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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 716024, 12 pages
High-Precision Continuation of Periodic Orbits
1Centro Universitario de la Defensa and IUMA, 50090 Zaragoza, Spain
2Departamento de Matemática Aplicada and IUMA, Universidad de Zaragoza, 50009 Zaragoza, Spain
Received 30 December 2011; Accepted 14 February 2012
Academic Editor: Muhammad Aslam Noor
Copyright © 2012 Ángeles Dena et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Obtaining periodic orbits of dynamical systems is the main source of information about how the orbits, in general, are organized. In this paper, we extend classical continuation algorithms in order to be able to obtain families of periodic orbits with high-precision. These periodic orbits can be corrected to get them with arbitrary precision. We illustrate the method with two important classical Hamiltonian problems.
Obtaining periodic orbits in dynamical systems is the main source of information about how the orbits in general are organized, and, in Poincaré’s words, they offer “the only opening through which we might try to penetrate the fortress (chaos) which has the reputation of being impregnable.” The stable periodic orbits explain the dynamics of bounded regular motion , while the unstable periodic orbits in a chaotic set (attractor or saddle) determine its structure  and, thus, the chaotic behavior of the system.
Therefore, it is not surprising the high number of works that use the detection of periodic orbits in the analysis of dynamical systems in many different fields. The analysis and control of chaotic dynamical systems [3, 4], the “scar” theory in quantum mechanics , electrical and magnetic fields , hydrodynamical flows , electrical circuits and optical systems , and astrodynamics [1, 9, 10] are just a few examples.
There are several works developing different numerical algorithms to compute periodic orbits [11–19], but most of them do not work with high-precision. This is a serious difficulty if our problem requires obtaining periodic orbits with many precision digits. Some important examples of this kind of problems are the complex pole location in physics [20–23], the study of the splitting of separatrices (numerical difficulties arise because this phenomenon can exhibit exponentially small splitting) , and the study of the fractal properties of the chaotic attractors . Currently, the only algorithms capable of computing high-precision periodic orbits are based on Lindstedt-Poincaré series  or Taylor series methods . The first one uses very clever algorithms, but it is quite difficult to use in a general setting. Based on the second method, we extend classical continuation algorithms in order to be able to continue families of periodic orbits with high-precision, generating with arbitrary precision the values of those orbits that are considered of special interest. This algorithm is based on the pseudo-arclength continuation method [26, 27], an optimized shooting method with the Newton method [11, 15], the Taylor series method (implemented by means of the free software TIDES [28, 29]), and the singular value decomposition, SVD [30, 31].
The organization of the paper is as follows. In Section 2, we review the correction method developed in  and we show an example to illustrate its use with arbitrary precision. Section 3 describes the continuation method explaining some technical aspects and examples. Finally, in Section 4, we present the combined use of methods described above to take advantage of each one of them.
2. Correction of Periodic Orbits with Arbitrary Precision
Let us consider the differential system: with and , and . And let be the solution of (2.1).
If we have an estimation of the initial conditions of a periodic orbit with estimated period , then . If we call and the corrections to the initial conditions and the period, respectively, then we have the periodicity condition: To obtain an approximation of and , we follow the strategy of the Newton method: we take the multivalued Taylor expansion up to the first order in (2.3) to get linear equations (with unknowns) to satisfy with the identity matrix, and approximations of and , respectively, and given by
At this point, we have to obtain numerically and . Usually, to compute , the variational equations of the differential system (2.1) are formulated and integrated. However, this task is not always easy and frequently cumbersome. Free software TIDES [28, 29] allows to compute simultaneously the solution of (2.1) and its partial derivatives automatically. Besides, this software is programmed implementing multiple-precision arithmetic (via the MPFR library ), which allows us to compute the former integrations with arbitrary precision.
To optimize each iteration of the method, we can add a condition to have a correction orthogonal to the vector field (2.1):
If our differential system (2.1) has an integral of the motion (, being a constant value) and we want to preserve it in the correction, we have to add an extra equation to the system. With the same idea described above, we take a multivalued Taylor expansion up to the first order: Note that, when (2.1) has independent integrals of the motion, the last equation becomes a set of equations with .
