30,347 research outputs found

    Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations

    Full text link
    We study the dynamics of the five-parameter quadratic family of volume-preserving diffeomorphisms of R^3. This family is the unfolded normal form for a bifurcation of a fixed point with a triple-one multiplier and also is the general form of a quadratic three-dimensional map with a quadratic inverse. Much of the nontrivial dynamics of this map occurs when its two fixed points are saddle-foci with intersecting two-dimensional stable and unstable manifolds that bound a spherical ``vortex-bubble''. We show that this occurs near a saddle-center-Neimark-Sacker (SCNS) bifurcation that also creates, at least in its normal form, an elliptic invariant circle. We develop a simple algorithm to accurately compute these elliptic invariant circles and their longitudinal and transverse rotation numbers and use it to study their bifurcations, classifying them by the resonances between the rotation numbers. In particular, rational values of the longitudinal rotation number are shown to give rise to a string of pearls that creates multiple copies of the original spherical structure for an iterate of the map.Comment: 53 pages, 29 figure

    Algorithms for computing normally hyperbolic invariant manifolds

    Get PDF
    An effcient algorithm is developed for the numerical computation of normally hyperbolic invariant manifolds, based on the graph transform and Newton's method. It fits in the perturbation theory of discrete dynamical systems and therefore allows application to the setting of continuation. A convergence proof is included. The scope of application is not restricted to hyperbolic attractors, but extends to normally hyperbolic manifolds of saddle type. It also computes stable and unstable manifolds. The method is robust and needs only little specification of the dynamics, which makes it applicable to e.g. Poincaré maps. Its performance is illustrated on examples in 2D and 3D, where a numerical discussion is included.

    Separating manifolds in slow-fast systems

    Get PDF

    Efficient Computation of Invariant Tori in Volume-Preserving Maps

    Full text link
    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

    A variational principle for computing slow invariant manifolds in dissipative dynamical systems

    Full text link
    A key issue in dimension reduction of dissipative dynamical systems with spectral gaps is the identification of slow invariant manifolds. We present theoretical and numerical results for a variational approach to the problem of computing such manifolds for kinetic models using trajectory optimization. The corresponding objective functional reflects a variational principle that characterizes trajectories on, respectively near, slow invariant manifolds. For a two-dimensional linear system and a common nonlinear test problem we show analytically that the variational approach asymptotically identifies the exact slow invariant manifold in the limit of both an infinite time horizon of the variational problem with fixed spectral gap and infinite spectral gap with a fixed finite time horizon. Numerical results for the linear and nonlinear model problems as well as a more realistic higher-dimensional chemical reaction mechanism are presented.Comment: 16 pages, 5 figure

    Hyperdeterminants as integrable discrete systems

    Full text link
    We give the basic definitions and some theoretical results about hyperdeterminants, introduced by A. Cayley in 1845. We prove integrability (understood as 4d-consistency) of a nonlinear difference equation defined by the 2x2x2-hyperdeterminant. This result gives rise to the following hypothesis: the difference equations defined by hyperdeterminants of any size are integrable. We show that this hypothesis already fails in the case of the 2x2x2x2-hyperdeterminant.Comment: Standard LaTeX, 11 pages. v2: corrected a small misprint in the abstrac

    Two-dimensional global manifolds of vector fields

    Get PDF
    We describe an efficient algorithm for computing two-dimensional stable and unstable manifolds of three-dimensional vector fields. Larger and larger pieces of a manifold are grown until a sufficiently long piece is obtained. This allows one to study manifolds geometrically and obtain important features of dynamical behavior. For illustration, we compute the stable manifold of the origin spiralling into the Lorenz attractor, and an unstable manifold in zeta(3)-model converging to an attracting limit cycle
    corecore