Using rigorous numerical methods we validate a part of the bifurcation diagram for a Poincaré map of the Rössler system [26] 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 LyapunovSchmidt reduction and uses the C-Lohner algorithm [31] to obtain rigorous bounds for the Rössler system.

We revisit the classical Poincaré inequality on closed surfaces, and prove its natural analogue for quadratic differentials. In stark contrast to the classical case, our inequality does not degenerate when we work on hyperbolic surfaces that themselves are degenerating, and this fact turns out to be essential for applications to the Teichmüller harmonic map flow.

We construct a rigorous renormalisation scheme for analytic vector fields on T of Poincaré type. We show that iterating this procedure there is convergence to a limit set with a “Gauss map” dynamics on it, related to the continued fraction expansion of the slope of the frequencies. This is valid for diophantine frequency vectors. Email: [email protected]

In this paper we propose an improved method for calculating Hénon’s stability parameter, which is based on the differential of the Poincaré map using the first variational equation. We show that this method is very accurate and give some examples where it gives correct results, while the previous method could not cope.

We investigate the bifurcations and basins of attraction in the Bogdanov map, a planar quadratic map which is conjugate to the Hénon area-preserving map in its conservative limit. It undergoes a Hopf bifurcation as dissipation is added, and exhibits the panoply of mode locking, Arnold tongues, and chaos as an invariant circle grows out, finally to be destroyed in the homoclinic tangency of the ...

This work is concerned with planar real analytic differential systems with an analytic inverse integrating factor defined in a neighborhood of a regular orbit. We show that the inverse integrating factor defines an ordinary differential equation for the transition map along the orbit. When the regular orbit is a limit cycle, we can determine its associated Poincaré return map in terms of the in...

Marsden and Scheurle [1993] studied Lagrangian reduction in the context of momentum map constraints—here meaning the reduction of the standard Euler-Lagrange system restricted to a level set of a momentum map. This provides a Lagrangian parallel to the reduction of symplectic manifolds. The present paper studies the Lagrangian parallel of Poisson reduction for Hamiltonian systems. For the reduc...

Domain decomposition methods are designed to deal with coupled or transmission problems for partial differential equations. Since the original boundary value problem is replaced by local problems in substructures, domain decomposition methods are well suited for both parallelization and coupling of different discretization schemes. In general, the coupled problem is reduced to the Schur complem...