The Shannon entropy: an efficient indicator of dynamical stability

Pablo M. Cincotta Grupo de Caos en Sistemas Hamiltonianos, Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata and Instituto de Astrofísica de La Plata (CONICET), La Plata, Argentina [email protected]

Claudia M. Giordano Grupo de Caos en Sistemas Hamiltonianos, Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata and Instituto de Astrofísica de La Plata (CONICET), La Plata, Argentina [email protected]

Raphael Alves Silva Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Universidade de São Paulo, Brasil [email protected]

Cristián Beaugé Instituto de Astronomía Teórica y Experimental (IATE), Observatorio Astronómico, Universidad Nacional de Córdoba, Argentina [email protected]

In this work it is shown that the Shannon entropy is an efficient dynamical indicator that provides a direct measure of the diffusion rate and thus a time-scale for the instabilities arising when deal- ing with chaos. Its computation just involves the solution of the Hamiltonian flow, the variatonal equations are not required. After a review of the theory behind this approach, two particular appli- cations are presented; a 4D symplectic map and the exoplanetary system HD 181433, approximated by the Planar Three Body Problem. Successful results are obtained for instability time-scales when compared with direct long range integrations (N-body or just iterations). Comparative dynamical maps reveal that this novel technique provides much more dynamical information than a classical chaos indicator.

Keywords: Chaotic diffusion; Shannon entropy; 4D maps; Three Body Problem 2

I. INTRODUCTION

Chaos indicators are powerful tools to investigate the global structure of phase space of dynamical systems. Most of them are based on the evolution of the tangent vector of a given trajectory as in case of the maximum Lyapunov Exponent (mLE). Let ϕ(t) be a given solution of a Hamiltonian flow. The mLE of ϕ is defined as

1 kδ(ϕ(t))k mLE(ϕ) = lim ln , (1) t→∞ t kδ(0)k kδ(0)k→0 where δ(ϕ(t)) is the tangent vector to ϕ(t) and it is the solution of the first variational equations of the Hamiltonian flow evaluated at ϕ(t), with initial condition δ(0). It is well known that in case of quasiperiodic motion, ϕq, after a motion time t the mLE converges to 0 as 4 −3 mLEt(ϕq) ≈ lnt/t, and for instance for t = 10 , mLEt(ϕq) ≈ 10 . µt On the other hand for a chaotic motion, ϕc(t), with mLE = µ > 0, kδ(ϕc(t))k ≈ kδ(0)ke . Thus by means of (1), −3 5 to distinguish ϕc(t) with µ ≤ 10 from ϕq(t), the computational time should be t & 10 . In the 90s, three techniques were widely used to investigate dynamics in phase space (particularly in Dynamical Astronomy): the mLE, the Frequency Map Analysis [22, 23] and the Poincaré Surface of Section. Computers were not fast enough to cope with the determination of the mLE for a large sample of orbits over long motion times. Thus fast dynamical indicators appear: the Fast Lyapunov Indicator, FLI [17, 20, 24]; the Mean Exponential Growth factor of Nearby Orbits, MEGNO [11, 14, 15]; the Smaller and the Generalized Alignment Indices, SALI–GALI [31–33]; the Orthogonal Fast Lyapunov Indicator, OFLI [3, 4, 16], among others. Fast dynamical indicators are then useful to display the global dynamical structure of phase space unveiling the chaotic and regular components as well as the resonance web. Moreover they are able to show up invariant manifolds and provide a measure of hyperbolicity of the chaotic regions. Though they provide information about the mLE in a given point of the phase space, it should be stressed that a positive mLE does not necessarily imply chaotic diffusion, i.e. a significant variation of the unperturbed actions or integrals of motion, the well known stable chaos is a typical phenomenon where the unstable motion is rather confined to small neighborhood of the initial values of the integrals over motion times larger than mLE−1 (see for instance [29] for an example in the Solar System). Therefore chaos indicators are effective tools to conduct further relevant dynamical studies, for instance how effective is chaos to erase correlations among the phase space variables, i.e. to obtain an estimate of the time-rate of the instabilities arising in the chaotic components of a divided phase space. In this work we take advantage of the Shannon entropy approach, already introduced in [5, 19], to show that the entropy besides being an effective dynamical indicator, it provides an accurate measure of the diffusion rate. In the above mentioned works the theoretical framework is provided when dealing with the action space of multidimensional systems. Successful applications of this novel technique to two coupled rational standard maps [12, 15], the Arnold Hamiltonian [1] and the restricted planar Three Body Problem were presented. On the other hand, in [9, 10, 12] it was shown analytically and numerically that the Shannon entropy is also a very powerful tool to measure correlations among the successive values of the phases involved in highly chaotic low dimensional maps, as the whisker mapping and its generalization to cope with diffusion in Arnold model [8], the standard map as well as the rational standard map, both for large values of the perturbation parameters. Herein we focus our effort in the derivation of a time-scale for the chaotic instability in a 4D symplectic map that model the dynamics around the junction of two resonances of different order and in the HD 181433 exoplanetary system that could be well represented by the planar Three Body Problem.

