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Tools to detect structures in dynamical systems using Jet Transport


This paper is devoted to the development of some dynamical indicators that allow the determination of regions and structures that separate different dynamic regimes in autonomous and non-autonomous dynamical systems. The underlying idea is closely related to the Lagrangian coherent structures concept introduced by Haller. In the present paper, instead of using the Cauchy–Green tensor, that determines the domains where the flow associated to a differential equation is expanding in the normal direction, the Jet Transport methodology is used. This is a semi-numerical tool, that has as basic ingredients a polynomial algebra package and a numerical integration method, allowing, at each integration step, the propagation under a flow of a neighbourhood U instead of a single initial condition. The output of the procedure is a polynomial in several variables that represents the image of U up to a selected order, containing high order terms of the variational equations. Using these high order representation, the places where the normal direction expands can be easily detected, in a similar manner as the procedures for calculating the Lagrangian coherent structures do. In order to illustrate the methodology, first the results obtained in the determination of the separatrices of the simple and the periodically perturbed pendulum are given. Later, the applications to the circular restricted three body problem are considered, where the aim is the detection of invariant manifolds of libration point orbits, as well as in the non-autonomous vector field defined by the elliptic restricted three body problem.

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The authors would like to thank to the anonymous reviewers for their comments and fruitful suggestions to improve the quality of the paper. This work has been supported by the Spanish Grants MTM2010-16425, MTM2013-41168-P (G.G.), MTM2012-31714, 2014SGR504 (J.J.M.), and MTM-2010-16425, MTM2013-41168-P, AP2010-0268 (D.P.).

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Correspondence to Daniel Pérez-Palau.

Appendix: Numerical results for the Jet Transport and the polynomial algebra

Appendix: Numerical results for the Jet Transport and the polynomial algebra

Before proceeding with the detection of dynamical structures, let us first present some of the tests we use to evaluate the efficiency of the implementation of the polynomial algebra package and its application to the Jet Transport.

To perform this comparative we have used Pari/GP (2012). This package implements two relevant things: from one side it supports multi-precision arithmetic. This is very suitable to check the accuracy of our computations, since the solution provided with Pari/GP has very low truncation errors and can be considered a reference. From another side, Pari/GP also includes polynomial algebra computations making the tests even simpler.

Then, as a first test, we have performed the basic algebraic operations using our own algebra package and Pari/GP. Subtracting the resulting polynomials we can see how much the coefficients differ. These tests have been done, for each algebraic operation, 1000 times (using randomly generated polynomials) in order to compute the averaged errors. As we have already mentioned, the accuracy of Pari/GP is higher than the machine precision, therefore the computations done with Pari/GP can be used to validate the ones of the algebra package.

Table 1 shows the decimal logarithm of the error for the power (\(P^\alpha (\xi )\)), the division (\(P(\xi )/Q(\xi )\)), and the trigonometric functions (\(\sin (P(\xi ))\), \(\cos (P(\xi ))\)), taking polynomials \(P(\xi )\) and \(Q(\xi )\) with two variables and degree three. It can be seen that the precision of the computations is close to the machine precision.

Table 1 Decimal logarithm of the difference of the coefficient associated to the exponents \(k_0, \, k_1\), computed using the polynomial algebra and Pari/GP for some basic functions: power, quotients, sinus and cosinus

Two more tests have been done to check the precision of the Jet Transport computations. For these ones we have used the simple pendulum and the CR3BP as reference models. The first one has to do with the accuracy of the derivatives of the flow map obtained by the Jet Transport compared with the values obtained by numerical differentiation using:

$$\begin{aligned} f'(x)\approx \frac{f(x+\delta )-f(x-\delta )}{2\delta } \end{aligned}$$
Fig. 15

Accuracy tests for the computation of the variational equations using the Jet Transport. Left: after 200 time units for a pendulum model (with initial conditions \((x,y)=(1,0)\)), and as a function of \(\delta \), values of the difference between the i-th order variational equations computed with the Jet Transport and the values obtained using numerical derivatives. For \(i=1,2,3,4,5\) (respectively red, green, blue, magenta and cyan). Right: After 100 time units in the CR3BP ( with initial conditions \((x,y,\dot{x}, \dot{y})= (0.9,0.3,0.1,0.4)\)), and as a function of \(\delta \), values of the difference between the first order variational equations computed with the Jet Transport and the values obtained using numerical derivatives for each one of the four variables of the system