Linear system (2.8) is not necessary squared. Thus, to find a solution that minimizes the residual error, we can use, for example, the Singular Value Decomposition (SVD) method [30, 31], which leads to a robust solver of the minimization problem. After solving the system, we define and . Now, we can apply the above steps again for a new iteration (with and ) that provides us a better approximation and iterate the correction algorithm to correct the initial approximation up to any precision level.
Particularizing now to the case of autonomous Hamiltonian systems with two degrees of freedom (), the th step is defined by the system where is the symplectic matrix, the desired value of Hamiltonian constant, and .
Sometimes, we are interested in a subset of the full phase space. Then we have to correct only some variables, fixing the remaining variables and removing their corresponding columns.
As an example, to show the applicability of the method described in this section, we show some results obtained for the classical Hénon-Heiles Hamiltonian : In this system, the energy is a first integral of the problem. This problem was introduced in the study of galactic dynamics to describe the motion of stars around a galactic center, assuming the motion is restricted to the plane.
The study of the chaotic and regular regions is traditionally done by means of the Poincaré surfaces of section (PSS)  and the Lyapunov exponents. To accelerate the analysis, some fast chaos indicators have been introduced the last few years. The one used in this paper, OFLI2 , is defined at the final time by where and are the first- and second-order sensitivities with respect to carefully chosen initial vectors and stands for the orthogonal component to the flow of the vector . In this case, the variational equations up to second order and the initial conditions are Note that the last line of (2.12) is written for a single -th component to simplify the notation. The OFLI2 tends to a constant value for the periodic orbits, behaves linearly (in log-log scale) for regular initial conditions, and grows exponentially for chaotic orbits .
Figure 1(a) shows the PSS (dark blue points) and the chaos indicator OFLI2 (different color regions, red color for chaotic behavior and blue color for regular one) for and . Both techniques give complementary information about the behavior of the system. Black circles sign two periodic orbits selected for the correction process. The left point corresponds to an unstable periodic orbit and the right point to a stable one. On the right side of Figure 1, we show the CPU time in seconds (on a personal computer PC Intel Core i7 CPU 860, 2.80 GHz under 2.6.32-29-generic SMP x86_64 GNU/Linux system) versus the number of digits for obtaining periodic orbits up to one thousand precision digits. We observe that the behavior is quite similar for the stable and the unstable orbits. The computational complexity behaves like being the requested number of digits. We can see that the CPU time for 1000 digits of precision is around 3 hours, a very acceptable time for this very high level of precision. The possibility of following orbits with high-precision enables to compute the Poincaré map and its first- and higher-order derivatives also with high-precision.
3. Continuation of Periodic Orbits with High-Precision
If (with ) is a smooth mapping and is a solution of then, by applying the implicit function theorem near , the solution set of former equation is a smooth -dimensional submanifold of . Therefore, when , if we want just a curve of solutions, we have to add further restrictions.
For autonomous Hamiltonian systems, once we have one elementary periodic orbit , it can be continued and so the periodic orbits appear in families.
The great advantage of the continuation methods is that, once we have one periodic orbit of a family of periodic orbits, the method gives us the complete family and bifurcations of it. There are several techniques to continue a family of periodic orbits. The simplest one consists of performing a prediction of the next point of the family and, then, a correction to have the new point up to the desired precision level. This technique is used in a great variety of research papers [15, 37] and programs [38–40]. In this work, we use the pseudo-arclength continuation method [26, 27, 41], which allows the continuation of the solution curve regardless of its direction. Therefore, there is no problem with folds or with “vertical” branches. This is an important feature when working in the calculation of periodic orbits of conservative systems [42, 43].
Given a point on the solution curve, the prediction of the next step is a point , where is the unit tangent vector to the curve at and is the fixed step size. To compute the unit tangent vector , we have to solve the system , with the restriction . For , we have two valid solutions, with the same direction but opposite senses, each of them will give us a side of the curve. For , we can approximate by choosing a unit secant vector, that is, the normalized vector of the difference . For practical implementations, this approach leads to a faster and easier to program algorithm.
Once we have , we can apply the correction method described in the previous section by adding the condition , where is the th approximation of obtained by the correction algorithm. That is, in the correction process, we impose that is on the hyperplane HP, perpendicular to passing through (see Figure 2). Here, again (see Section 2), the use of the free software TIDES and SVD allows us to automate the process and to work with any arbitrary precision.