II. THE SHANNON ENTROPY FORMULATION

In this section we summarize and extend the formulation given in [5, 10, 13, 19] regarding the Shannon entropy as a dynamical indicator as well as a measure of the diffusion rate in action space of multidimensional Hamiltonian systems or symplectic maps. For a general background on the Shannon entropy we refer to [25, 30] as well as [2]. Let us consider an N-dimensional system defined by actions (I1,...,IN ) and phases (ϑ1,...,ϑN ). For simplicity and 2 2 due to formal aspects of this presentation we assume a 4D map with (I1,I2) ∈ R ,(ϑ1,ϑ2) ∈ T and a given section 0 0 0 0 S = {(I1,I2): |ϑ1 − ϑ1| + |ϑ2 − ϑ2| < δ  1} where ϑ1,ϑ2 are some fixed values of the phases that define S. A given trajectory φ = {(I1(t),I2(t)),t = 1,...,∞} ⊂ S leads to a surface distribution density on S, ρ(I1,I2) assumed 3 normalized, such that introducing a partition of S, α = {ak, k = 1,...,q}, q  1, the (disjoint) elements have a measure Z µ(ak) = ρ(I1,I2)dI1dI2. (2) ak

For finite but large motion times, t ≤ Ns, that will be the scenario hereafter, the above measure reads

nk µ(ak) = . Ns where nk is the number of action values (I1,I2) in the cell ak. Thus the entropy of φ for the partition α is defined as

q0 q0 X 1 X S(φ,α) = − µ(ak)ln(µ(ak)) = lnNs − nk lnnk. (3) Ns k=1 k=1 where 1  q0 ≤ q denotes the non-empty elements of the partition. It is simple to show that 0 ≤ S ≤ lnq0. The minimum appears when nk = Ns,nj = 0 ∀j 6= k; a trajectory lying on a torus, while the maximum corresponds to ergodic motion, nk = Ns/q0 ∀k. Thus, as it was shown in for instance [19], the entropy is in fact an effective indicator of the stability of the motion, comparisons with other fast dynamical indicators were given. Let us focus first on nearly random motion. As it was discussed in [9, 10], if nk follows a Poissonian distribution 2 with mean λ = Ns/q0  1, setting nk = λ+ξk with |ξk|  λ, then up to O((ξk/λ) ), the entropy (3) for uncorrelated motion, say γr, reduces to

q0 r 1 1 X 2 S(γ ,α) ≈ lnq0 − 2 ξk. (4) 2λ q0 k=1 √ Recalling that the Poissonian fluctuations obey a normal distribution with mean value 0 and standard deviation λ, then

q0 1 X 2 ξk = λ, (5) q0 k=1 and the entropy (4) reduces to 1 S(γr,α) ≈ lnq − . (6) 0 2λ −1 Therefore for random motion |S − lnq0| = O(λ ) being λ  1, the entropy can be well approximated by r S(γ ,α) ≈ S0 = lnq0. ˜ In case of a strong unstable, chaotic but non-random trajectory, γ, we write nk = λ + ξk where we assume that ˜ |ξk| < |ξk|  λ. Then accordingly to (4)

q0 1 1 X ˜2 S(γ,α) ≈ lnq0 − 2 ξk, (7) 2λ q0 k=1 recalling (5) and defining β such that

q0 q0 X ˜2 X 2 ξk = β ξk, k=1 k=1 it follows then ˜2 q0 hξki ˜2 1 X ˜2 β = , hξki = ξk. (8) λ q0 k=1 Thus, from (7) β |S(γ,α) − lnq | ≈ . (9) 0 2λ 4

