J. D. Mireles James

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 ...

Computation of Heteroclinic Arcs with Application to the Volume Preserving Henon Family

Let

Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach

f:\mathbb{R}^3\rightarrow\mathbb{R}^3

Quadratic Volume-Preserving Maps: (Un)stable Manifolds, Hyperbolic Dynamics, and Vortex-Bubble Bifurcations

be a diffeomorphism with

Validated Computation of Heteroclinic Sets

p_0,p_1\in\mathbb{R}^3

Connecting Orbits for Compact Infinite Dimensional Maps: Computer Assisted Proofs of Existence

distinct hyperbolic fixed points. Assume that

Fourier–Taylor Approximation of Unstable Manifolds for Compact Maps: Numerical Implementation and Computer-Assisted Error Bounds

W^u(p_0)

Computer Assisted Fourier Analysis in Sequence Spaces of Varying Regularity

and

Adaptive Set-Oriented Computation of Topological Horseshoe Factors in Area and Volume Preserving Maps

W^s(p_1)

Analytic Continuation of Local (Un)Stable Manifolds with Rigorous Computer Assisted Error Bounds

are two-dimensional manifolds which intersect transversally at a point q. Then the intersection is locally a one-dimensional smooth arc