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Parameterization Method for Invariant Manifolds

Goal: Computational methods for efficient high order computation of invariant manifolds.

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Rafael De la Llave
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We provide evidence of the existence of KAM quasi-periodic attractors for a dissipative model in Celestial Mechanics. We compute the attractors extremely close to the breakdown threshold. We consider the spin–orbit problem describing the motion of a triaxial satellite around a central planet under the simplifying assumption that the center of mass of the satellite moves on a Keplerian orbit, the spin-axis is perpendicular to the orbit plane and coincides with the shortest physical axis. We also assume that the satellite is non-rigid; as a consequence, the problem is affected by a dissipative tidal torque that can be modeled as a time-dependent friction, which depends linearly upon the velocity. Our goal is to fix a frequency and compute the embedding of a smooth attractor with this frequency. This task requires to adjust a drift parameter. We have shown in Calleja et al. (2020) [14] that it is numerically efficient to study Poincaré maps; the resulting spin–orbit map is conformally symplectic, namely it transforms the symplectic form into a multiple of itself. In Calleja et al. (2020) [14], we have developed an extremely efficient (quadratically convergent, low storage requirements and low operation count per step) algorithm to construct quasi-periodic solutions and we have implemented it in extended precision. Furthermore, in Calleja et al. (2020) [15] we have provided an “a-posteriori” KAM theorem that shows that if we have an embedding and a drift parameter that satisfy the invariance equation up to an error which is small enough with respect to some explicit condition numbers, then there is a true solution of the invariance equation. This a-posteriori result is based on a Nash-Moser hard implicit function theorem, since the Newton method incurs losses of derivatives. The goal of this paper is to provide numerical calculations of the condition numbers and verify that, when they are applied to the numerical solutions, they will lead to the existence of the torus for values of the parameters extremely close to the parameters of breakdown. Computing reliably close to the breakdown allows to discover several interesting phenomena, which we will report in Calleja et al. (2020) [28]. The numerical calculations of the condition numbers presented here are not completely rigorous, since we do not use interval arithmetic to estimate the round off error and we do not estimate rigorously the truncation error, but we implement the usual standards in numerical analysis (using extended precision, checking that the results are not affected by the level of precision, truncation, etc.). Hence, we do not claim a computer-assisted proof, but the verification is more convincing that standard numerics. We hope that our work could stimulate a computer-assisted proof.
Jason Mireles James
added a research item
We make a detailed numerical study of the dynamics of a three dimensional dissipative vector field exhibiting a Neimark-Sacker bifurcation. Our main goals are to follow the attracting invariant torus born out of this bifurcation to its destruction in subsequent appearance of a chaotic attractor, and also to study the stable/unstable manifolds of the equilibrium solutions which act as separatrices/transport barriers for the system. Computing the periodic orbits -- and the stable/unstable manifolds -- which make up the resonance torus provides a reliable method for visualization, especially in the regime where the torus is only a $C^0$ invariant object. Collisions between the stable/unstable manifolds of the periodic orbits signal the destruction of the invariant torus and the onset of chaos.
Jason Mireles James
added 2 research items
Let f : R 3 → R 3 be a diffeomorphism with p0, p1 ∈ R 3 distinct hyperbolic fixed points. Assume that W u (p0) and W s (p1) are two-dimensional manifolds which intersect transversally at a point q. Then the intersection is locally a one-dimensional smooth ar γ through q, and points oñ γ are orbits heteroclinic from p0 to p1. We describe and implement a numerical scheme for computing the jets o γ to arbitrary order. We begin by computing high order polynomial approximations of some functions Pu, Ps : R 2 → R 3 , and domain disks Du, Ds ⊂ R 2 , such that W u loc (p0) = Pu(Du) and W s loc (p1) = Ps(Ds) with W u loc (p0) ∩ W s loc (p1) = ∅. Then the intersection ar γ solves a functional equation involving Ps and Pu. We develop an iterative numerical scheme for solving the functional equation, resulting in a high order Taylor expansion of the ar γ. We present numerical example computations for the volume preserving Hénon family and compute some global invariant branched manifolds.
We implement a semi-analytic scheme for numerically computing high order polynomial approximations of the stable and unstable manifolds associated with the fixed points of the normal form for the family of quadratic volume-preserving diffeomorphisms with quadratic inverse. We use this numerical scheme to study some hyperbolic dynamics associated with an invariant structure called a vortex bubble. The vortex bubble, when present in the system, is the dominant feature in the phase space of the quadratic family, as it encloses all invariant dynamics. Our study focuses on visualizing qualitative features of the vortex bubble such as bifurcations in its geometry, the geometry of some three-dimensional homoclinic tangles associated with the bubble, and the "quasi-capture" of homoclinic orbits by neighboring fixed points. Throughout, we couple our results with previous qualitative numerical studies of the elliptic dynamics within the vortex bubble of the quadratic family.