Thus defined, β ≥ 1 is the ratio between the variance of the fluctuations of nk and the mean value λ for a non-Poissonian distribution. Thus, also for γ, S(γ,α) ≈ S0 provided that β/λ  1. On the other hand, in case of a trajectory γc confined to a small domain of S, as it was discussed in [9, 13], the distribution of the nk approaches a delta, δ(nk − λ), and thus estimating |ξk| ≈ 1/2 it follows from (4) that 1 |S(γc,α) − lnq | ≈ (10) 0 8λ2 and thus it is also true that S ≈ S0 even though λ ∼ 1. Following [5, 13, 19], a local diffusion coefficient for γ in the interval (t,t + δt) can be estimated from the time derivative of S whenever dS/dt ≈ dS0/dt, Σ dS Σ δq D (γ,t) := q (t) (t) ≈ 0 (t), (11) S q 0 dt q δt Σ being the area of S where the partition is defined, so that Σ/q provides the size of the cells in action dimensions (see below for an alternative definition of Σ). The estimate (11) rests on the assumption that locally any increase in the number of occupied cells is entirely due to changes in the actions in the interval (t,t + δt), in such a way that

2 2 δq0(t) ∝ hδI1 (t) + δI2 (t)i ≈ Dtδt, where h·i denotes space average and Dt ≡ D(I1(t),I2(t)) is a local diffusion coefficient in action space, when γ is restricted to the region (I1,I1 + δI1) × (I2,I2 + δI2). Thus, a global diffusion coefficient for γ can be defined as

1 Z t DS(γ) := lim DS(γ,t)dt ≈ hDS(γ,t)it≤N , (12) t→∞ s t t0 where the last approximation applies in case of finite but large enough motion times. This formulation has a free parameter, the number of elements of the partition q. In any case, the condition q0  q is required so that q0(t) could increases with time. However its value mainly depends on the nature of the motion. If σ denotes the area covered by the diffusion in S, we consider two different limiting situations, when (i) σ  Σ and (ii) σ ≈ Σ. In case (i), the area of the unit cell Σ/q, should be small with respect to σ in such a way the non-empty elements of the partition would have nearly the same invariant measure, so q  Σ/σ  1. In other words, very small cells are required in order to have enough resolution such that the q0 cells properly cover σ. When (ii) applies, q  1 still holds. ˆ 1/SL ˆ In [13] it was shown that the optimal choice of q in order to (11) and (12) work is that Ns . q < Ns , where SL is some threshold value of Sˆ = S/lnq, such that SˆL < 1. Let us discuss in more detail the above condition. At first sight, the statistical approach would require that Ns/q  1. However, as discussed above, the average λ involves q0 not q, such that q0(t)  q ∀t, so the condition Ns/q  1 can be relaxed allowing Ns/q . 1 but Ns/q0  1. For the upper limit, being σ the area covered by the diffusion, then the mean (discrete) density is ρ0 = Ns/σ. p Therefore the mean distance distance between the iterates is d ≈ σ/Ns. On the other hand linear size of the unit cell is ∆ = pΣ/q. If the diffusion is confined to a small region of S, σ  Σ, we can assume that the nk follows a nearly δ distribution, the density ρ(I1,I2) ≈ ρ0 ∀(I1,I2) ∈ σ, is large and therefore q can be taken in such a way that d < ∆. This condition leads to q < (Σ/σ)Ns, with Σ/σ  1. 1−Sˆ In this case of a nearly uniform distribution, the factor Σ/σ can be estimated as q/q0 ≈ q , with Sˆ ≤ SˆL < 1 and ˆ 1/SL therefore the above condition reduces to q < Ns . Thus, this upper bound for q implies that no empty cells appear in σ due to discrete character of ρ. Therefore whenever ρ(I1,I2) ≈ ρ0 and Σ/q  1, DS is almost invariant under a partition change while S increases with q (see [13]). On the other hand, if the extension of the diffusion region in S is large, σ ≈ Σ, a nearly Poissonian distribution p applies. The density now is smaller (for the same number of iterates), the fluctuations are large (∼ Ns/q0) and thus in general d > ∆ except if q is small enough, but small values of q are not allowed in this formulation since we require that q0 grows with time. The estimate Σ/σ ≈ q/q0 is no longer true and thus no additional restriction appear to q. In this scenario DS is not invariant under a change of the partition. In the next Sections we present applications of this approach to two different dynamical systems. We refer to [5, 13, 19] for particular examples concerning the time evolution of S,DS for several initial conditions on different multidimensional systems. 5

III. APPLICATIONS

In this section we present applications of the Shannon entropy approach to measure the diffusion rate in quite different models: a system of discrete time consistent in a 4D map and a system of continuous time, the Three Body Problem for a particular planetary system.

