Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Computer assisted proofs for transverse heteroclinics by the parameterization method

Maxime Murray, J. D. Mireles James

TL;DR

This work develops a rigorous computer-assisted framework for proving transverse heteroclinic connections between hyperbolic periodic orbits in ODEs by applying the parameterization method to their local stable/unstable manifolds. It combines Fourier–Taylor representations of manifolds with Chebyshev-based two-point boundary-value problem solvers and an a-posteriori radii polynomial analysis to obtain fully rigorous existence and transversality results. The authors validate the approach on two nontrivial problems—the dissipative Lorenz system and the Hamiltonian Hill’s restricted four-body problem—producing explicit manifold parameterizations and heteroclinic connections with quantified error bounds. The techniques cover both dissipative and Hamiltonian settings, using polynomial embeddings where necessary and providing a clear pathway for applying computer-assisted proofs to a broad class of nonlinear dynamical systems. Overall, the paper advances the toolbox for rigorous CAPs in nonlinear dynamics and demonstrates concrete, high-precision validations of intricate heteroclinic structures.

Abstract

This work develops a functional analytic framework for making computer assisted arguments involving transverse heteroclinic connecting orbits between hyperbolic periodic solutions of ordinary differential equations. We exploit a Fourier-Taylor approximation of the local stable/unstable manifold of the periodic orbit, combined with a numerical method for solving two point boundary value problems via Chebyshev series approximations. The a-posteriori analysis developed provides mathematically rigorous bounds on all approximation errors, providing both abstract existence results and quantitative information about the true heteroclinic solution. Example calculations are given for both the dissipative Lorenz system and the Hamiltonian Hill Restricted Four Body Problem.

Computer assisted proofs for transverse heteroclinics by the parameterization method

TL;DR

This work develops a rigorous computer-assisted framework for proving transverse heteroclinic connections between hyperbolic periodic orbits in ODEs by applying the parameterization method to their local stable/unstable manifolds. It combines Fourier–Taylor representations of manifolds with Chebyshev-based two-point boundary-value problem solvers and an a-posteriori radii polynomial analysis to obtain fully rigorous existence and transversality results. The authors validate the approach on two nontrivial problems—the dissipative Lorenz system and the Hamiltonian Hill’s restricted four-body problem—producing explicit manifold parameterizations and heteroclinic connections with quantified error bounds. The techniques cover both dissipative and Hamiltonian settings, using polynomial embeddings where necessary and providing a clear pathway for applying computer-assisted proofs to a broad class of nonlinear dynamical systems. Overall, the paper advances the toolbox for rigorous CAPs in nonlinear dynamics and demonstrates concrete, high-precision validations of intricate heteroclinic structures.

Abstract

This work develops a functional analytic framework for making computer assisted arguments involving transverse heteroclinic connecting orbits between hyperbolic periodic solutions of ordinary differential equations. We exploit a Fourier-Taylor approximation of the local stable/unstable manifold of the periodic orbit, combined with a numerical method for solving two point boundary value problems via Chebyshev series approximations. The a-posteriori analysis developed provides mathematically rigorous bounds on all approximation errors, providing both abstract existence results and quantitative information about the true heteroclinic solution. Example calculations are given for both the dissipative Lorenz system and the Hamiltonian Hill Restricted Four Body Problem.
Paper Structure (13 sections, 13 theorems, 154 equations, 7 figures)

This paper contains 13 sections, 13 theorems, 154 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Two Example Problems
  3. A Dissipative Example: The Lorenz System
  4. A Hamiltonian Example: A Lunar Hill Problem
  5. Validated numerics for the parameterization method
  6. Contraction Argument for the tail of the Fourier-Taylor expansion
  7. Connections between periodic orbits
  8. Examples
  9. Cycle-to-cycle connections in the Lorenz equations
  10. Cycle-to-cycle connections in Hill's lunar problem
  11. A-posteriori analysis: radii polynomials for nonlinear operators between Banach spaces
  12. Some Banach spaces of infinite sequences
  13. Cauchy Bounds

Key Result

Theorem 2.6

For $\alpha \geq N$, consider Then, for all $\alpha \geq N$, there exists a invertible and linear operator $\mathcal{G}_{\alpha}: (\ell_{\nu}^1)^n \to (\ell_{\nu}^1)^n$ such that $a_\alpha$ satisfies and the operator $\mathcal{G}_\alpha^{-1} R_\alpha : (\ell_{\nu}^1)^n \to (\ell_{\nu}^1)^n$ is Fréchet differentiable for each $\alpha$.

