Abstracts of contributed presentations

Computation of lyapunov exponents via spatial integration with application to blowout bifurcations

A new method for the numerical approximation of the largest Lyapunov exponent is described, based on the computation of a spatial average with respect to an underlying (natural) invariant measure rather than on a long term simulation of the dynamical system. This approach is particularly advantageous for the detection of blowout bifurcations of a synchronous chaotic state, and we illustrate this for a system of two coupled Duffing oscillators.

Numerically obtaining breathers and homoclinic orbits

A numerical method is presented for obtaining all homoclinic orbits of invertible maps in any (finite) dimension. This method is then used to obtain breather solutions of 1-dimensional Klein-Gordon and FPU lattices, using a new numerical scheme generalizing a work by G.P. Tsironis.

Stability of large-scale ocean circulation

Transitions between different ocean circulation states are one of the potential sources of (past) climate change. In this presentation, an overview will be given of theoretical developments to determine the sensitivity of the present ocean circulation to perturbations. Most of these results have been obtained from numerical bifurcation analyses of a hierarchy of ocean models. Focus of the talk will be on the methods to compute branches of steady states for the large-dimensional (up to 100,000 degrees of freedom) dynamical systems, to determine the stability of these steady states and to handle details such as continental geometry and bottom topography.

Practical continuation of invariant subspaces for bifurcations problems

We present a practical version of the Continuation of Invariant Subspaces (CIS) algorithm for a parameter dependent matrix A(s). It includes robust procedures for updating, automatically, the invariant subspace of interest in the generic situations of bifurcations and when the eigenvalue set from it coalesces and/or overlaps with the eigenvalue set from its complement during the continuation process. As an application, we consider stability analysis of a simulation model of the single stage to orbit reusable launch vehicle called the X-33. The key problem is choosing the points in flight that capture critical events which affect stability and performance of the vehicle.

The principle of Least Action and the figure-8 solution of the three body problem.

The stability properties and the minimizing character of the recently discovered figure 8 solution of the three body problem with equal masses in celestial mechanics has renewed the interest in variational methods in dynamical systems. In this work we apply a numerical continuation method valid for periodic orbits, relative equilibria and relative periodic orbits in symmetric Hamiltonian to this remarkable solution. We investigate the local and global bifurcation behaviour of this orbit and try to unravel the puzzling properties of the real minimizer of the action by studying the second variation of the functional. This study is a benchmark for a more general scheme to continue periodic orbits and relative periodic orbits in symmetric Hamiltonian systems.

Stability problems in periodic delayed systems

Joint work with Gábor Stépán & Róbert Szalai.

The stability of the delayed Mathieu equation is investigated as a basic problem of periodic delayed systems. The bifurcation types are identified according to the location of the relevant characteristic multipliers. The dynamic behaviour of the milling process is investigated analytically and experimentally. Period one, period two (flip) and secondary Hopf bifurcations are identified.

A Newton-Picard iterative solver for collocation for delay differential equations

When using Gauss-Legendre collocation to compute periodic solutions of DDEs, the linearised equations in Newton's method have less structure than in the ODE case. As a result, direct solvers are not always efficient. In this talk, we will present a Newton-Picard iterative method to solve the collocation equations. This method simultaneously computes a periodic solution and its dominant Floquet multipliers.

Spectral analysis of optical bistability in finite and semi-infinite nonlinear photonic gratings

Joint work with Arnd Scheel (University of Minnesota). 

We study optical bistability in nonlinear periodic structures of finite and semi-infinite length. For finite-length structures, the system exhibits instability mechanisms typical for dissipative dynamical systems like delay-differential equations. We show that the Leray-Schauder degree equals the sign of the Evans function in = 0. Using the Evans function methods, we rigorously prove that the stationary solutions with optically bistable transmission characteristics are indeed spectrally unstable. For semi-infinite structures, the system exhibits two pulse solutions, when the infinite Hamiltonian dynamical system has soliton solutions, and one pulse solution, otherwise. We show that the non-monotonic pulse solution is always spectrally unstable. Numerical computations of the linear stability problem and time-dependent coupled-mode equations confirm the analytical results.