A. I. A system of discrete time

0 0 0 0 1 Following [18], we consider the 4-D symplectic map M: (I1,I2,ϑ1,ϑ2) → (I1,I2,ϑ1,ϑ2), Ij ∈ R,ϑj ∈ S defined as

0 I1 = I1 + η sinϑ1, 0 I2 = I2 + ηεsinϑ2, 0 0 0 ϑ1 = ϑ1 + η(I1 + a2I2), (13) 0 0 0 ϑ2 = ϑ2 + η(a2I1 + a3I2); where |ε|  1,η . 2 are real parameters and a2,a3 ∈ Q. Actually, this map can be thought as a 4D generalization of the well known 2D standard map. The application M can be regarded as the time-η map associated to the flow of the Hamiltonian

I2 I2 H(I ,I ,ϑ ,ϑ ) = 1 + a 2 + a I I + cosϑ + εcosϑ . (14) 1 2 1 2 2 3 2 2 1 2 1 2

Actually, the map M not only provides the successive values of (Ij(tl),ϑj(tl)) at tl = lη, l = 0,1,...,N generated by the Hamiltonian (14) but also the evolution of the actions and angles due to H plus a periodic time-dependent perturbation. Indeed, the discrete system derives from the differential equations

I˙1 = cosϑ1 × 2πδ2π(τ), I˙2 = εcosϑ2 × 2πδ2π(τ), ϑ˙1 = I1 + a2I2, ϑ˙2 = a2I1 + a3I2,

−1 where τ = 2πη t and δ2π is the 2π-periodic delta function defined through its Fourier expansion. The above set of equations corresponds to the flow of the Hamiltonian (see [12] for details concerning the numerical equivalence between the map and the Hamiltonian flow)

∞ I2 I2 X H(I ,I ,ϑ ,ϑ ,τ) = 1 + a 2 + a I I + [cos(ϑ − kτ) + εcos(ϑ − k0τ)]. (15) 1 2 1 2 2 3 2 2 1 2 1 2 k,k0=−∞

Thus H reduces to H when keeping only the terms in the sum with k,k0 = 0. The frequencies of the system being

−1 ω1(I1,I2) = I1 + a2I2, ω2(I1,I2) = a3I2 + a2I1, 2πη . (16)

The parameter η, besides being the time step of the flow, defines the frequency of the external perturbation and thus it plays an important role in the dynamics of the system as we discuss below. Both, the Hamiltonian (14) and the map (13) were introduced in [18] to investigate the dynamics near the inter- section of two resonances of different order. In fact, H is a truncated normal form around the intersection of the resonances I1 + a2I2 = 0 and a2I1 + a3I2 = 0. 0 0 0 0 The map M has four fixed points p = (I1 ,I2 ,ϑ1,ϑ2) located at p1 = (0,0,0,0), p2 = (0,0,π,0), p3 = (0,0,0,π), p4 = 2 (0,0,π,π). As it was shown in [18], denoting c = a3 − a2 then for εc > 0 and η . 2, p1 is hyperbolic-hyperbolic, p2 is elliptic-hyperbolic, p3 is hyperbolic-elliptic and p4 is a elliptic-elliptic fixed point. From (15) and (16), the full set of first order resonances are

0 0 R = {(I1,I2): I1 + a2I2 = 2πk/η, a3I2 + a2I1 = 2πk /η, k,k ∈ Z}, (17) where the double resonance model H given by (14) corresponds to the resonances with k = k0 = 0. 6

m m m m Note that the map is invariant under the transformation I1 → I1 + I1 ,I2 → I2 + I2 , with I1 = 2πq/(ηa2),I2 = 2πp/(ηa3), where q,p are integer numbers such that q/a2 ∈ Z, pa3 = ra2 with r an integer number. Thus we can m m m m m m restrict the action space to D = (−I1 ,I1 ) × (−I2 ,I2 ) with opposite side identified and therefore Σ = 4I1 I2 . In what follows we take a2 = 1/2,a3 = 5/4, so q = 1,p = 5, and thus the model corresponds to the crossings of resonances of order 3 and 7. The separation between resonances depends on η,a2 and a3, being the latter 2π 2π dk = , dk0 = , q 2 q 2 2 η 1 + a2 η a3 + a2 for the lower and higher order resonances respectively. Large values of η would lead to a highly chaotic map due to the strong resonance interaction. After the canonical transformations, (I1,I2,ϑ1,ϑ2) → (J1,J2,ϕ1,ϕ2) defined by

