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Foreword

Applications of the odd symplectic group in Hamiltonian systems

KAM à la R

The reversible context 2 in KAM theory: the first steps

Dynamics and geometry near resonant bifurcations

Quasi-periodic bifurcations in reversible systems

Resonance tongues in the quasi-periodic Hill-Schrödinger equation with three frequencies

New directions in algebraic dynamical systems

Exponential mixing for smooth hyperbolic suspension flows

Hamiltonicity and integrability of the Suslov problem

A free energy based mathematical study for molecular motors

Piecewise linear hamiltonian flows associated to zero-sum games: Transition combinatorics and questions on ergodicity

Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems

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Resonance tongues in the quasi-periodic Hill-Schrödinger equation with three frequencies

Content Provider	Springer Nature Link
Author	Puig, Joaquim Simó, Carles
Copyright Year	2011
Abstract	In this paper we investigate numerically the following Hill’s equation x″ + (a + bq(t))x = 0 where $$ q(t) = \cos t + \cos \sqrt {2t} + \cos \sqrt {3t} $$ is a quasi-periodic forcing with three rationally independent frequencies. It appears, also, as the eigenvalue equation of a Schrödinger operator with quasi-periodic potential.Massive numerical computations were performed for the rotation number and the Lyapunov exponent in order to detect open and collapsed gaps, resonance tongues. Our results show that the quasi-periodic case with three independent frequencies is very different not only from the periodic analogs, but also from the case of two frequencies. Indeed, for large values of b the spectrum contains open intervals at the bottom. From a dynamical point of view we numerically give evidence of the existence of open intervals of a, for large b, where the system is nonuniformly hyperbolic: the system does not have an exponential dichotomy but the Lyapunov exponent is positive. In contrast with the region with zero Lyapunov exponents, both the rotation number and the Lyapunov exponent do not seem to have square root behavior at endpoints of gaps. The rate of convergence to the rotation number and the Lyapunov exponent in the nonuniformly hyperbolic case is also seen to be different from the reducible case.
Starting Page	61
Ending Page	78
Page Count	18
File Format	PDF
ISSN	15603547
Journal	Regular and Chaotic Dynamics
Volume Number	16
Issue Number	1-2
e-ISSN	14684845
Language	English
Publisher	SP MAIK Nauka/Interperiodica
Publisher Date	2010-10-13
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	quasi-periodic Schrödinger operators quasi-periodic cocycles and skew-products spectral gaps resonance tongues rotation number Lyapunov exponent numerical explorations Dynamical Systems and Ergodic Theory
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Statistical and Nonlinear Physics Mathematical Physics Mechanical Engineering Modeling and Simulation

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Resonance Tongues and Spectral Gaps in Quasi-Periodic Schrödinger Operators with One or More Frequencies. A Numerical Exploration

Resonance Tongues and Spectral Gaps in Quasi-Periodic Schrödinger Operators with One or More Frequencies. A Numerical Exploration

Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena

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Resonance tongues in the quasi-periodic Hill-Schrödinger equation with three frequencies

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