Approximate renormalization for the break-up of invariant tori with three frequencies

We construct an approximate renormalization transformation for Hamiltonian systems with three degrees of freedom in order to study the break-up of invariant tori with three incommensurate frequencies which belong to the cubic field $Q(\tau)$, where $\tau^3+\tau^2-2\tau-1=0$. This renormalization has two fixed points~: a stable one and a hyperbolic one with a codimension one stable manifold. We compute the associated critical exponents that characterize the universality class for the break-up of the invariant tori we consider.Comment: 5 pages, REVTe

Incomplete Dirac reduction of constrained Hamiltonian systems

First-class constraints constitute a potential obstacle to the computation of a Poisson bracket in Dirac's theory of constrained Hamiltonian systems. Using the pseudoinverse instead of the inverse of the matrix defined by the Poisson brackets between the constraints, we show that a Dirac-Poisson bracket can be constructed, even if it corresponds to an incomplete reduction of the original Hamiltonian system. The uniqueness of Dirac brackets is discussed

Determination of the threshold of the break-up of invariant tori in a class of three frequency Hamiltonian systems

We consider a class of Hamiltonians with three degrees of freedom that can be mapped into quasi-periodically driven pendulums. The purpose of this paper is to determine the threshold of the break-up of invariant tori with a specific frequency vector. We apply two techniques: the frequency map analysis and renormalization-group methods. The renormalization transformation acting on a Hamiltonian is a canonical change of coordinates which is a combination of a partial elimination of the irrelevant modes of the Hamiltonian and a rescaling of phase space around the considered torus. We give numerical evidence that the critical coupling at which the renormalization transformation starts to diverge is the same as the value given by the frequency map analysis for the break-up of invariant tori. Furthermore, we obtain by these methods numerical values of the threshold of the break-up of the last invariant torus.Comment: 18 pages, 4 figure

Envelope-driven recollisions triggered by an elliptically polarized laser pulse

Increasing ellipticity usually suppresses the recollision probability drastically. In contrast, we report on a recollision channel with large return energy and a substantial probability, regardless of the ellipticity. The laser envelope plays a dominant role in the energy gained by the electron, and in the conditions under which the electron comes back to the core. We show that this recollision channel eciently triggers multiple ionization with an elliptically polarized pulse

Strange attractor for the renormalization flow for invariant tori of Hamiltonian systems with two generic frequencies

We analyze the stability of invariant tori for Hamiltonian systems with two degrees of freedom by constructing a transformation that combines Kolmogorov-Arnold-Moser theory and renormalization-group techniques. This transformation is based on the continued fraction expansion of the frequency of the torus. We apply this transformation numerically for arbitrary frequencies that contain bounded entries in the continued fraction expansion. We give a global picture of renormalization flow for the stability of invariant tori, and we show that the properties of critical (and near critical) tori can be obtained by analyzing renormalization dynamics around a single hyperbolic strange attractor. We compute the fractal diagram, i.e., the critical coupling as a function of the frequencies, associated with a given one-parameter family.Comment: 10 pages, 5 figure

Kolmogorov-Arnold-Moser Renormalization-Group Approach to the Breakup of Invariant Tori in Hamiltonian Systems

We analyze the breakup of invariant tori in Hamiltonian systems with two degrees of freedom using a combination of KAM theory and renormalization-group techniques. We consider a class of Hamiltonians quadratic in the action variables that is invariant under the chosen KAM transformations, following the approach of Thirring. The numerical implementation of the transformation shows that the KAM iteration converges up to the critical coupling at which the torus breaks up. By combining this iteration with a renormalization, consisting of a shift of resonances and rescalings of momentum and energy, we obtain a much more efficient method that allows to determine the critical coupling with high accuracy. This transformation is based on the physical mechanism of the breakup of invariant tori. We show that the critical surface of the transformation is the stable manifold of codimension one of a nontrivial fixed point, and we discuss its universality properties.Comment: 9 pages, 5 figures, RevTe