ϕ1 = ϑ1, ϕ2 = ϑ2 − a2ϑ1,J1 = I1 + a2I2,J2 = I2

or (I1,I2,ϑ1,ϑ2) → (P1,P2,ψ1,ψ2) such that

ψ1 = ϑ1 − a2ϑ2/a3, ψ2 = ϑ2,P1 = I1,P2 = I2 + a2I1/a3

the Hamiltonian (15) can be written in terms of the resonant Hamiltonian corresponding to the resonances I1 +a2I2 = 0 1 or a2I1 + a3I2 = 0 , as J 2 bJ 2 H¯(J ,J ,ϕ ,ϕ ,τ) = 1 + 2 + cosϕ + εcos(ϕ + a ϕ ), 1 2 1 2 2 2 1 2 2 1 or 2 2 a3P2 bP1 H˜(P1,P2,ψ1,ψ2) = + + εcosψ2 + cos(ψ1 + a2ψ2/a3), 2 2a3 p revealing that the resonance half-widths are 2 and 2 ε/a3 respectively. A massive overlap of the low order primary resonances takes place when their separation is of the order of two times their half width, that is when η > ηc = 2 −1/2 0.5π(1+a2) , that for the a2 value here considered (a2 = 0.25), leads to ηc ≈ 1.52. On the other hand, the overlap 2 2 2 2 −1 of the high order resonances takes place when (η ε)c = 0.25π a3(a2 + a3) . For instance, setting η = 1(η < ηc), εc ≈ 1.7 Therefore under the condition η < ηc, ε < εc and away from resonance crossings, the motion around the center of the resonances should be stable. Moreover, since we consider values of a2,a3 and ε such that bε > 0, around the resonance intersection the dynamics is also stable since the fixed point p4 is totally elliptic.

1

0.5 m 2 0

/ I Figure 1. Contour plot of the MEGNO 2 I for the map (13) for a2 = 0.5,a3 = 1.25 for ε = 0.3 and η = 0.6 after N = 600 iterates. -0.5 The initial values of the phases are fixed to ϑ1 = ϑ2 = π such that the elliptic-elliptic fixed point at (I1,I2) = (0,0) belongs to -1 the section. The red line corresponds to -1 -0.5 0 0.5 1 m the resonance I1 + a2I2 = 0 while the yel- I1 / I1 low one to a2I1 + a3I2 = 0.

Considering the reduced map M, i.e., (I1,I2) ∈ D with opposite sides identified and adopting η = 0.6,ε = 0.3, the resonance web is shown in Fig. 1, where a fast dynamical indicator (the MEGNO, see [11, 14, 15]) was used for separate stable, regular motion from the chaotic regime.

1All the resonances with k,k0 6= 0 are identical to those with k,k0 = 0. 7

The center of the lower order resonance is drawn in red (I1 +a2I2 = 0) while the one for the higher order resonance appears in yellow (a2I1 + a3I2 = 0). The figure is a contour plot of the final values of the MEGNO after N = 600 iterates for an equispaced grid of 2000×2000 pixels for (I1,I2) ∈ D, with ϑ1(0) = ϑ2(0) = π such that the elliptic-elliptic fixed point at (I1,I2) = (0,0) lies on this section, since εc > 0 for the considered values of a2 and a3. The final values of the MEGNO, hY i, are displayed such that light colors represent regular, periodic or quasiperiodic trajectories, hY i ≤ 2, while dark colors indicate unstable chaotic motion hY i ≈ µN/2  2, where µ is the mLE of the corresponding trajectory. The actual resonance web is quite similar to the expected theoretical one. Besides the intersection at the origin between the low order resonance and the higher order one, several other resonances are present, those with k,k0 6= 0 that show up parallel to the latter. Note that all the crossings between these primary resonances are identical, their dynamical properties around each junction being the same as the one at the origin. Many other resonances, which are linear combinations of the three involved frequencies,

−1 m1ω1(I1,I2) + m2ω2(I1,I2) + 2πm3η = 0, mi ∈ Z can also be identified as very narrow channels. For the adopted values of a2 and a3, the lower order resonance, n0 = 3, and the higher one n1 = 5, satisfy n1 < 2n0 a required condition for H to be a valid representation of the normal form around a resonance crossing (see [18]). A relevant aspect of this map is that diffusion along resonances occurs and thus it turns out interesting to investigate the time rates of the instabilities in M, particularly along the primary resonances.

B. Diffusion

In this section we focus on the diffusion that takes place in the map (13) along the homoclinic tangle of the primary resonances, after adopting a section that includes the hyperbolic-hyperbolic fixed point p4 and for values of the parameters such that η < ηc, ε < εc. Thus in what follows we adopt a section defined as S = {(I1,I2) ∈ D : ϑ1 = ϑ2 = 0}. We consider somewhat larger values of the parameters, ε = 0.6,η = 0.7 but smaller than the critical ones, and take 20 initial conditions along the two main resonances. Fig. 2 (left) presents a MEGNO contour plot of M on S for the adopted values of the parameters as well as the selection of the initial conditions.

