclinic Networks of Equilibria and Periodic Orbits with Tangencies
This article concerns arbitrary finite heteroclinic networks in any phase
space dimension whose vertices can be a random mixture of equilibria and
periodic orbits. In addition, tangencies in the intersection of un/stable
manifolds are allowed. The main result is a reduction to algebraic equations of
the problem to find all solutions that are close to the heteroclinic network
for all time, and their parameter values. A leading order expansion is given in
terms of the time spent near vertices and, if applicable, the location on the
non-trivial tangent directions. The only difference between a periodic orbit
and an equilibrium is that the time parameter is discrete for a periodic orbit.
The essential assumptions are hyperbolicity of the vertices and transversality
of parameters. Using the result, conjugacy to shift dynamics for a generic
homoclinic orbit to a periodic orbit is proven. Finally,
equilibrium-to-periodic orbit heteroclinic cycles of various types are
considered
Parallel transport along Seifert manifolds and fractional monodromy
The notion of fractional monodromy was introduced by Nekhoroshev,
Sadovski\'{i} and Zhilinski\'{i} as a generalization of standard (`integer')
monodromy in the sense of Duistermaat from torus bundles to singular torus
fibrations. In the present paper we prove a general result that allows to
compute fractional monodromy in various integrable Hamiltonian systems. In
particular, we show that the non-triviality of fractional monodromy in 2
degrees of freedom systems with a Hamiltonian circle action is related only to
the fixed points of the circle action. Our approach is based on the study of a
specific notion of parallel transport along Seifert manifolds
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