Stability of the phase motion in race-track microtrons

We model the phase oscillations of electrons in race-track microtrons by means of an area preserving map with a fixed point at the origin, which represents the synchronous trajectory of a reference particle in the beam. We study the nonlinear stability of the origin in terms of the synchronous phase —the phase of the synchronous particle at the injection. We estimate the size and shape of the stability domain around the origin, whose main connected component is enclosed by an invariant curve. We describe the evolution of the stability domain as the synchronous phase varies. We also clarify the role of the stable and unstable invariant curves of some hyperbolic (fixed or periodic) points.Peer ReviewedPostprint (author's final draft

Rotation Vectors for Torus Maps by the Weighted Birkhoff Average

In this paper, we focus on distinguishing between the types of dynamical behavior that occur for typical one- and two-dimensional torus maps, in particular without the assumption of invertibility. We use three fast and accurate numerical methods: weighted Birkhoff averages, Farey trees, and resonance orders. The first of these allows us to distinguish between chaotic and regular orbits, as well as to calculate the frequency vectors for the regular case to high precision. The second method allows us to distinguish between the periodic and quasiperiodic orbits, and the third allows us to distinguish among the quasiperiodic orbits to determine the dimension the resulting attracting tori. We first consider the well-studied Arnold circle map, comparing our results to the universal power law of Jensen and Ecke. We next consider quasiperiodically forced circle maps, inspired by models introduced by Ding, Grebogi, and Ott. We use the Birkhoff average to distinguish between "strong" chaos (positive Lyapunov exponents) and "weak" chaos (strange nonchaotic attractors). Finally, we apply our methods to 2D torus maps, building on the work of Grebogi, Ott, and Yorke. We distinguish incommensurate, resonant, periodic, and chaotic orbits and accurately compute the proportions of each as the strength of the nonlinearity grows. We compute generalizations of Arnold tongues corresponding to resonances and to periodic orbits, and we show that chaos typically begins before the map becomes noninvertible. We show that the proportion of nonresonant orbits does not obey a universal power law like that seen in the 1D case.Comment: Keywords: Circle maps, Quasiperiodic forcing, Arnold tongues, Resonance, Birkhoff averages, Strange nonchaotic attractor

Birkhoff Averages and the Breakdown of Invariant Tori in Volume-Preserving Maps

In this paper, we develop numerical methods based on the weighted Birkhoff average for studying two-dimensional invariant tori for volume-preserving maps. The methods do not rely on symmetries, such as time-reversal symmetry, nor on approximating tori by periodic orbits. The rate of convergence of the average gives a sharp distinction between chaotic and regular dynamics and allows accurate computation of rotation vectors for regular orbits. Resonant and rotational tori are distinguished by computing the resonance order of the rotation vector to a given precision. Critical parameter values, where tori are destroyed, are computed by a sharp decrease in convergence rate of the Birkhoff average. We apply these methods for a threedimensional generalization of Chirikov&rsquo;s standard map: an angle-action map with two angle variables. Computations on grids in frequency and perturbation amplitude allow estimates of the critical set. We also use continuation to follow tori with fixed rotation vectors. We test three conjectures for cubic fields that have been proposed to give locally robust invariant tori, but are not able to provide compelling evidence that one of these three fields is more robust than the other two.</p

Breakdown of tori in symplectic maps

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2014, Director: Carles SimóMost of physical phenomena can be explained in terms of Hamiltonian systems. These continuous dynamical systems can be related with symplectic maps. Under certain hypothesis one can see that these maps present some invariant tori. Then, it is really interesting to understand how these tori behave. One of the most important properties these tori present is that they persist under small perturbation of our initial systems but that for higher perturbations they are going to break down. These perturbations are usually related with equations depending on a parameter, K. For 2D symplectic maps, renormalization techniques allow to understand the mechanisms concerning the destruction of invariant circles. Rotation numbers of these circles play a key role in the analysis of their breakdown. Throughout this work we will show some of the most important tools to deal with these invariant circles

Quasiperiodicity and Chaos

In this work, we investigate a property called ``multi-chaos'' is which a chaotic set has densely many hyperbolic periodic points of unstable dimension $k$ embedded in it, for at least 2 different values of $k$. We construct a family of maps on the torus having this property. They serve as a paradigm for multi-chaos occurring in higher dimensional systems. One of the factors that leads to this strong form of chaos is the occurrence of a quasiperiodic orbit transverse to an expanding sub-bundle of the tangent bundle. Hence, a key step towards identifying multi-chaos numerically is finding quasiperiodic orbits in high dimensional systems. To analyze quasiperiodic orbits, we develop a method of weighted ergodic averages and prove that these averages have super-polynomial convergence to the Birkhoff average. We also show how this accelerated convergence of the ergodic averages over quasiperiodic trajectories enable us to compute the rotation number, Fourier series and Lyapunov exponents of quasiperiodic orbits with a high degree of precision ($\approx 10^{-30}$)