Figures (7)

  • Figure 1: (Top) Illustration of the Lorenz attractor obtained by integrating a single initial condition. Here we chose an initial condition near the origin, but this is incidental. (Bottom) Three periodic orbits of the Lorenz system at the classic parameter values. Note that these three periodic orbits already give a strong impression of the shape of the Lorenz attractor.
  • Figure 2: Family of periodic orbits at $\mathcal{L}_1$ and $\mathcal{L}_2$ for $\mu=0.00095$. The orbits displayed have Jacobi integral between $C=2.00$ (in blue) and $C=4.30$ (in red). The periodic orbits at $C=4$ and $C=2.5$ (shown in green), are the basis of the computations discussed in Section \ref{['sec:HillResults']}. That is, we will establish the existence of heteroclinic connections between these.
  • Figure 3: Stable and unstable manifolds attached to the three shortest periodic solutions for the Lorenz equation at the classical parameters $\sigma=10, \rho=28, \beta=\frac{8}{3}$. Recall that the periodic orbits were illustrated in Figure \ref{['fig:LorenzAttractor']}. The stable manifold attached to each of the three periodic solutions are illustrated on the left (green), and the unstable manifolds are illustrated on the right (red). All parameterizations are computed with $k=75$ Fourier coefficients and $N=8$ Taylor coefficients.
  • Figure 4: A pair of periodic solutions to the HRFBP at $\mu= 0.00095$ and Jacobi constant of $C=4$. The Lyapunov orbit at $\mathcal{L}_1$ is displayed with a parameterization of its stable manifold (green) The Lyapunov orbit at $\mathcal{L}_2$ is displayed with its unstable manifold (red). Both manifold are computed with $N=8$ Taylor coefficients and $m=30$ Fourier coefficient. To illustrate the dynamic on each manifold, we apply the conjugacy relation \ref{['eq:parmMethod']} to points evenly distributed on the boundary $|\sigma|=1$. The resulting trajectories are displayed in blue and accumulate, in forward time for the stable case and backward time for the unstable case, to the periodic orbit at the center of the cylinder.
  • Figure 5: Six solutions to \ref{['eq:BVPinitial']}. The first row represents the connecting orbit $I$ (left) and $II$ (right). Both connecting orbits accumulate to the periodic solution $AB$ at $-\infty$, connection $I$ accumulates to $AAB$ at $+\infty$ while $II$ accumulates to $ABB$. The Chebyshev arcs are represented in blue, while the arcs in red and green are integration-free (that is, these portions of the orbit are on the parameterized stable/unstable manifolds). and are calculated via the appropriate conjugacy relation. The second row represents the connecting orbit $III$ (left, between $AAB$ and $AB$) and $IV$ (right, between $AAB$ and $ABB$). The third row represents the connecting orbit $V$ (left, between $ABB$ and $AB$) and $VI$ (right, between $ABB$ and $AAB$).
  • ...and 2 more figures

Theorems & Definitions (40)

  • Remark 1.1: Solving Equation \ref{['eq:parmMethod']}
  • Remark 1.2: Literature on CAP for connecting orbits
  • Remark 1.3: Polynomial Embedding
  • Example 2.1: Periodic solutions of the Lorenz system in Fourier space
  • Example 2.2: Periodic solutions of Hill's three-body problem in Fourier space
  • Example 2.3: Normal bundles in the example system
  • definition 1
  • Example 2.4: The Fourier operator for Taylor coefficients of higher order
  • definition 2
  • definition 3
  • ...and 30 more