For the pendulum and the CR3BP cases, Fig. 15 shows the differences between the first order derivatives, which are related to the first order variational equations, (the five first derivatives for the pendulum and the first order ones for the CRBP) obtained from the Jet Transport and the numerical values of the variational equations computed using the previous differentiation formula. As usual with this kind of numerical computations, when we decrease the value of \(\delta \) the errors become smaller until a certain value of \(\delta \) from which the round-off error in the calculation of the difference becomes larger than the truncation error. To improve this results we implement a Richardson extrapolation applying the recursive formulae:

$$\begin{aligned} F_1(h)= & {} f'(h),\\ F_{i+1}(h)= & {} F_i(h)\dfrac{F_i(h)-F_i(2h)}{2^{i+1}-1}. \end{aligned}$$

Using these formulae we are able to get better approximations of the numerical derivatives. Figure 16 shows the results of five steps of Richardson extrapolation when we consider the first variable x of the CR3BP. From this figure, it is clear that at each extrapolation step the errors decrease and that the approximation given by the jet is very good, the error is of the order of \(10^{-13}\) using double precision arithmetic.

Fig. 16

Accuracy test for the first order variational terms in the CR3BP after 2 time units. The colour curves display the difference between the first order variational equations of the first coordinate \(\left( \frac{\partial \phi _{t_0}^T(x_0)]_1}{\partial x}\right) \) computed with the Jet Transport and the values obtained using the Richardson extrapolation step \(F_i\) as a function of \(\delta \) in log10 scale. The black lines show the trends expected according to the order of the formula

The last test that has been done is related to the conservation of a first integral. Instead of checking its conservation as a scalar value, we check it as a polynomial. First we compute an initial polynomial giving the variations of the values of the first integral associated to small deviations of the variables. Then, at some point during the integration, we can compute again the first polynomial integral and check its components against the initial values. Figure 17 shows the difference between the first, second and third order terms of the polynomial first integral along the integrations of the pendulum and the CR3BP differential equations. As expected, the longer the time the bigger the error, but always within a good accuracy.

Fig. 17

Accuracy tests for the conservation of the first integral (left: pendulum with initial condition \((x,y)=(1,0)\), Right: CR3BP). As a function of time, differences between the initial energy polynomial with initial conditions \((x,y,\dot{x}, \dot{y})= (0.9,0.3, 0.1,0.4)\) and the one computed using the jet for the independent term (in red), the first order terms (in green) and second order terms (in blue)

The above results ensure that it is possible to extract information about the high order variational terms from the output of the Jet Transport computations. The main drawback of the Jet Transport is that, in case that some information is needed for a single point, computations will be slow compared to a single standard integration. However, if we are looking for information of neighbouring points, as is the case in many Monte Carlo simulations, or we need high order variational terms, then the Jet Transport becomes very useful. Figure 18 shows the amount of CPU time needed to integrate different amounts of initial conditions of the CR3BP, during 20 time units, in front of the CPU time required by the Jet Transport to get the images under the flow of the same initial conditions for the same amount of time. We can appreciate the almost constant behaviour of the Jet in front of the linear growth using a RK78 classical procedure.

Fig. 18

In the CR3BP, CPU Time in seconds spent as a function of the number of initial conditions propagated using the Jet-Taylor method with orders 1,2,3 (respectively red, green and blue) and a usual Taylor Method without jet propagation (magenta). Each integration is done in a randomly chosen orbit close to the initial condition at (1, 0, 0) with initial velocity \(v_0=(1,1,1)\) during 20 adimensional time units. The mass parameter is \(\mu =0.012\). The computations have been carried out with an Intel Xeon(R), CPU E5645, 2.40 GHz

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Pérez-Palau, D., Masdemont, J.J. & Gómez, G. Tools to detect structures in dynamical systems using Jet Transport. Celest Mech Dyn Astr 123, 239–262 (2015).

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  • Lagrangian coherent structures (LCS)
  • Jet Transport
  • Invariant structures
  • Simple pendulum
  • Perturbed pendulum
  • Circular restricted three-body problem
  • Elliptic restricted three-body problem
  • Finite time Lyapunov exponents