1 1

0.5 0.5 m m 2 0 2 0 / I / I 2 2 I I

-0.5 -0.5

-1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 m m I1 / I1 I1 / I1

Figure 2. Action space of the map M for ε = 0.6,η = 0.7 and the selection of initial conditions, in magenta on the resonance I1 + a2I2 = 0 and in green on a2I1 + a3I2 = 0 (left). Observed diffusion for an initial ensemble located on the resonance a2I1 + a3I2 = 0 (right).

The considered values of η,ε are larger than the ones in Fig. 1 and the numerical experiments show that the diffusion m m spreads beyond the region Σ = 4I1 I2 but confined to the homoclinic tangles of the main resonances as Fig. 2 (right) shows for an ensemble located on the resonance a2I1 + a3I2 = 0 with the largest value of I1(0). In this example m |I1| > I1 , so the normalization constant Σ/q should be modified in such a way that it takes into account that σ could exceed Σ. Let us proceed with a series of numerical experiments. 8

9 First we iterate each of the initial conditions, (I1(0),I2(0)), on both resonances up to N ≤ 10 and compute the m m time (or number of iterates) after which |I1| ≥ I1 or |I2| ≥ I2 on the section S = {I1(k),I2(k): |ϑ1|+|ϑ2| < 0.02}. In m m m m other words, we determine the escape time, tesc, as the time when the trajectory leaves D = (−I1 ,I1 ) × (−I2 ,I2 ). Later, we compute the average escape time htesci over ensembles of np = 100 initial conditions centered around 8 (I1(0),I2(0)) for N ≤ 5 × 10 . The use of an ensemble to determine an average time would reduce stickiness effects and should provide a smooth dependence of htesci on the initial conditions. Afterwords we compute DS by means of (11) considering an ensemble of np = 1000 initial conditions around each 6 m of the 40 values of (I1(0),I2(0)) and after N = 5 × 10 , with q = 2000 × 2000 using Ij(t) mod(Ij ) but also keeping the values of Ij(t) ∈ R in order to modify the normalization constant. To compute the derivative of the entropy, we 3 5 take intervals δt = 5 × 10 , leading then to Ns ≈ 2.5 × 10 iterates on S. m m m As mentioned since |Ij| could exceeds Ij , we replace Σ = 4I1 I2 → Σe ≈ σ, that we estimate by means of the max min max min maximum and minimum values attained by the actions, σ ≈ (I1 − I1 )(I2 − I2 ). Indeed, whenever σ > Σ, the normalization constant in (12) should be modified in such a way that σ/q (instead of Σ/q) provides the effective area of the unit cell. Thus, an escape time can be estimated as

m 2 m 2 S (I1 − I1(0)) + (I2 − I2(0)) tesc = K , (18) DS

S where the factor K ∼ 1 takes into account the fact that tesc depends on the escape route in action space on the section S. Indeed, if for instance the escape occurs only along the resonance a2I1 + a3I2 through I1, the second term in (18) m m m should be modified as I2 → −a2I1 /a3 < I2 and therefore the above definition of S with K = 1 would overestimate the actual escape time. It is clear that this factor depends on the dynamics of the system for the given values of the parameters.

0.9 b (R1) b (R2)

0.85

0.8

Figure 3. Exponents obtained after a linear 0.75 b 2 2 2 fit of Var(If ) = Dt with If = I1 +I2 for the selected initial conditions, in magenta on the 0.7 -0.4 -0.2 0 0.2 0.4 resonance R1 = I1 +a2I2 = 0 and in green on I (0) 1 R2 : a2I1 + a3I2 = 0.

6 Finally, we also compute the ensemble variance over the np = 1000 initial conditions after N = 5 × 10 iterates and numerically determine both, the exponent b and the coefficient D by recourse to a mean square fit on a power law b 2 2 2 Var(If ) = Dt , where If is a fast action, in this case If = I1 + I2 . Whenever b ≈ 1, D would lead to the expected diffusion coefficient provided that correlations among the phases are negligible. V m 2 m 2 Thus an escape time can also be derived from the estimate of D, tesc = K((I1 − I1(0)) + (I2 − I2(0)) )/D, for initial conditions on both resonances. Since the linear fit was performed for each ensemble of initial conditions, we expect a non smooth behavior of D V V or tesc. This particular behavior of tesc would be mostly determined by the fit of b, the smaller b leads to the larger D. In any case we found 0.71 < b < 0.88, as Fig. 3 reveals, so the diffusion, at least for the considered motion times, is not normal. Therefore the diffusion coefficient D obtained by a numerical fit on the variance evolution would not provide a good measure of the actual diffusion rate. S V m Fig. 4 shows the results for tesc,htesci,tesc,tesc. We set K = 1/4 in such a way that in (18) (Ij − Ij(0))/2 is the S average distance traveled by the trajectories before the particles escape from D. We observe that tesc provides a good S and smooth estimate of the actual escape time in comparison with tesc. The values of tesc for initial conditions on both resonances is nearly the same, consistent with the periodicity of the map and the fact that the diffusion spreads over the same resonances. The fluctuations in tesc are clearly due to stickiness, htesci is nearly constant, it does not present significant oscil- S lations and in any case tesc is quite close to htesci. V The oscillations in tesc are similar to that observed in the exponent b and consequently opposite to D. In any case V tesc underestimate htesci, in some cases in about two orders of magnitude. 9

