Daniel Wilczak

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

Abstract: The restricted circular three-body problem is considered for the following parameter values C=3.03, μ=0.0009537 – the values for the Oterma comet in the Sun-Jupiter system. We present a computer assisted proof of an existence of a homo- and heteroclinic cycle between two Lyapunov orbits and an existence of symbolic dynamics on four symbols built on this cycle.

Rigorous verification of cocoon bifurcations in the Michelson system

We prove the existence of cocoon bifurcations for the Michelson system where and 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.

Uniformly Hyperbolic Attractor of the Smale–Williams Type for a Poincaré Map in the Kuznetsov System

We propose a general algorithm for computer assisted verification of uniform hyperbolicity for maps which exhibit a robust attractor. The method has been successfully applied to a Poincare map for a system of coupled nonautonomous van der Pol oscillators. The model equation has been proposed by Kuznetsov [Phys. Rev. Lett., 95 (2005), paper 144101], and the attractor seems to be of the Smale–Williams type.

Symmetric Heteroclinic Connections in the Michelson System: A Computer Assisted Proof ∗

In this paper we present a new technique of proving the existence of an infinite number of symmet- ric heteroclinic and homoclinic solutions. This technique combines the covering relations method introduced by Zgliczynski (Topol. Methods Nonlinear Anal., 8 (1996), pp. 169-177; Nonlinearity ,1 0 (1997), pp. 243-252) with symmetry properties of a dynamical system. As an example we present a computer assisted proof of the existence of an infinite number of heteroclinic connections be- tween equilibrium points in the Kuramoto-Sivashinsky ODE (D. Michelson, Phys. D, 19 (1986), pp. 89-111). Moreover, we present the proof of the existence of an infinite number of heteroclinic connections between periodic orbits and equilibrium points. 1. Introduction. The aim of this paper is to present a new method for proving the ex- istence of symmetric homoclinic or heteroclinic solutions in systems possessing the reversing symmetry property. In (5) (and references given there) a method of proving the existence of time reversing symmetric homoclinic and heteroclinic solutions for dynamical systems is presented and it is called the fixed set iteration method. It applies to dynamical systems with continuous and discrete time. The basic idea of such a method is to search for the points u which are invariant under the symmetry and whose trajectories converge to an equilibrium point or a periodic orbit. This allows us to conclude that the trajectories of the points u must be homoclinic or heteroclinic. Galias and Zgliczynski (3) presented the method for proving the existence of homoclinic and heteroclinic solutions for maps R 2 → R 2 . This result was applied to the planar circular restricted three body problem (1, 16), where the existence of an infinite number of homoclinic and heteroclinic connections between periodic orbits was shown. In this paper we demonstrate how to combine these two methods for proving the existence of symmetric homoclinic or heteroclinic orbits in systems possessing the reversing symmetry property. Moreover, we present some generalization of the Galias-Zgliczynski method. We show how to prove the existence of heteroclinic orbits between objects possessing unequal dimensions—for example, the equilibrium points and periodic orbits.

Computer Assisted Proof of the Existence of Homoclinic Tangency for the Hénon 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.

Period Doubling in the Rössler System—A Computer Assisted Proof

Using rigorous numerical methods, we validate a part of the bifurcation diagram for a Poincaré map of the Rössler system (Rössler in Phys. Lett. A 57(5):397–398, 1976)—the existence of two period-doubling bifurcations and the existence of a branch of period two points connecting them. Our approach is based on the Lyapunov–Schmidt reduction and uses the Cr-Lohner algorithm (Wilczak and Zgliczyński, available at http://www.ii.uj.edu.pl/~wilczak) to obtain rigorous bounds for the Rössler system.

Coexistence and dynamical connections between hyperchaos and chaos in the 4D Rössler system: A computer-assisted proof

It has recently been reported [P. C. Reich, Neurocomputing, 74 (2011), pp. 3361--3364] that it is quite difficult to distinguish between chaos and hyperchaos in numerical simulations which are frequently “noisy.” For the classical four-dimensional (4D) Rossler model [O. E. Rossler, Phys. Lett. A, 71 (1979), pp. 155--157] we show that the coexistence of two invariant sets with different nature (a global hyperchaotic invariant set and a chaotic attractor) and heteroclinic connections between them give rise to long hyperchaotic transient behavior, and therefore it provides a mechanism for noisy simulations. The same phenomena is expected in other 4D and higher-dimensional systems. The proof combines topological and smooth methods with rigorous numerical computations. The existence of (hyper)chaotic sets is proved by the method of covering relations [P. Zgliczynski and M. Gidea, J. Differential Equations, 202 (2004), pp. 32--58]. We extend this method to the case of a nonincreasing number of unstable directio...

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

An algorithm for validated computation of monodromy matrices for ODEs is provided.Smaller truncation error allows larger time steps making the computation faster.The existence of a chaotic and hyperbolic set for the Rossler system is proved via computer-assisted proofs techniques. 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 Zgliczynski 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 Rossler 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.

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 a specific type of connecting orbits in nonautonomous singular ODEs. We apply the method to a certain singular nonautonomous real Ginzburg--Landau type equation, which arises from the problem of formation of spots in the Swift--Hohenberg equation.

Systematic Computer-Assisted Proof of Branches of Stable Elliptic Periodic Orbits and Surrounding Invariant Tori

We present a concurrent algorithm for rigorous validation of the existence of continuous branches of stable elliptic fixed points for area-preserving planar maps. The method utilizes a classical theorem of Siegel and Moser combined with computed-assisted estimation of higher order derivatives of maps, continuation along the parameter range, and concurrent scheduling of tasks. We apply the algorithm to certain exemplary Poincare maps coming from reversible or Hamiltonian systems: the periodically forced pendulum equations, the Michelson system, and the Henon--Heiles Hamiltonian. Moreover, our algorithm provides at once a computer-assisted proof of the existence of wide branches of stable elliptic periodic solutions and the existence of invariant tori surrounding them.