High-Precision Continuation of Periodic Orbits

Abstract

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.

1. Introduction

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 [1], while the unstable periodic orbits in a chaotic set (attractor or saddle) determine its structure [2] 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 [5], electrical and magnetic fields [6], hydrodynamical flows [7], electrical circuits and optical systems [8], 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) [24], and the study of the fractal properties of the chaotic attractors [25]. Currently, the only algorithms capable of computing high-precision periodic orbits are based on Lindstedt-Poincaré series [14] or Taylor series methods [19]. 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 [19] 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:(),𝐲𝑡𝐲′=𝐟𝐲,𝑡₀=𝐲₀, (2.1) with 𝐲=(𝑦₁,𝑦₂,…,𝑦_𝑠)^𝑇 and 𝐲₀=(𝑦⁰₁,𝑦⁰₂,…,𝑦⁰_𝑠)^𝑇∈ℝ^𝑠, and 𝑡∈ℝ. And let𝐲=𝐲𝑡;𝐲₀,𝑡≥𝑡₀, (2.2) be the solution of (2.1).

If we have an estimation of the initial conditions 𝐲₀ of a periodic orbit with estimated period 𝑇₀, then 𝐲(𝑡₀+𝑇₀;𝐲₀)≈𝐲₀. If we call Δ𝐲 and Δ𝑇 the corrections to the initial conditions and the period, respectively, then we have the periodicity condition:𝐲𝑡₀+𝑇₀+Δ𝑇;𝐲₀+𝚫𝐲=𝐲₀+𝚫𝐲. (2.3) To obtain an approximation of Δ𝑇 and Δ𝐲, we follow the strategy of the Newton method: we take the multivalued Taylor expansion up to the first order in (2.3) to get 𝑠 linear equations (with 𝑠+1 unknowns) to satisfy𝑀₀−𝐼_𝑠𝜕𝐲𝑡𝜕𝑡⁰+𝑇₀;𝐲₀𝚫𝐲₀Δ𝑇₀=𝐲₀−𝐲𝑡₀+𝑇₀;𝐲₀, (2.4) with 𝐼_𝑠∈ℝ^𝑠×𝑠 the identity matrix, Δ𝑇₀ and Δ𝐲₀ approximations of Δ𝑇 and Δ𝐲, respectively, and 𝑀₀∈ℝ^𝑠×𝑠 given by𝑀₀=𝑚⁰_𝑘,𝑙,where𝑚⁰_𝑘,𝑙=𝜕𝑦_𝑘𝜕𝑦⁰_𝑙𝑡₀+𝑇₀;𝐲₀,1≤𝑘,𝑙≤𝑠. (2.5)

At this point, we have to obtain numerically 𝐲(𝑡₀+𝑇₀;𝐲₀) and 𝑀₀. Usually, to compute 𝑀₀, 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 [32]), 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 (Δ𝐲₀,Δ𝑇₀) orthogonal to the vector field (2.1):𝜕𝐲^𝑇𝑡𝜕𝑡⁰;𝐲₀0𝚫𝐲₀Δ𝑇₀=0. (2.6)

If our differential system (2.1) has an integral of the motion (𝐼𝑀(𝑡;𝐲₀)≡𝑖_𝑚, being 𝑖_𝑚∈ℝ 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:𝐼𝑀𝑡₀+𝑇₀+Δ𝑇₀;𝐲₀+𝚫𝐲₀−𝑖_𝑚≈𝐼𝑀𝑡₀+𝑇₀;𝐲₀−𝑖_𝑚+𝜕𝐼𝑀𝑡𝜕𝐲⁰+𝑇₀;𝐲₀𝚫𝐲₀+𝜕𝐼𝑀𝑡𝜕𝑡⁰+𝑇₀;𝐲₀Δ𝑇₀=0. (2.7) Note that, when (2.1) has 𝑘 independent integrals of the motion, the last equation becomes a set of 𝑘 equations with 𝑖_𝑚∈ℝ^𝑘.

Putting together (2.4), (2.6), and (2.7), we arrive to the system𝑀₀−𝐼_𝑠𝜕𝐲𝑡𝜕𝑡⁰+𝑇₀;𝐲₀𝜕𝐲^𝑇𝑡𝜕𝑡⁰;𝐲₀0𝜕𝐼𝑀𝑡𝜕𝐲⁰+𝑇₀;𝐲₀𝜕𝐼𝑀𝑡𝜕𝑡⁰+𝑇₀;𝐲₀𝚫𝐲₀Δ𝑇₀=𝐲₀−𝐲𝑡₀+𝑇₀;𝐲₀0𝑖_𝑚−𝐼𝑀𝑡₀+𝑇₀;𝐲₀. (2.8)

