A phase transition from integrability to nonintegrability in two-dimensional Hamiltonian mappings is described and characterized in terms of scaling arguments. The mappings considered produce a mixed structure in the phase space in the sense that, depending on the combination of the control parameters and initial conditions, KAM islands which are surrounded by chaotic seas that are limited by invariant tori are observed. Some dynamical properties for the largest component of the chaotic sea are obtained and described in terms of the control parameters. The average value and the deviation of the average value for chaotic components of a dynamical variable are described in terms of scaling laws, therefore critical exponents characterizing a scaling function that describes a phase transition are obtained and then classes of universality are characterized. The three models considered are: The Fermi-Ulam accelerator model, a periodically corrugate waveguide, and variant of the standard nontwist map.

Dynamical systems described by two-dimensional nonlinear area preserving mappings have been considered for many years as prototype for the study of low-dimensional physical systems. Applications of the description of dynamical systems by mappings are observed in the study of magnetic field lines in toroidal plasma devices with reversed shear (like tokamaks), channel flows, waveguide, Fermi acceleration and many other [

In this paper we consider a special class of two-dimensional area preserving mappings exhibiting a phase transition from integrable to nonintegrable. For certain values of the control parameter, the phase space of the system has only periodic and quasiperiodic orbits, thus the phase space is filled by straight lines. As this control parameter varies, the phase space exhibits a mixed form containing both Kolmogorov-Arnold-Moser (KAM) islands surrounded by a chaotic sea (conservative chaos) and a set of invariant spanning curves limiting the size of the chaotic sea. This form of the phase space is generic for nondegenerate Hamiltonian systems [

The main approach considered in this paper is the description of the behavior of some average dynamical variables as a function of the control parameter, like the first momenta and the deviation around the average variable. This characterization is made along the chaotic sea near a transition from integrable to nonintegrable. This formalism furnishes critical exponents that can be used to define classes of universality.

This paper is organized as follows. In Section

In this section we present and discuss some dynamical properties for a two-dimension mapping. We assume that there is a two-dimensional integrable system that is slightly perturbed. Then, the Hamiltonian function that describes the system is written as

Since the mapping (

for the case of

considering

In this section, we shall consider the following expression for the two-dimensional area-preserving mapping:

Phase space generated by mapping (

The fixed points of the mapping (

We can see in Figure

Behavior of

The behavior shown in Figure

For low iterations and after a short initial transient, that is,

For large enough

The number of iterations which characterizes the changeover from growth to the saturation is given

Plotting the behavior of

Plot of: (a)

Rescaling properly, we can see that all curves coalesce together, as they are shown in Figure

Collapse of different curves of

Let us now discuss in this section some properties of a periodically corrugated waveguide. This section is a review of papers [

There are many different ways to describe problems involving waveguides. One of them consists in considering the well-known billiard approach (A billiard problem consists of a system in which a point-like particle moves freely inside a bounded region and suffers specular reflections with the boundaries.). Generally, the dynamics is described using the formalism of discrete maps. Therefore, depending on the combinations of the control parameters as well as on the initial conditions, the phase space for such mappings might be included in three different classes namely: (i) regular, (ii) ergodic, and (iii) mixed. Roughly speaking, the integrability of the regular cases is generally related to the angular momentum preservation and the static circular billiard is a typical example. On the other hand, for the completely ergodic billiards, only chaotic and therefore unstable periodic orbits are present in the dynamics. The well-known Bunimovich stadium [

In this section, scaling arguments are used to describe the behavior of the deviation of the average reflection angle within the chaotic sea, for the problem of a

Let us discuss now all the details needed to describe the model and obtain the equations that describe the dynamics of the problem. The model thus consists in obtaining a mapping

(a) Reflection from the corrugated surface of a light ray coming from the flat surface at

The relative corrugation is assumed to be small, so that

In the limit of small corrugation, it is also assumed that

Let us now argue on the approximation of small relative corrugation. It is easy to see that, for

It is shown in Figure

Phase space for the mapping (

The positive Lyapunov exponent obtained via triangularization algorithm for the control parameter

Let us now discuss one of the consequences of the variation of the control parameter

This section is devoted to discuss a scaling property which is present in the chaotic sea. As it was discussed in Section

(a) Plot of

(i) For a short iteration number, say

(ii) For large enough iteration number, say

(iii) The number of iterations that characterizes the crossover, that is, the iterations that marks the change from growth to the saturation is written as

This scaling formalism is commonly used in the description of phase transitions in critical phenomena. It is also very useful in studies of problems of surface sciences (see, e.g., [

Plot of (a)

Given that the values of the critical exponents are now obtained, the scaling hypotheses can be verified. In this sense, it is shown in Figure

(a) Different curves of

In this section we will use the formalism described in Section

Illustrative picture of the one-dimensional Fermi accelerator model.

The phase space generated for iteration of the mapping (

Phase space for the mapping (

It is shown in Figure

(a) Behavior of

The exponent

(a) Plot of

(a) Evolution of

Since the critical exponents are the same of those obtained for the corrugated waveguide, we can conclude that the one-dimensional Fermi accelerator model belongs to the same class of universality of the corrugated waveguide and experiences the same transition from integrability to nonintegrability when the control parameter

In summary, we have studied a phase transition from integrability to nonintegrability in three different two-dimensional mappings. Critical exponents for the behavior of the chaotic time series were obtained and used to define classes of universality. Even though the mappings are similar in their forms, the models discussed in Sections

Financial support from CNPq, FAPESP and FUNDUNESP, Brazilian agencies are gratefully acknowledged. The present paper contains totally new results, those shown in Section