Chaotic motions in the restricted four body problem via Devaney's saddle-focus homoclinic tangle theorem

Abstract

We prove the existence of chaotic motions in a planar restricted four body problem, establishing that the system is not integrable. The idea of the proof is to verify the hypotheses of a topological forcing theorem. The forcing theorem applies to two freedom Hamiltonian systems where the stable and unstable manifolds of a saddle-focus equilibrium intersect transversally in the energy level set of the equilibrium. We develop a mathematically rigorous computer assisted argument which verifies the hypotheses of the forcing theorem, and we implement our approach for the restricted four body problem. The method is constructive and works far from any perturbative regime and for non-equal masses. Being constructive, our argument results in useful byproducts like information about the locations of transverse connecting orbits, quantitative information about the invariant manifolds, and upper/lower bounds on transport times.

Torus knot choreographies in the $n$-body problem

“…In this section, we turn the non-polynomial DDE (13) into a higher dimensional DDE with polynomial nonlinearities, using the automatic differentiation technique as in [31,32,33].…”

“…This reduction greatly simplifies the analysis of the delay differential equation in Fourier space, but requires adding an unfolding parameter to balance the system. In addition we utilize automatic differentiation as in [31,32,33], and reformulate (5) as a problem with polynomial nonlinearities. The polynomial problem is amenable to straight forward analysis exploiting the Banach algebra properties of the solution space and the use of the FFT algorithm as in [34].…”

“…The relevant literature is rich and we direct the interested reader to the works of [44,45,16,46,47] for a much more complete view of the literature. Other authors have studied center manifolds [48], transverse intersections of stable/unstable manifolds [31,49], Melnikov theory [50], Arnold diffusion and transport [51,52,53], and existence/continuation/bifurcation of Halo orbits [32,54] -all in gravitational n-body problems and all using computer assisted arguments. Especially relevant to the present work are the computer assisted existence and KAM stability proofs for n-body choreographies in [13,14,15,16].…”

Chebyshev–Taylor Parameterization of Stable/Unstable Manifolds for Periodic Orbits: Implementation and Applications

“…The mapping R used in the automatic differentiation is defined [Kepley and Mireles James, 2017], but is similar to the mapping discussed above for the three body case.…”

On polynomial forms of nonlinear functional differential equations

“…Over the following decades, this argument has been used in various contexts (cf. [13,10,11,12,5,4,7] for ODE, [9,8] for computer-assisted proofs in ODE and [15] for computer-assisted proofs in DDE).…”