Piotr Zgliczyński

C 1 Lohner Algorithm

Abstract. We present a modification of the Lohner algorithm for the computation of rigorous bounds for solutions of ordinary differential equations together with partial derivatives with respect to initial conditions. The modified algorithm requires essentially the same computational effort as the original one. We applied the algorithm to show the existence of several periodic orbits for Rössler equations and the 14-dimensional Galerkin projection of the Kuramoto—Sivashinsky partial differential equation.

Rigorous Numerics for Partial Differential Equations: The Kuramoto—Sivashinsky Equation

Abstract We present a new topological method for the study of the dynamics of dissipative PDEs. The method is based on the concept of the self-consistent a priori bounds, which permit the rigorous justification of the use of Galerkin projections. As a result, we obtain a low-dimensional system of ODEs subject to rigorously controlled small perturbation from the neglected modes. To these ODEs we apply the Conley index to obtain information about the dynamics of the PDE under consideration. We applied the method to the Kuramoto—Sivashinsky equation

Fixed point index for iterations of maps, topological horseshoe and chaos

u_t = \left( {u^2 } \right)_x - u_{xx} - vu_{xxxx} , u(x,t) = u(x + 2\pi , t), u(x,t) = - u( - x,t).

Computer assisted proof of chaos in the Lorenz equations

We obtained a computer-assisted proof of the existence of several fixed points for various values of ν > 0 .

Rigorous Numerics for Dissipative Partial Differential Equations II. Periodic Orbit for the Kuramoto–Sivashinsky PDE—A Computer-Assisted Proof

We introduce horseshoe-type mappings which are geometrically similar to Smales horseshoes. For such mappings we prove by means of the fixed point index the existence of chaotic dynamics - the semi-conjugacy to the shift on a finite number of symbols. Our theorem does not require any assumptions concerning derivatives, it is a purely topological result. The assumptions of our theorem are then rigorously verified by computer assisted computations for the classical Henon map and for classical Rossler equations.

Abundance of homoclinic and heteroclinic orbits and rigorous bounds for the topological entropy for the Hénon map

There are many examples of complicated or chaotic dynamics, but the set of examples for which chaos has been rigorously demonstrated is quite small. In most cases where chaotic dynamics has been proven, the strategy has involved analysing a simple singular map or integrable problem and then perturbing the results (see [2], [5]). This usually required some estimates on the derivatives of mappings under consideration. Another strategy to tackle such problems is to appropriately homotope the given system to a model problem for which some algebraic invariants could be explicitly computed and show that these invariants remain unchanged. Nontriviality of the algebraic invariant provides a minimal description of the complexity of the dynamics of the system. In [3], [4] with the help of the discrete Conley index introduced in [6], this strategy has been applied to the Henon map and the Lorenz equations. In applying this strategy to a concrete problem we must answer three closely related questions: what algebraic invariants we will use, what is the model map, what are the appropriate homotopies.

The existence of simple choreographies for the N-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

Abstract In this paper we prove with computer assistance the existence of chaos in a suitable Poincare map generated by the Lorenz system of equations. By chaos we mean the existence of symbolic dynamics with infinite number of periodic trajectories. The proof combines abstract results based on the fixed point index and finite rigorous computer calculations. Discussion concerning numerical algorithms is also included.