Initial Value Problems
Analytic Continuation of Local (Un)Stable Manifolds with Rigorous Computer Assisted Error Bounds
The Parameterization Method
Chart maps and atlases
Automatic Differentiation
The CRFBP
Space-periodic orbits in the equilateral circular constrained four-body problem: computer-aided existence proofs. submitted to Celestial Mechanics and Dynamic Astronomy), 2018. Parameterization of invariant manifolds for periodic orbits (ii): a-posterior analysis and computer-aided error bounds. A posteriori verification of invariant objects of evolution equations: periodic orbits in Kuramoto-Sivashinsky PDE.
Construction of quasi-periodic solutions of state-dependent delay differential equations by the parameterization method II: analytical case. Construction of quasi-periodic solutions of state-dependent delay differential equations using the parameterization method I: finitely differentiable, hyperbolic case. Spatial periodic orbits in the equilateral circle constrain the four-body problem: computer-aided proofs of existence.
Investigation on chaotic behavior of the constrained four-body problem with an equilateral triangle configuration. Chaotic motions in the constrained four-body problem via deveney's saddle-focused homoclinic tangle theorem. Chaotic motions in the constrained four-body problem via Devaney's saddle-focused homoclinic tangle theorem.
Homoclinic dynamics in a restricted four-body problem: a multiparameter study of transverse connections for the saddle-focus equilibrium solutions. submitted to Celestial Mechanics and Dynamical Astronomy), 2018. Inflating complex solutions of the 3D Navier-Stokes system and renormalization group method.
COMPUTING MANIFOLDS FOR PERIODIC ORBITS OF MAPS 1795
Fourier-Taylor approximation of unstable manifolds for compact maps: numerical implementation and computer-aided error bounds. -Taylor parameterization of unstable manifolds for parabolic partial di↵erential equations: Formalism, implementation and rigorous proof. If P is a conjugate covering map in the sense of Definition B.4, then the image of P is an unstable local manifold for .
A computationally convenient equivalent condition is given in the following proposition, which forms the core of the parameterization method for periodic orbits of differential equations. The proof for the case of finite dimensional vector fields is found in [11], and can be adapted to the present case of an unbounded ordinary differential equation defined densely on a Banach space. X is a conjugative covering map for a locally unstable manifold if and only if P satisfies the linear constraints of equation (42) and P solves the partial differential equation.
Note that this is exactly the same notion of non-resonance as occurs in the equilibrium case (see definition B.3), except with unstable eigenvalues replaced by unstable Floquet exponents.
CiAP in Nonlinear Dynamics Introduction and the Method of Radius Polynomials[89] J. Mireles James and Konstantin Mischaikow. Parameterization of slow-stable manifolds and their invariant vector bundles: theory and numerical implementation. It also corresponds to the situation encountered when parameterizing local stable/unstable manifolds associated with periodic orbits with finite-dimensional ODEs.
PARAMETERIZATION METHOD FOR DDES 47
Krauskopf, Resonance phenomena in a scalar delay di↵erential equation with two state-dependent delays, (Submitted). Campbell, Computation of central manifolds for delay di↵erential equations using MapleTM, in Delay Di↵erential Equations, Springer, New York. Capi´nski, Computer-assisted existence proof of Lyapunov orbits in L2 and cross-crossings of Jupiter-Sun invariant manifolds PCR3BP, SIAM J.
Mireles James, Parameterization of invariant manifolds for periodic orbits (ii): a-posterior analysis and computer-aided error bounds, (To appear in Journal of Dynamics and Di↵Equations). Hale, Theory of di↵ functional equations, 2nd ed., Springer-Verlag, New York-Heidelberg, 1977, Applied Mathematical Sciences, vol. Wu, Di↵erential equations with state-dependent delays: theory and applications, in Handbook of di↵ercial equations: ordinary di↵ercial equations.
Lessard, Delayed di↵erential equations and continuation, Lecture Notes for the AMS Short Course on Rigorous Numericals in Dynamics. James, Fourier-Taylor parametrization of unstable manifolds for parabolic partial di↵erential equations: Formalism, implementation and rigorous proof, (submitted). Michiels, Software for stability and bifurcation analysis of delay di↵erential equations and applications to stabilization, in Advances in time-delay systems, vol.
In Advances in Differential Equations and Applications, Volume 4 SEMA SIMAI Springer Ser., Pages 85–94. Computational evidence for the existence of Lyapunov orbits on L2 and transverse intersections of invariant manifolds in the Jupiter-Sun PCR3BP. Zgliczy´ nski, An Algorithm for Rigorous Integration of Delay Differential Equations and a Computer-Assisted Proof of Periodic Orbits in the Mackey-Glass Equation, (Submitted) URL http: // ww2.
FPTM08
Mireles James, Parametrization of slow-stable manifolds and their invariant vector bundles: theory and numerical implementation, Discrete Contin. Goodman, High-order adaptive method for computing two-dimensional invariant manifolds from three-dimensional maps, Commun. This suggests that the method could be useful in computer-aided existence proofing, a topic that will be the subject of future research.
Indeed, it seems possible to combine the techniques of [32] with the recent work of [20, 43] and the parameterization method of the present study to obtain computer-aided proofs of homoclinic and heteroclinic connection dynamics for infinite-dimensional systems. . Another interesting project would be to apply the methods of the present work to the difficult stable/unstable manifold calculations of the period five point discussed in [7]. Unfortunately, the explicit form of the map used for that study is not given in the reference (however, the authors note that the map is an 11th-order polynomial, a fact that suggests that the multiple-shooting approach of the present work can be a big help).
Another interesting direction of future research would be to apply rigorous globalization methods such as those of [66, 78] to grow the local manifolds studied here. This can lead to a better understanding of the connection pathway structure and topological entropy for discrete time systems. The authors would like to thank the three anonymous referees who read the submitted version of the manuscript for making a number of helpful suggestions.
Osinga, Exploring the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields, Discrete Contin. Wittig, Computer-assisted evidence for the existence of highly periodic fixed points, in Proceedings of the Fields Institute, http://bt.pa.msu.edu/cgi-bin/display.pl?name=.
CONSTRUCTION OF QUASI-PERIODIC SOLUTIONS: II 41
Jan Bouwe van den Berg, Maxime Breden, Jean-Philippe Lessard, and Maxime Murray
Boundary value problem formulations for the computation of invariant manifolds and connecting trajectories in the circular constrained three-body problem. Bifurcations and enumeration of classes of relative equilibria in the planar restricted four-body problem.
Jan Bouwe van den Berg and Jean-Philippe Lessard. Rigorous numerics in dynamics
Mireles James, Computation of the largest local (un)stable patches of a manifold by a parameterization method, Indag.
COMPUTING MANIFOLDS FOR PERIODIC ORBITS OF MAPS 1793
Alessandra Celletti and Luigi Chierchia. On the stability of realistic three-body problems
