The restricted circular three-body problem is considered for the following parameter values C = 3.03, µ = 0.0009537 -the values for Oterma comet in the Sun-Jupiter system. We present a computer assisted proof of an existence of homo-and heteroclinic cycle between two Lyapunov orbits and an existence of symbolic dynamics on four symbols built on this cycle.
In this paper we present a new topological tool which allows us to prove the existence of Shilnikov homoclinic or heteroclinic solutions. We present an application of this method to the Michelson system y + y + 0.5y 2 = c 2 [16]. We prove that there exists a countable set of parameter values c for which a pair of the Shilnikov homoclinic orbits to the equilibrium points (±c √ 2, 0, 0) appear. This result was conjectured by Michelson [16]. We also show that there exists a countable set of parameter values for which there exists a heteroclinic orbit connecting the equilibrium (−c √ 2, 0, 0) possessing a one-dimensional unstable manifold with the equilibrium (c √ 2, 0, 0) possessing a one-dimensional stable manifold. The method used in the proof can be applied to other reversible systems.To verify the assumptions of the main topological theorem for the Michelson system, we use rigorous computations based on interval arithmetic.
We prove the existence of cocoon bifurcations for the Michelson system ẋ = y,, where (x, y, z) ∈ R 3 and c ∈ R + is a parameter, based on the theory given in (Dumortier et al 2006 Nonlinearity 19 305-28). The main difficulty lies in the verification of the (topological) transversality of some invariant manifolds in the system. The proof is computer-assisted and combines topological tools including covering relations and the smooth ones using the cone conditions. These new techniques developed in this paper will have broader applicability to similar global bifurcation problems.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.