Note that the phase space where we look for the curve of periodic orbits can be given by variables of the system or by functions of them (the Hamiltonian, e.g.). In the latter case, the value of depends on other corrected variables:
Following the example of Section 2, we are going to look for families of periodic orbits of the classical Hénon-Heiles Hamiltonian with and given by and . In this case, we have three variables to continue (we also have to predict next period ) and In Figure 3, we show two curves corresponding to two different families of periodic orbits. The blue curve corresponds to orbits of multiplicity , while the red curve corresponds to orbits of multiplicity . The stability of the orbits [18, 44] is discriminated on the figure by using continuous or dashed lines for stable and unstable periodic orbits, respectively. Both curves have been obtained with and 15 digits of precision. In Table 1, we show CPU times (in seconds) for the computation of the unstable branch of the family with multiplicity 5 (m5u) in Figure 3 for different values of and precision level. These CPU times are still acceptable, although obviously we cannot take values of delta of the size of the precision considered in the correction since the number of points calculated on the curve would soar.
Summarizing the method, we start from a point in the family of periodic orbits (we can obtain it by correcting an approximation with the algorithm of Section 2). Now, if we want to compute the new initial conditions at a distance from point , we define the first approximation, , of the new point as the starting point. The correction of the provisional conditions will be in the previously defined hyperplane HP: Then,(i)we compute corrections in position , momentum and period . The corrected orbit will satisfy(a)correction conditions (2.9) with ,(b)restriction of being in the hyperplane HP (3.3);(ii)we output (according to (3.2)) from and ;(iii)we iterate previous stages until the desired precision level.
As a second example, we consider the Copenhagen problem, a simplification of the three-body problem supposing that the mass of one of the three bodies is negligible and the other two masses are equal [10, 44]. The equations of motion of this problem are with and . The only known integral of motion of this problem is the Jacobian constant that can be used to define the effective energy as . Note that now we do not use the Hamiltonian formalism and so the function (3.1) is the Jacobi constant.
In , Hénon made an exhaustive analysis of classes of periodic orbits of the problem. Several families of such orbits were drawn. Here, we use the necessary precision to redraw one of them and to check that the end of the curve is a spiral (see Figure 4). This situation cannot be seen from the pictures in  and provides a clear application of the present approach. Note that with this approach we may obtain the curves with any desired precision level, up to the hardware and CPU time restrictions, whereas the classical continuation programs, like AUTO or MATCONT, are limited to double-precision (enough for most of the situations).
4. Continuation of Periodic Orbits with Arbitrary Precision
As stated in the previous section, the CPU time for the families of periodic orbits with high-precision is acceptable as long as we do not take a too small value of or a too high-precision level. If you also want to make an exhaustive continuation of the different families of periodic orbits that can appear in a region of the phase space, restrictions must be more severe. However, in most cases, these restrictions are acceptable because we do not need to get the data of all the orbits of the different families with arbitrary precision. Double- or quadruple-precision is usually sufficient for most of them. Only a few of these orbits (being of special interest) have to be further corrected with arbitrary precision.
This idea has been implemented in the calculations of Figure 5 where the two families of periodic orbits have been computed in double-precision. Subsequently, the orbits of the family of multiplicity 5, corresponding to values of and 0.14, or , (values taken just as an example) have been corrected to obtain 100 digits of precision. On the right plots, the obtained periodic orbits are shown: unstable orbits above and below the stable ones. In Table 2, we show the corrected values up to 50 digits for these selected orbits.
In this paper, we introduce a numerical scheme able to continue families of periodic orbits in high-precision and to correct selected orbits up to any arbitrary precision level. The method is based on the combined use of the pseudo-arclength continuation method, an optimized shooting method with the Newton method, the Taylor series method (implemented by means of the free software TIDES), and the singular value decomposition (SVD). We illustrate the scheme on the paradigmatic Hénon-Heiles Hamiltonian and the Copenhagen problem. On the Copenhagen problem, we have been able to show that a classical family of periodic orbits found by Hénon ends in a spiral structure, which cannot be observed without high-precision.
The authors have been supported for this research by the Spanish Research Grant MTM2009-10767. A. Dena thanks Professor G. Lord (Heriot-Watt University, Edinburgh) for his hospitality and his help during her stay at his department.
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