Synchronization of van der Pol oscillators with delayed coupling

The synchronization of self-sustained oscillators such as the van der Pol oscillator is a model for the adjustment of rhythms of oscillating objects due to their weak interaction and has wide applications in natural and technical processes. That these oscillators adjust their frequency or phase to an external forcing or mutually between several oscillators is a phenomenon which can be used in sound synthesis for various purposes. In this paper we focus on the influence of delays on the synchronization properties of these oscillators. As there is no general theory yet on this topic, we mainly present simulation results, together with some background on the non-delayed case. Finally, the theory is also applied in Neukom’s studies 21.1-21.9

Nekhoroshev theorem for the Dirichlet Toda chain

In this work, we prove a Nekhoroshev theorem for the Toda chain with Dirichlet boundary conditions, i.e., fixed ends. The Toda chain is a special case of a Fermi-Pasta-Ulam (FPU) chain, and in view of the unexpected recurrence phenomena observed numerically in these chains, it has been conjectured that theory of perturbed integrable systems could be applied to these chains, especially since the Toda chain has been shown to be a completely integrable system. Whereas various results have already been obtained for the periodic lattice, the Dirichlet chain is more important from the point of view of applications, since the famous numerical experiments have been performed for this type of system. Mathematically, the Dirichlet chain can be treated by exploiting symmetries of the periodic chain. Precisely, by considering the phase space of the Dirichlet chain as an invariant submanifold of the periodic chain, namely the fixed point set of a certain symmetry of the periodic chain, the results obtained for the periodic chain can be used to obtain similar results for the Dirichlet chain. The Nekhoroshev theorem is a perturbation theory result which does not have the probabilistic character of other results such as those of the KAM theorem

Nekhoroshev stability for the Dirichlet Toda lattice

In this work, we prove a Nekhoroshev-type stability theorem for the Toda lattice with Dirichlet boundary conditions, i.e., with fixed ends. The Toda lattice is a member of the family of Fermi-Pasta-Ulam (FPU) chains, and in view of the unexpected recurrence phenomena numerically observed in these chains, it has been a long-standing research aim to apply the theory of perturbed integrable systems to these chains, in particular to the Toda lattice which has been shown to be a completely integrable system. The Dirichlet Toda lattice can be treated mathematically by using symmetries of the periodic Toda lattice. Precisely, by treating the phase space of the former system as an invariant subset of the latter one, namely as the fixed point set of an important symmetry of the periodic lattice, the results already obtained for the periodic lattice can be used to obtain analogous results for the Dirichlet lattice. In this way, we transfer our stability results for the periodic lattice to the Dirichlet lattice. The Nekhoroshev theorem is a perturbation theory result which does not have the probabilistic character of related theorems, and the lattice with fixed ends is more important for applications than the periodic one

Results on Normal Forms for FPU Chains

In this paper we prove, among other results, that near the equilibirum position, any periodic FPU chain with an odd number N of particles admits a Birkhoff normal form up to order 4, whereas any periodic FPU chain with N even admits a resonant normal form up to order 4. This resonant normal form of order 4 turns out to be completely integrable. Further, for N odd, we obtain an explicit formula of the Hessian of its Hamiltonian at the fixed poin

Nekhoroshev theorem for the periodic Toda lattice

The periodic Toda lattice with $N$ sites is globally symplectomorphic to a two parameter family of $N-1$ coupled harmonic oscillators. The action variables fill out the whole positive quadrant of $\R^{N-1}$. We prove that in the interior of the positive quadrant as well as in a neighborhood of the origin, the Toda Hamiltonian is strictly convex and therefore Nekhoroshev's theorem applies on (almost) all parts of phase space.Comment: 28 page

Global action-angle variables for the periodic Toda lattice

In this paper we construct global action-angle variables for the periodic Toda lattice.Comment: 43 pages, 1 figur