R1: I1+a2I2=0 R2: a2I1+a3I2=0

9 tesc, np=1, t=1 x 10 8 < tesc >, np=100, t=5 x 10 8 8 S 10 10 t esc V t esc

7 7 10 10

6 106 10

105 105 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

I1(0) I1(0)

Figure 4. Escape times in the map M for ε = 0.6,η = 0.7 and 40 initial conditions on the homoclinic tangle of the resonances I1 + a2I2 = 0 and a2I1 + a3I2 = 0 after setting K = 1/4.

C. II. A system of continuous time: The planar 3BP

Herein we present as an application the general stability analysis of the exoplanetary system HD 181433 [6, 7, 21], in the context of the planar three-body problem (3BP). We follow some considerations already exposed in [13], but now focus on the relationship between a computed diffusion coefficient DS and a global instability time-scale associated to an specific initial condition (IC) of a given phase space. A preliminary architecture for HD 181433 was firstly proposed in [6]: a three-planetary system with minimum masses of 0.02MJup, 0.64MJup and 0.54MJup, where MJup denotes Jupiter’s mass. These planets are orbiting a K- type star with a mass of 0.86M [6], close to the Solar mass M . However, the obtained values for the eccentricities locate the two outer planets in trajectories of rather unstable character. Later on, new nominal solutions for the system were derived in [21] revealing an almost 7/1 mean motion resonance (MMR) between the two massive planets. According to [7] instead, such planets are placed near a 5/2 MMR. In the present work, we adopt the solution given in [21] derived taking into account further data from recent observations. The concomitant orbital parameters are displayed in Table I, the initial mean anomalies Mi (i = 1,2,3) being obtained from the indicated values of T0.

Table I. Three-planet solution for HD 181433 given in [21]. Parameter Unit HD 181433 b HD 181433 c HD 181433 d

m × sini [MJup] 0.0223 ±0.0003 0.674 ±0.003 0.612 ±0.004 a [AU] 0.0801 ±0.0001 1.819 ±0.001 6.60 ±0.22 e 0.336 ±0.014 0.235 ±0.003 0.469 ±0.013 P [day] 9.37452 ±0.0002 1014.5 ±0.6 7012 ±276 ω [deg] 210.4 ±2.5 8.6 ±0.7 241.4 ±2.4

T0 [day] 52939.16 ±0.06 52184.3 ±1.9 46915 ±239

The proposed dynamical architecture of this system, with a small inner planet very close to the host star and two giant planets in wider orbits, may be approximated to a simpler model where the inner body is neglected; in fact, it is possible to verify, through numerical integrations, that the presence of the lighter body does not globally disturb the motion of the two external ones in long-term time-scales. Indeed, notice that planet b’s mass barely amounts ∼ 3% of the remaining masses and, as a consequence, its presence has almost no disturbative effect on the heavier bodies, which is specially true on taking into account the distance ratio between the inner planet and the external ones. Therefrom, the HD 181433 system can be studied in the framework of the 3BP: the host star with two orbiting bodies of commensurable masses, namely HD 181433 c and d (hereafter, we will use the subscripts 1 and 2 to indicate each one, respectively). Herein we illustrate the application of the Shannon entropy approach to perform a study of planetary stability and compare the attained results to those outcoming from other well known current tools. 10

D. ICs in a line segment of a2

Firstly, we took a set of ICs on a segment of the outer semi-major axis, being 4AU ≤ a2 ≤ 6.5AU, and fixed all the other orbital parameters to their nominal values (see Table I). Though we have excluded from the interval the nominal position of the system (a20 = 6.6 AU), there is no loss of meaning in regards to our illustrative purpose. Using the Ncorp code [5] developed by our group, we integrated a set of 600 ICs inside the defined range of the outer planet semi-major axis. We used a Bulirsh-Stoer integrator with a precision ll = 12, for a total integration time T = 109 years and a sampling step h = 102 years, in order to monitoring the instability time-scales of the system in the considered region of its phase space. Fig. 5 presents the results of such integrations: the y-axis shows the corresponding escape times2 of each IC of the considered interval, with the upper limit of 109 years and the grey dots representing each IC that was numerically solved.