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 𝐲₁=𝐲₀+Δ𝐲₀ and 𝑇₁=𝑇₀+Δ𝑇₀. Now, we can apply the above steps again for a new iteration (with 𝐲₁ and 𝑇₁) 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 (𝐲=(𝑥,𝑦,𝑋,𝑌)^𝑇), the (𝑖+1)th step is defined by the system𝑀_𝑖−𝐼₄𝒥∇_𝐲ℋ𝐲^∗_𝑖−∇_𝐲ℋ𝐲𝑡₀;𝐲_𝑖^𝑇∇𝒥0_𝐲ℋ𝐲^∗_𝑖0𝚫𝐲_𝑖Δ𝑇_𝑖=𝐲_𝑖−𝐲^∗_𝑖0ℎ−ℋ𝐲^∗_𝑖, (2.9) where 𝒥 is the symplectic matrix, ℎ the desired value of Hamiltonian constant, and 𝐲^∗_𝑖=𝐲(𝑡₀+𝑇_𝑖;𝐲_𝑖).

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 [33]:ℋ=1𝑥,𝑦,𝑋,𝑌2𝑋²+𝑌²+12𝑥²+𝑦²+𝑦𝑥²−13𝑦³. (2.10) In this system, the energy 𝐸≡ℋ(𝑥,𝑦,𝑋,𝑌) 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 𝑥𝑦 plane.

The study of the chaotic and regular regions is traditionally done by means of the Poincaré surfaces of section (PSS) [34] 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 [35], is defined at the final time 𝑡_𝑓 by OFLI2∶=sup_{0<𝑡<𝑡𝑓}𝝃(𝑡)+1ln2𝛿𝝃(𝑡)^⟂, (2.11) where 𝝃 and 𝛿𝝃 are the first- and second-order sensitivities with respect to carefully chosen initial vectors and 𝐱^⟂ stands for the orthogonal component to the flow of the vector 𝐱. In this case, the variational equations up to second order and the initial conditions are 𝑑𝝆()(0)𝑑𝑡=𝐟𝑡,𝝆,𝝆=𝝆₀,𝑑𝝃=()𝑑𝑡𝜕𝐟𝑡,𝝆(0)=𝐟𝜕𝝆𝝃,𝝃0,𝝆₀𝐟0,𝝆₀,𝑑𝛿𝜉_𝑗=𝑑𝑡𝜕𝑓_𝑗𝜕𝝆𝛿𝝃+𝝃^𝑇𝜕²𝑓_𝑗𝜕𝝆²𝝃,𝛿𝝃(0)=𝟎. (2.12) Note that the last line of (2.12) is written for a single 𝑗-th component 𝛿𝜉_𝑗 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 [35].

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 𝑥=0 and 𝐸=1/8. 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 𝒪(𝑑⁴) being 𝑑=−log₁₀(TOL) 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.

(a)

(b)

Figure 1 

(a): OFLI2 and PSS section of the Hénon-Heiles problem for 𝑥=0 and Energy 𝐸=1/8. (b): Computational relative error versus CPU time for the correction method applied to the two marked periodic orbits on the left picture.

3. Continuation of Periodic Orbits with High-Precision

If 𝐹∶ℝ^𝑛+𝑘→ℝ^𝑛 (with 𝑘>0) is a smooth mapping and 𝑥^∗∈ℝ^𝑛+𝑘 is a solution of 𝐹(𝑥)=0 then, by applying the implicit function theorem near 𝑥^∗, the solution set of former equation is a smooth 𝑘-dimensional submanifold of ℝ^𝑛+𝑘. Therefore, when 𝑘>1, 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 [36], 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 𝐱_𝑖 on the solution curve, the prediction of the next step is a point 𝐱_𝑖+1=𝐱_𝑖+𝛿⋅𝐮_𝑖, where 𝐮_𝑖 is the unit tangent vector to the curve at 𝐱_𝑖 and 𝛿 is the fixed step size. To compute the unit tangent vector 𝐮_𝑖, we have to solve the system 𝐹_𝐱(𝐱_𝑖)𝐮_𝑖=0, with the restriction ‖𝐮_𝑖‖=1. For 𝐱₀, we have two valid solutions, with the same direction but opposite senses, each of them will give us a side of the curve. For 𝑖>0, we can approximate 𝐮_𝑖 by choosing a unit secant vector, that is, the normalized vector of the difference 𝐱_𝑖−𝐱_𝑖−1. For practical implementations, this approach leads to a faster and easier to program algorithm.

