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

Homoclinic points of 2-D and 4-D maps via the Parametrization Method

An interesting problem in solid state physics is to compute discrete breather solutions in $\mathcal{N}$ coupled 1--dimensional Hamiltonian particle chains and investigate the richness of their interactions. One way to do this is to compute the homoclinic intersections of invariant manifolds of a saddle point located at the origin of a class of $2\mathcal{N}$--dimensional invertible maps. In this paper we apply the parametrization method to express these manifolds analytically as series expansions and compute their intersections numerically to high precision. We first carry out this procedure for a 2--dimensional (2--D) family of generalized Henon maps ($\mathcal{N}$=1), prove the existence of a hyperbolic set in the non-dissipative case and show that it is directly connected to the existence of a homoclinic orbit at the origin. Introducing dissipation we demonstrate that a homoclinic tangency occurs beyond which the homoclinic intersection disappears. Proceeding to $\mathcal{N}=2$, we use the same approach to determine the homoclinic intersections of the invariant manifolds of a saddle point at the origin of a 4--D map consisting of two coupled 2--D cubic H\'enon maps. In dependence of the coupling the homoclinic intersection is determined, which ceases to exist once a certain amount of dissipation is present. We discuss an application of our results to the study of discrete breathers in two linearly coupled 1--dimensional particle chains with nearest--neighbor interactions and a Klein--Gordon on site potential.Comment: 24 pages, 10 figures, videos can be found at https://comp-phys.tu-dresden.de/supp

On the computation of reducible invariant tori on a parallel computer

We present an algorithm for the computation of reducible invariant tori of discrete dynamical systems that is suitable for tori of dimensions larger than 1. It is based on a quadratically convergent scheme that approximates, at the same time, the Fourier series of the torus, its Floquet transformation, and its Floquet matrix. The Floquet matrix describes the linearization of the dynamics around the torus and, hence, its linear stability. The algorithm presents a high degree of parallelism, and the computational effort grows linearly with the number of Fourier modes needed to represent the solution. For these reasons it is a very good option to compute quasi-periodic solutions with several basic frequencies. The paper includes some examples (flows) to show the efficiency of the method in a parallel computer. In these flows we compute invariant tori of dimensions up to 5, by taking suitable sections

Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations

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

Invariant manifolds and the parameterization method in coupled energy harvesting piezoelectric oscillators

Energy harvesting systems based on oscillators aim to capture energy from mechanical oscillations and convert it into electrical energy. Widely extended are those based on piezoelectric materials, whose dynamics are Hamiltonian submitted to different sources of dissipation: damping and coupling. These dissipations bring the system to low energy regimes, which is not desired in long term as it diminishes the absorbed energy. To avoid or to minimize such situations, we propose that the coupling of two oscillators could benefit from theory of Arnold diffusion. Such phenomenon studies $O(1)$ energy variations in Hamiltonian systems and hence could be very useful in energy harvesting applications. This article is a first step towards this goal. We consider two piezoelectric beams submitted to a small forcing and coupled through an electric circuit. By considering the coupling, damping and forcing as perturbations, we prove that the unperturbed system possesses a $4$-dimensional Normally Hyperbolic Invariant Manifold with $5$ and $4$-dimensional stable and unstable manifolds, respectively. These are locally unique after the perturbation. By means of the parameterization method, we numerically compute parameterizations of the perturbed manifold, its stable and unstable manifolds and study its inner dynamics. We show evidence of homoclinic connections when the perturbation is switched on

Parameterization of Invariant Manifolds for Periodic Orbits I: Efficient Numerics via the Floquet Normal Form

We present an efficient numerical method for computing Fourier-Taylor expansions of (un)stable manifolds associated with hyperbolic periodic orbits. Three features of the method are that (1) we obtain accurate representation of the invariant manifold as well as the dynamics on the manifold, (2) it admits natural a posteriori error analysis, and (3) it does not require numerically integrating the vector field. Our approach is based on the parameterization method for invariant manifolds, and studies a certain partial differential equation which characterizes a chart map of the manifold. The method requires only that some mild nonresonance conditions hold. The novelty of the present work is that we exploit the Floquet normal form in order to efficiently compute the Fourier-Taylor expansion. A number of example computations are given including manifolds in phase space dimension as high as ten and manifolds which are two and three dimensional. We also discuss computations of cycle-to-cycle connecting orbits which exploit these manifolds

``Critical'' phonons of the supercritical Frenkel-Kontorova model: renormalization bifurcation diagrams

The phonon modes of the Frenkel-Kontorova model are studied both at the pinning transition as well as in the pinned (cantorus) phase. We focus on the minimal frequency of the phonon spectrum and the corresponding generalized eigenfunction. Using an exact decimation scheme, the eigenfunctions are shown to have nontrivial scaling properties not only at the pinning transition point but also in the cantorus regime. Therefore the phonons defy localization and remain critical even where the associated area-preserving map has a positive Lyapunov exponent. In this region, the critical scaling properties vary continuously and are described by a line of renormalization limit cycles. Interesting renormalization bifurcation diagrams are obtained by monitoring the cycles as the parameters of the system are varied from an integrable case to the anti-integrable limit. Both of these limits are described by a trivial decimation fixed point. Very surprisingly we find additional special parameter values in the cantorus regime where the renormalization limit cycle degenerates into the above trivial fixed point. At these ``degeneracy points'' the phonon hull is represented by an infinite series of step functions. This novel behavior persists in the extended version of the model containing two harmonics. Additional richnesses of this extended model are the one to two-hole transition line, characterized by a divergence in the renormalization cycles, nonexponentially localized phonons, and the preservation of critical behavior all the way upto the anti-integrable limit.Comment: 10 pages, RevTeX, 9 Postscript figure

Numerical periodic normalization for codim 2 bifurcations of limit cycles : computational formulas, numerical implementation, and examples

Explicit computational formulas for the coefficients of the periodic normal forms for codimension 2 (codim 2) bifurcations of limit cycles in generic autonomous ODEs are derived. All cases (except the weak resonances) with no more than three Floquet multipliers on the unit circle are covered. The resulting formulas are independent of the dimension of the phase space and involve solutions of certain boundary-value problems on the interval [0, T], where T is the period of the critical cycle, as well as multilinear functions from the Taylor expansion of the ODE right-hand side near the cycle. The formulas allow one to distinguish between various bifurcation scenarios near codim 2 bifurcations of limit cycles. Our formulation makes it possible to use robust numerical boundary-value algorithms based on orthogonal collocation, rather than shooting techniques, which greatly expands its applicability. The implementation is described in detail with numerical examples, where numerous codim 2 bifurcations of limit cycles are analyzed for the first time