Chaos Indicators

The Chaos Indicators, CIs, are numerical techniques to accomplish one of the most important aspects in the study of the behavior of dynamical systems: the differentiation of trajectories of regular and chaotic nature. Such task is of difficult realization as the difference between two trajectories of different nature can be very subtle (Maffione et al., 2011). This was the motivation behind the development of CIs.

There are two main types of CIs: those which are based on the analysis of the deviation vector (and related to the concept of exponential divergence), and those which are based on the analysis of the orbit itself. Under the first category, one of the first and most used CIs are the Lyapunov Characteristic Exponents, LCEs. These can be studied through the maximum LCE, mLCE, or through their spectra (the distribution of all the LCEs). The method may take a huge amount of time to determine the nature of a trajectory, especially in orbits that remain regular for a long period of time before presenting any chaotic behavior. The mLCE, however, does much more than to determine the nature of a trajectory, it also quantifies the notion of chaosticity by providing a timescale for the studied dynamical system, namely the Lyapunov time (Skokos, 2010).

In (Maffione et al., 2011), a comparison of some methods based on the mLCE is performed. In this article, the FLI, the Lyapunov indicator, LI, the MEGNO, the SALI, the Dynamical Spectra of stretching numbers, SSN, and the corresponding spectral distance, and the RLI are compared in terms of robust- ness, speed of convergence, final values, and behavior under complex scenarios. In our work we are interested in the test of robustness (i.e., capacity to distinguish chaos) and speed of convergence. The first is important to have confidence in our results and the second because we compute the value of the CI for a large set of initials conditions which can reveal to be time-consuming, hence, we need an indicator that takes as less time for its computation as possible. The FLI showed the better relation between these two tests and, thus, it was selected as the CI to be used in our work.

2.6.1. Lyapunov Characteristic Exponents

Following (Skokos, 2010) we introduce the theoretical background on the LCEs and the numerical methods for their computation. We also review the computation method of the mLCE, which is useful to define the FLI computation method.

The knowledge of the spectrum of the LCEs provides the basic information on the behavior of a dynamical system. The LCEs are asymptotic measures characterizing the average rate of growth (or shrinking) of small perturbations to the solutions of a dynamical system. The value of the mLCE is an indicator of the chaotic or regular nature of orbits, while the whole spectrum of LCEs is related to the underlying dynamics of a system.

The computation of the mLCE, χ1, of a trajectory allows us to characterize its nature as regular or chaotic. For regular orbits we haveχ1 = 0 whereas chaotic orbits haveχ1 >0, implying exponential divergence. Furthermore, the mLCE has also the ability to quantify the orbit’s chaosticity. It defines a specific timescale for the considered dynamical system as the inverse of the mLCE, the so-called Lyapunov time,

t_L= 1 χ1

(2.65) which gives an estimate of the time needed for a dynamical system to become chaotic. It measures the time needed for nearby orbits of the system to diverge by e(Neper’s number, do not confuse with eccentricity).

The evaluation of the mLCE of an orbit with initial conditionx(0)requires the evaluation of the orbit’s time evolution and of the deviation vector’s time evolution with initial conditionw(0), i.e., the Hamilton’s equations of motion and the variational equations must be solved simultaneously.

In order to prevent the increase of the deviation vector to extreme large values causing numerical overflow, one may fix a small time intervalτ and define the mLCE for timet=kτ, k= 1,2, . . .. First, let us recall equation 2.10

w(t) =∇_xΦ^tw(0) and have the initial deviation vectorw(0)with norm

D0=kw(0)k (2.66)

We denote by

w((ib −1)τ) = ∇_x(0)Φ^(i−1)τw(0) ∇_x(0)Φ^(i−1)τw(0)

D0, (2.67)

the deviation vector at the pointΦ^(i−1)τ(x(0))having the same direction withw((i−1)τ)and normD0, and byDi its norm after its evolution forτtime units

Di=

∇_Φ(i−1)τ(x(0))Φ^τw((ib −1)τ)

. (2.68)

We define now the local coefficient of expansion of the deviation vector, α_i, for a time interval τ when

the orbit evolves fromΦ^(i−1)τ(x(0))toΦ^iτ(x(0))as

ln

∇_x(0)Φ^iτw(0) ∇x(0)Φ^(i−1)τw(0)

= ln kw(iτ)k

kw((i−1)τ)k = ln D_i D0

= lnα_i. (2.69)

The valuelnαi/τ is also called stretching number.

The mLCE is then computed by

χ1= lim

k→∞

1 kτ

k

X

i=1

ln Di

D₀ = lim

k→∞

1 kτ

k

X

i=1

lnαi (2.70)

2.6.2. Fast Lyapunov Indicator

The Fast Lyapunov Indicator, FLI, was introduced in (Froeschl ´e et al., 1997) motivated by the need to have a quicker method to distinguish between chaotic and regular orbits. In some of the FLI definitions, it can also distinguish resonant from non-resonant motion (Skokos, 2010).

The main difference of the FLI to the evaluation of the mLCE is that the FLI registers the current value of the norm of the deviation vector whereas the mLCE computes the limit value,t → ∞, of the mean of the stretching numbers. By dropping the time average requirement of the stretching numbers, FLI succeeds in determining the nature of orbits faster than the computation of the mLCE (Skokos, 2010).

Since the initial definition of the FLI by Froeschl ´e et al. there has been an evolution of this definition. In their pioneer article, they used and tested the FLI definitionsΨ1,Ψ2, andΨ3and later on, in (Froeschl ´e and Lega, 2000), the newΨ4was developed

Ψ₁= 1

kw1(t)kⁿ; Ψ₂= 1 Qn

j=1kwj(t)k; Ψ₃= 1

sup_jkwj(t)kⁿ ; Ψ₄= sup_t≤t

flnkw(t)k (2.71) wherew_jis a basis of deviation vectors andwa randomly chosen deviation vector.

In our work we use an adapted FLI definition from (Villac and Aiello, 2005) F LI = sup

t≤t_f

sup

i

lnkw_i(t)k (2.72)

or in the notation used in the definition of the mLCE

F LI= sup

t≤tf

sup

i

lnkαi(t)k (2.73)

wheretfis the final time of integration, andwi(t)is a basis ofndeviation vectors with initial conditions

wi(0) = (w1(0),w2(0), . . . ,wn(0)) =In (2.74) andα_i(t)the respective expansion coefficients withn = 6. The FLI is the largest logarithmic variation between two consecutive steps in all six coordinates. This definition was developed not to distinguish resonant motion but to depend as little as possible on the choice of the initial deviation vectors basis.

This definition provides only one value per set of initial conditions, making it possible to construct FLI maps where chaotic and regular regions are easily distinguishable.

There is still one issue in the definition of the FLI: the normalization. It is stated and proved in (Skokos, 2010) that, although for different chosen norms the value of the FLI changes, its capacity to distinguish regular from chaotic motion remains intact, i.e., the norm choice only affects the FLI value quantitatively but not qualitatively. In the computation of the FLI we use the norm

kwk= r1

r w²_x+w_y²+w²_z +1

p

w²_px+w²_py+w²_pz

(2.75)

whererandprepresent the Euclidean norms of the position and momenta of the third body state at the current timet, respectively.

It is far better to foresee even without certainty than not to foresee at all.

(Henri Poincar ´e1854 - 1912)

C