Heteroclinic Connections between Periodic Orbits in Planar Restricted Circular Three Body Problem - Part II

We present a method for proving the existence of symmetric periodic, heteroclinic or homoclinic orbits in dynamical systems with the reversing symmetry. As an application we show that the Planar Restricted Circular Three Body Problem (PCR3BP) corresponding to the Sun-Jupiter-Oterma system possesses an infinite number of symmetric periodic orbits and homoclinic orbits to the Lyapunov orbits. Moreover, we show the existence of symbolic dynamics on six symbols for PCR3BP and the possibility of resonance transitions of the comet. This extends earlier results by Wilczak and Zgliczynski - Heteroclinic Connections between Periodic Orbits in Planar Restricted Circular Three Body Problem - A Computer Assisted Proof, Commun. Math. Phys. 234, 37-75 (2003).Comment: 18 pages, 5 figure

An implicit algorithm for validated enclosures of the solutions to variational equations for ODEs

We propose a new algorithm for computing validated bounds for the solutions to the first order variational equations associated to ODEs. These validated solutions are the kernel of numerics computer-assisted proofs in dynamical systems literature. The method uses a high-order Taylor method as a predictor step and an implicit method based on the Hermite-Obreshkov interpolation as a corrector step. The proposed algorithm is an improvement of the $C^1$-Lohner algorithm proposed by Zgliczy\'nski and it provides sharper bounds. As an application of the algorithm, we give a computer-assisted proof of the existence of an attractor set in the R\"ossler system, and we show that the attractor contains an invariant and uniformly hyperbolic subset on which the dynamics is chaotic, that is, conjugated to subshift of finite type with positive topological entropy.Comment: 33 pages, 11 figure

Connecting orbits for a singular nonautonomous real Ginzburg-Landau type equation

We propose a method for computation of stable and unstable sets associated to hyperbolic equilibria of nonautonomous ODEs and for computation of specific type of connecting orbits in nonautonomous singular ODEs. We apply the method to a certain a singular nonautonomous real Ginzburg-Landau type equation, which that arises from the problem of formation of spots in the Swift-Hohenberg equation.Comment: 36 pages, 6 figure

Heteroclinic Connections between Periodic Orbits in Planar Restricted Circular Three Body Problem - A Computer Assisted Proof

The restricted circular three-body problem is considered for the following parameter values $C=3.03$, $\mu=0.0009537$ - the values for {\em 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.Comment: 40 pages, 11 figure

Computer assisted proof of the existence of homoclinic tangency for the Henon map and for the forced-damped pendulum

We present a topological method for the efficient computer assisted verification of the existence of the homoclinic tangency which unfolds generically in a one-parameter family of planar maps. The method has been applied to the Henon map and the forced damped pendulum ODE.Comment: 34 pages, 3 figure