Research paper
A new method to compute periodic orbits in general symplectic maps

https://doi.org/10.1016/j.cnsns.2021.105838Get rights and content

Highlights

  • Parameterization method used to find periodic orbits without use of symmetries.

  • Newton method coupled with a Gaussian reduction used to refine periodic orbits.

  • Renormalization theory results confirmed for mappings without the use of symmetries.

Abstract

The search of high-order periodic orbits has been typically restricted to problems with symmetries that help to reduce the dimension of the search space. Well-known examples include reversible maps with symmetry lines. The present work proposes a new method to compute high-order periodic orbits in twist maps without the use of symmetries. The method is a combination of the parameterization method in Fourier space and a Newton–Gauss multiple shooting scheme. The parameterization method has been successfully used in the past to compute quasi-periodic invariant circles. However, this is the first time that this method is used in the context of periodic orbits. Numerical examples are presented showing the accuracy and efficiency of the proposed method. The method is also applied to verify the renormalization prediction of the residues’ convergence at criticality (extensively studied in reversible maps) in the relatively unexplored case of maps without symmetries.

Introduction

The systematic study of dynamical systems with low number of degrees of freedom started at the beginning of last century with the discovery of heteroclinic phenomena in periodic orbits. Poincaré and Birkhoff studied in detail the existence of periodic orbits in discrete dynamical systems, in particular twist maps that preserve area. The existence and uniformity of these periodic orbits with respect to their rotation number allowed their use as a tool to analyze other invariant objects that persist in almost integrable maps. From the results obtained by Birkhoff [1] it was possible to implement a procedure to approximate invariant tori and determine their properties. Later on, with the arrival of digital computers it was possible to implement search algorithms for these invariant objects through periodic orbits of rotation numbers that approximate the invariant object. Studies from the 60’s and 70’s, including Chirikov [2], Greene [3], Kadanoff [4] and others obtained interesting results relying on numerical experiments performed on increasingly more powerful computers. In particular, Greene’s residue criterion [3], arguably the most accurate and most used method to determine the persistence or destruction of invariant circles, relies on finding high order periodic orbits for critical parameter values.

As a result from these studies, the renormalization theory of twist maps was introduced [5] and it was possible to show the existence of universal behavior for parameters close to the destruction of invariant curves [4], [6]. All these results were obtained with the help of periodic orbits and, as the numerical capabilities increased, the numerical experiments were performed with periods reaching orders of tens of millions and limited only by the arithmetic precision used. This opened the possibility of numerically study in detail the scenarios predicted by the renormalization theory and the Aubry–Mather theory.

However, the numerical study of periodic orbits in area preserving twist maps has been mostly limited to a particular kind of maps known as reversible maps. The fundamental property of reversible maps is that they can be written as the composition of two involutions which are maps with the property that their composition with themselves is the identity. The invariant sets of involution maps are usually one-dimensional sets known as symmetry lines. Using the theory of DeVogelaere [3], [7] it can be proved that two iterates of every symmetric periodic orbit lay in these invariant sets. This result allows to simplify the search of periodic orbits with the use of one-dimensional methods (e.g., 1D-quasi-Newton methods). The robust and stable behavior of these 1D methods allows the computation of periodic orbits of very high order, up to 107, with an accuracy limited only by machine precision Ref. [8]. This is one of the reasons why only in reversible maps, and in particular the standard map introduced by Chirikov and Taylor [1], [2], [9], [10], that it has been possible to study critical dynamical phenomena in detail. It is important to point out that reversibility is independent of the twist condition and that there are non-twist maps that are reversible, an extensively studied example being the standard non-twist map Refs. [11], [12], [13], [14], [15], [16]. Like in the standard map, the use of involutions and symmetry lines in the non-twist map have allowed the possibility of studying in great detail the criticality and renormalization of the destruction of shearless invariant circles using periodic orbits of very high order.

Unfortunately many dynamical systems that can be reduced to area preserving maps can not be studied in the same way as reversible maps either because the lack of knowledge or the actual impossibility of writing them as products of involutions. In this case, the search for periodic orbits must be done in two or more dimensions and, as it is well known, this might compromise the convergence of Newton methods because they typically exhibit very small, in some cases fractal, basins of attraction. This problem is exacerbated in high dimensional maps an example being the 4D Froeschlé map for which the computation of periodic orbits is typically limited to relatively low periods [17], [18].