Once we have 𝐱_𝑖+1, we can apply the correction method described in the previous section by adding the condition 𝐮^𝑇_𝑖⋅(𝐱^𝑗_𝑖+1−𝐱_𝑖+1)=0, where 𝐱^𝑗_𝑖+1 is the 𝑗th approximation of 𝐱_𝑖+1 obtained by the correction algorithm. That is, in the correction process, we impose that 𝐱_𝑖+1 is on the hyperplane HP, perpendicular to 𝐮_𝑖 passing through 𝐱_𝑖+1 (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.

Figure 2 

Sketch of one step in pseudo-arclength continuation method.

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 𝒢(𝐲) of them (the Hamiltonian, e.g.). In the latter case, the value of Δ𝒢(𝐲) depends on other corrected variables:Δ𝒢(𝐲)=^𝑠_𝑙=1𝜕𝒢𝜕𝑦_𝑙Δ𝑦_𝑙. (3.1)

Following the example of Section 2, we are going to look for families of periodic orbits of the classical Hénon-Heiles Hamiltonian with 𝑥₀=𝑌₀=0 and 𝑋₀ given by 𝑦₀ and ℋ. In this case, we have three variables to continue {𝑦₀,ℋ,𝑇} (we also have to predict next period 𝑇) andΔℋ=𝑋Δ𝑋+𝑦−𝑦²Δ𝑦. (3.2) In Figure 3, we show two curves corresponding to two different families of periodic orbits. The blue curve corresponds to orbits of multiplicity 𝑚=1, while the red curve corresponds to orbits of multiplicity 𝑚=5. 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 𝛿=0.01 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 𝛿 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.

Table 1 

CPU times for the computation of the unstable branch of the family with multiplicity 5 (m5u) in Figure 3.

Figure 3 

Continuation of two families of periodic orbits (multiplicities 𝑚=1 and 5) of the classical Hénon-Heiles Hamiltonian with 𝑥0=𝑌0=0 depending on the coordinate 𝑦 and the energy 𝐸.

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 (𝑦_new,ℋ_new,𝑇_new) at a distance 𝛿 from point (𝑦_old,ℋ_old,𝑇_old), we define the first approximation, (𝑦,ℋ,𝑇)=(𝑦_old,ℋ_old,𝑇_old)+𝛿⋅𝐮, of the new point as the starting point. The correction of the provisional conditions (𝑦+Δ𝑦,ℋ+Δℋ,𝑇+Δ𝑇) will be in the previously defined hyperplane HP:HP=𝑦,ℋ,𝑇∣𝑦−𝑦,ℋ−ℋ,𝑇−𝑇⋅𝐮=0. (3.3) Then,(i)we compute corrections in position Δ𝑦, momentum Δ𝑋 and period Δ𝑇. The corrected orbit will satisfy(a)correction conditions (2.9) with Δ𝑥=Δ𝑌=0,(b)restriction of being in the hyperplane HP (3.3);(ii)we output Δℋ (according to (3.2)) from Δ𝑦 and Δ𝑋;(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 are𝑥̈−2𝑦̇=𝑥−𝑥+1/22𝑟³₁−𝑥−1/22𝑟³₂,𝑦𝑦̈+2𝑥̇=𝑦−2𝑟³₁−𝑦2𝑟³₂, (3.4) with 𝑟²₁=(𝑥+1/2)²+𝑦² and 𝑟²₂=(𝑥−1/2)²+𝑦². The only known integral of motion of this problem is the Jacobian constant 𝐶𝐶=−𝑥̇²+𝑦²̇+1𝑟₁+1𝑟₂+𝑥²+𝑦², (3.5) that can be used to define the effective energy as 𝐸_𝐽=−𝐶/2. Note that now we do not use the Hamiltonian formalism and so the function 𝒢 (3.1) is the Jacobi constant.

In [44], 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 [44] 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).

Figure 4 

Family of periodic orbits for the Copenhagen problem depending on the coordinate 𝑥 and the Jacobi constant 𝐶. Picture (a) is extracted from Exploration numérique du problème restreint. I. Masses égales; orbites périodiques, Hénon [44]; Picture (b) shows the same curve obtained with the method described above. Bottom: magnifications of the squared regions showing the spiral structure.

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 𝛿 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 𝐸=0.12 and 0.14, or 𝑦=0.5, (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.

Table 2 

Corrected values up to 50 digits for the selected orbits in Figure 5.

(a)

(b)

Figure 5 

(a) continuation of two families of periodic orbits of the classical Hénon-Heiles Hamiltonian with 𝑥0=𝑌0=0. Orbits with some selected values of the energy and coordinate 𝑦 are pointed out. (b) corrected arbitrary precision periodic orbits corresponding to surrounded points following the color code; top, unstable orbits, and bottom, the stable ones.

5. Conclusions

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.

Acknowledgments

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.