Transport phenomena and anomalous diffusion in conservative systems of low dimension

Abstract

Apart from this introductory chapter, the contents of the thesis is splitted among four more chapters. Chapters 2, 3 and 4 deal with the planar case, while chapter 5 deals with the 3D volume preserving case. More specifically,

- In Chap. 2 we start by considering conservative quadratic Hénon maps (both orientation preserving and orientation reversing cases). First, we study the main features of the domain of stability of these two maps, mainly from the point of view of the area that they occupy, and how it does evolve as parameters change. To be as exhaustive as possible, we review the theory that allows to explain what one can observe in the phase space of these maps. We finish the chapter by considering the Chirikov standard map (1.13) in the 2-torus T2 for large values of the parameter, k ≥ 1. The most prominent sources of regular area in this setting are accelerator modes that appear periodically in k, and scaled somehow. We give numerical evidence of such a scaling, and guided by the experimental evidence, we derive limit representations for the dynamics in some compact set containing these islands, which turn out to be conjugated to the orientation preserving quadratic Hénon map or conjugated to the square of the orientation reversing quadratic Hénon map. Some of these islands are the accelerator modes we checked that appeared in Sect. 1.3. This motivates the following chapter.

- Chap. 3 is devoted to study the role of these islands of stability that ’jump’ when the standard map is considered in the cylinder. The stability domain of these islands is determined and studied independently from the standard map Mk in Chap. 2, and is recovered in some regions in the phase space of Mk under suitable scalings. We focus in two main observables: the squared mean displacement of the action under iteration of Mk and the trapping time statistics. We study them both in an adequate range of the parameters, where we can see the effect of considering more and more iterations and the fact that we change parameters and the size of the gaps of a Cantorus change. We provide evidence of the fact that the trapping time statistics behave as the superposition of the effect of two distinctive phenomena: the one of the stickiness, detected as power-law statistics, and the one of the outermost Cantorus, detected as bumps. These bumps change their position in the time axis accordingly to the change of the size of the largest gap in the Cantorus.

First, assuming that the stickiness effect gives rise to power law statistics with a certain value of the exponent, and under some other mild conditions (that also are suggested by the simulations), we are able to give a lower bound on the growth of the mean squared displacement of the actions. This is the way these two phenomena are related to each other in this context.

Then, the fact that we can identify the source of the bumps as being due to the effect of the outermost Cantorus, motivates the topic of the next chapter: studying this effect by its own in a proper context.

- In Chap. 4 we return to the Chirikov standard map, but for values of the parameter close to the destruction of the last RIC, that is, for value of the parameter close but larger than k(G) and approaching it from above. In this setting, we study escape rates across this Cantorus, and we deal with this problem from two different points of view.

First, as k decreases to k(G). In this setting, it is known that the mean escape ratio

across the Cantorus, that we will denote as hNki, behaves essentially as (k−kG)−B,B ≈ 3. The Greene-MacKay renormalisation theory, and the interpretation of DeltaW as an area justify that, in fact, hNki (k − kG)B should eventually be periodic in a suitable logarithmic scale, as k → kG. In this chapter we give the first evidence of the shape of this periodic behaviour, and perform a numerical study of a region surrounding the Cantorus that allows to give a first (partial) explanation of it.

Second, we consider a problem related to the previous topic but for each fixed value of k: the probability that an orbit crosses the Cantorus in a prescribed time. We explain how to compute these statistics, and we show that in logarithmic scale in the number of iterates, as k → kG, they seem to behave the same way, but shifted in this log-scale in time.

- Finally, Chap. 5 is devoted to study the stickiness problem in the 3D volume preserving setting. To do so, a map inspired in the Standard map is constructed following the scheme in Sect. 1.3. This map depends on various parameters, one of them, say ε, being a distance-to-integrable one. The map is considered in such a way that:

1. Invariant tori subsist until moderate values of ε, and

2. At integer values of the parameter the origin becomes an accelerator mode, and that exactly at integer values it undergoes a Hopf-Saddle-Node bifurcation, giving rise to a stability bubble.

The normal form of the unfolding of this bifurcation justifies that, in fact, there are just two relevant parameters (since it is a co-dimension 2 bifurcation). An analysis inspired in that of Chap. 3 is performed by fixing one of them. Also in this case one can observe a power law decay of the trapping time statistics, but with slightly different values of the exponent in different ranges of the number of iterates. Preliminary results of more massive simulations seem to indicate that the effect decreases as the number of iterates increases.