4,284 research outputs found
Variational method for locating invariant tori
We formulate a variational fictitious-time flow which drives an initial guess
torus to a torus invariant under given dynamics. The method is general and
applies in principle to continuous time flows and discrete time maps in
arbitrary dimension, and to both Hamiltonian and dissipative systems.Comment: 10 page
Efficient computation of quasiperiodic oscillations in nonlinear systems with fast rotating parts
We present a numerical method for the investigation of quasiperiodic oscillations in applications modeled by systems of ordinary differential equations. We focus on systems with parts that have a significant rotational speed. An important element of our approach is to change coordinates into a co-rotating frame. We show that this leads to a dramatic reduction of computational effort in the case that gravitational forces can be neglected. As a practical example we study a turbocharger model for which we give a thorough comparison of results for a model with and without gravitational forces
Efficient Computation of Invariant Tori in Volume-Preserving Maps
In this paper we implement a numerical algorithm to compute codimension-one
tori in three-dimensional, volume-preserving maps. A torus is defined by its
conjugacy to rigid rotation, which is in turn given by its Fourier series. The
algorithm employs a quasi-Newton scheme to find the Fourier coefficients of a
truncation of the series. This technique is based upon the theory developed in
the accompanying article by Blass and de la Llave. It is guaranteed to converge
assuming the torus exists, the initial estimate is suitably close, and the map
satisfies certain nondegeneracy conditions. We demonstrate that the growth of
the largest singular value of the derivative of the conjugacy predicts the
threshold for the destruction of the torus. We use these singular values to
examine the mechanics of the breakup of the tori, making comparisons to
Aubry-Mather and anti-integrability theory when possible
Fast iteration of cocyles over rotations and Computation of hyperbolic bundles
In this paper, we develop numerical algorithms that use small requirements of
storage and operations for the computation of hyperbolic cocycles over a
rotation. We present fast algorithms for the iteration of the quasi-periodic
cocycles and the computation of the invariant bundles, which is a preliminary
step for the computation of invariant whiskered tori
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