This paper presents a new numerical method to find periodic orbits in general area preserving twist maps without the use of symmetries. Among the physics motivations for this study is our recent work on self-consistent transport phenomena in a reduced plasma physics model consisting of a large number of standard-like maps coupled by a mean-field [19], [20]. The study of the global stability properties of this model required the finding of high order periodic orbits in nonautonomous maps without known symmetries.

Our methodology is based on the implementation of the parameterization method developed by de la Llave and Calleja [21], [22] based on Ref. [23] (see also Ref. [24]), that allows the approximation of continuous invariant objects, e.g. invariant circles, in two-dimensional twist maps. The numerical implementation of the method is based on the computation of Fourier modes of an invariant circle with a fixed rotation number and it provides lower bounds for the critical values of the parameters for which the invariant circle exists as a continuous set. Among the advantages of this method is that it provides a suitable change of coordinates that allows to reduce the number of operations from O(N2) to O(Nlog(N)), where N is the number of points used to represent the invariant curve. This favorable scaling enables the computation of periodic orbits with periods of the order O(107). The method proposed in the present paper consists of two steps. In the first step, a modified version of the parameterization method is used to obtain a suitable seed for the use in the second step consisting of a refinement based on the use of a Newton–Gauss method.

The rest of the paper is organized as follows. Section 2 presents review material and introduces the rational harmonic map that will be used in the numerical examples. Section 3 discusses the main ingredients of the standard parameterization method introduced in Refs. [23], [25]. Section 4 presents the new compound parameterization method for the computation of periodic orbits. Section 5 discusses the numerical implementation of the proposed method in the context of the standard map and the rational harmonic map including a comparison with the renormalization theory prediction of the residues’ convergence at criticality. In Section 6 we present a summary discussion and possible future applications of the new method.

Section snippets

Preliminaries

This section reviews basic definitions and concepts, further details can be found in standard references including Refs. [1], [26], [27]. The maps of interest in the present work are symplectic diffeomorphisms on the cylinder, T:S×RS×R. A periodic orbit with rotation number p/q on the lift of map T, T˜:R2R2, satisfies the relation,T˜q(z0)=z0+P,wherez0R2,P=(p0).

A map T is called reversible [17], [28], [29] if it can be written as the composition, T=I2I1, of two functions I1 and I2 with the

The parameterization method

The parameterization method was originally introduced by de la Llave et al.[23] to find an approximate conjugation function between an invariant torus and the rigid rotation over a ideal torus. The rationale of the method is best understood in the constructive proof of the KAM theorem in Ref. [23], which relies among other things on a Newton iteration in the spirit of Nash–Moser theory, see Ref. [38]. An extra ingredient of the particular implementation described here is related to the area

A new compound method

The proposed compound method consists of the composition of two methods: a modified parameterization method, described in Sections 4.1 and 4.2, and a Newton–Gauss method, described in Section 4.3.

Numerical implementation of the compound method

In this section we discuss the application of the compound method for the computation of periodic orbits in the Chirikov–Taylor standard map in Eq. (4) and the rational harmonic map in Eq. (7). Although the periodic orbits in the standard map can be computed very efficiently using symmetry lines, this map is a good starting point to illustrate the implementation and test the accuracy of the new compound method against well-established numerical results. Conceptually speaking, the rational

Discussion

The method proposed in this work can be applied to a wide variety of maps for the efficient and accurate computation of periodic orbits without the use of symmetries to simplify the search. In the cases considered, the only requirements were the symplectic and twist conditions needed to justify some hypothesis of the parameterization method. The method as presented does not include the non-twist case when the periodic orbits are close to a shearless invariant circle, see Refs. [11], [15].

CRediT authorship contribution statement

R. Calleja: Conceptualization, Methodology, Software, Writing - review & editing. D. del-Castillo-Negrete: Visualization, Writing - review & editing. D. Martínez-del-Río: Software, Visualization, Writing - review & editing. A. Olvera: Conceptualization, Methodology, Software, Validation, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was founded by PAPIIT IN112920, IN110317, IA102818 and IN101020, FENOMEC-UNAM and by the Office of Fusion Energy Sciences of the US Department of Energy at Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U.S. Department of Energy under contract DE-AC05-00OR22725. This material is based upon work supported by the National Science Foundation under Grant no. DMS-1440140 while R.C. and D.M. were in residence at the Mathematical Sciences Research Institute in Berkeley,

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    Current address: Mathematics Institute, University of Warwick

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