Jason Mireles James
added a research item
\In this work, we develop a high order polynomial approximation scheme for the lo- cal unstable manifold attached to a linearly unstable traveling wave. The wave profile is itself an equilibrium solution of a nonlinear parabolic partial differential equation (PDE) formulated on the real line, and its unstable manifold describes the dynamics of small perturbations. Our approach is based on the parameterization method and studies an invariance equation describing a local chart map for the invariant manifold. We show that this invariance equation is a PDE posed on the product of a disk and the line. The dimension of the disk is equal to the Morse index of the wave. We de- velop a formal series solution for the invariance equation, where the Taylor coefficients of the formal series are solutions of certain asymptotically constant linear boundary value problems (BVPs) on the line. By solving these BVPs numerically we obtain a polynomial approximation of the unstable manifold in a macroscopic neighborhood of the nonlinear wave. We demonstrate this method for a number of example problems. Truncation and numerical errors are quantified via a posteriori indicators.
Jason Mireles James
added a research item
A new method for approximating unstable manifolds for parabolic PDEs is introduced, which combines the parameterization method for invariant manifolds with finite element analysis and formal Taylor series expansions, and is applicable to problems posed on irregular spatial domains. The param-eterization method centers on an infinitesimal invariance equation for the unstable manifold, which we solve via a power series anzats. A power matching argument leads to linear elliptic PDEs-the so called homological equations-describing the jets of the manifold. These homological equations are solved recursively to any desired order using finite element approximation. The end result is a polynomial expansion of the manifold whose coefficients lie in an appropriate finite element space. We implement the method for a variety of example problems having both polynomial and non-polynomial nonlinearities posed on non-convex two dimensional polygonal domains. The parameterization method admits a natural notion of a-posteriori error and we provide numerical evidence in support of the claim that the manifolds are computed accurately.
Jason Mireles James
added a research item
We make a detailed numerical study of the dynamics of a three dimensional dissipative vector field exhibiting a Neimark-Sacker bifurcation. Our main goals are to follow the attracting invariant torus born out of this bifurcation to its destruction in subsequent appearance of a chaotic attractor, and also to study the stable/unstable manifolds of the equilibrium solutions which act as separatrices/transport barriers for the system. Computing the periodic orbits – and the stable/unstable manifolds – which make up the resonance torus provides a reliable method for visualization, especially in the regime where the torus is only a C0 invariant object. Collisions between the stable/unstable manifolds of the periodic orbits signal the destruction of the invariant torus and the onset of chaos.
Jason Mireles James
added 15 research items
This paper develops Chebyshev-Taylor spectral methods for studying stable/unstable manifoldsattached to periodic solutions of differential equations. The work exploits the parameterizationmethod – a general functional analytic framework for studying invariant manifolds. Useful fea-tures of the parameterization method include the fact that it can follow folds in the embedding,recovers the dynamics on the manifold through a simple conjugacy, and admits a natural no-tion of a-posteriori error analysis. Our approach begins by deriving a recursive system of lineardifferential equations describing the Taylor coefficients of the invariant manifold. We representperiodic solutions of these equations as solutions of coupled systems of boundary value problems.We discuss the implementation and performance of the method for the Lorenz system, and forthe planar circular restricted three and four body problems. We also illustrate the use of themethod as a tool for computing cycle-to-cycle connecting orbits.
We consider the problem of computing stable/unstable manifolds attached to periodic orbits of maps, and develop quasi-numerical methods for polynomial approximation of the manifolds to any desired order. The methods avoid function compositions by exploiting an idea inspired by multiple shooting schemes for periodic orbits. We consider a system of conjugacy equations which characterize chart maps for the local stable/unstable manifold segments attached to the points of the periodic orbit. We develop a formal series solution for the system of conjugacy equations, and show that the coefficients of the series are determined by recursively solving certain linear systems of equations. We derive the recursive equations for some example problems in dimension two and three, and for examples with both polynomial and transcendental nonlinearities. Finally we present some numerical results which illustrate the utility of the method and highlight some technical numerical issues such as controlling the decay rate of the coefficients and managing truncation errors via a-posteriori indicators.
In this paper we develop mathematically rigorous computer assisted techniques for studying high order Fourier-Taylor parameterizations of local stable/unstable mani-folds for hyperbolic periodic orbits of analytic vector fields. We exploit the numerical methods developed in [1] in order to obtain a high order Fourier-Taylor series expansion of the parameterization. The main result of the present work is an a-posteriori Theorem which provides mathematically rigorous error bounds. The hypotheses of the theorem are checked with computer assistance. The argument relies on a sequence of preliminary computer assisted proofs where we validate the numerical approximation of the periodic orbit, its stable/unstable normal bundles, and the jets of the manifold to some desired order M. We illustrate our method by implementing validated computations for two dimensional manifolds in the Lorenz equations in R 3 and a three dimensional manifold of a suspension bridge equation in R 4 .
Jason Mireles James
added a project goal
Computational methods for efficient high order computation of invariant manifolds.