Figure 5. Distribution of 600 ICs integrated in the range [4.0,6.5] AU of the outer semi-major axis a2, with their corresponding system lifetime. The straight lines detach the nominal position of some resonances present in the region. Our numerical solutions shows with reasonable resolution the changing in the instability time-scales of the system as one approaches the resonances’ locations.

We notice these prominent structures indicating a fast increase in the predicted lifetime of the system which coincide with the nominal positions of high-order MMRs, namely the 4/1, 5/1 and 6/1 MMR, highlighted with red lines in Fig. 5. Such resonances seem to provide a protective mechanism, keeping conditions that are inside their width apart from the quick instabilities that can be detected by the surrounding dots. With a lighter tone of red, we have also indicated in the picture the nominal position of the weaker 9/2 and 11/2 MMRs. Furthermore, one may also note a considerable dispersion between adjacent points: see e.g. the region separating the nominal resonances (4.5 AU < a2 < 5.5 AU), or ICs with a2 & 5.5 AU. Notice that there are dots separated by less than 0.1 AU in the x-axis that differ up to almost two orders of magnitude in the y-axis. Even considering the intrinsic numerical errors due to the numerical integration, such a dispersion points out the intrinsic chaoticity associated to the dynamics of the system around this region. The same set of ICs was integrated using the Shannon entropy methodology. Firstly, our routine operates a rescaling ∗ of the system time-space dimensions, such that the initial outer semi-major axis is taken as a2 = 1 AU (the "∗" symbol indicates a rescaled quantity): let η > 0 be a factor that either can expand or compress the system’s real architecture, ∗ i.e ai = ηai. By our proposition, the routine admits that η = 1AU/a2. It is easily verifiable that the intrinsic time- scales of the system (orbital periods and therefore, the secular and resonant periods) are also rescaled by a factor that goes as ∼ η3/2. We defined the partition box based on the concepts of a macroscopic orbital stability (in the Hill’s criteria) [26]. The partition box for each planet can be pictured as a rectangular area in the system’s phase space, with extensions [−∆ai,∆ai]×[−∆ei,∆ei], and where the center is occupied by the specific pair (ai,ei) of the IC that is being evaluated. The subindex i = 1,2 corresponds to the inner and outer planet respectively. Notice that a global diffusion coefficient

2In the present work, we call "escape time" the instant at which the system is destroyed as a consequence of the dynamical features of the trajectories: either both orbits approach each other to distances with high potential of collision, or the system is driven into planetary scattering processes, causing the ejection of the outer body or the inner one to "fall onto" the central star. 11

Figure 6. Comparison between the "pure" numerical integrations (Fig. 5) and estimations of instability timescales using the Shannon approach. We show the results considering two values for the proportionality factor K. The blue line highlights the nominal position of the semi-major axis a20 corresponding to the outer planet.

DS is estimated for the trajectories described by each orbiting body. We took ∆a1 = ∆a2 = ∆h, where  1/3 √ (m1 + m2) (a1 + a2) ∆h = 2 3RH ; RH = , (19) 3m0 2

RH being the mutual Hill’s radius of the planets and where m0 is the star mass. In regards to the eccentricities, we set ∆e1 = ∆e2 = 0.5 (the singular cases ei − ∆ei < 0 or ei + ∆ei > 1 are "naturally" avoided through internal conditions of the routine). Afterwards, we used these values of ∆ai and ∆ei to reconstruct the box in terms of the Delaunay- like variables Li and Gi, such that overlooking the respective mass factors and gravitational constant, Li = ai and 2 Gi = ai(1 − ei ), i = 1,2. Thus defined, L,G are the square of the classical Delaunay variables (factors aside). In fact, throughout this work, it proved more adequate to deal with diffusion and to estimate a diffusion coefficient in terms of variables sharing the same dimensions. We used a partition size of q = 1600×1600 cells and the total integration time T was defined as the minimum value between forty times the (rescaled) secular period of the system3 and 105 yrs. In this first experiment, we allowed the routine to establish a sampling rate h such that the total number of orbital points N = T/h be ten times the value of q (hence resulting in an rigorous relation N/q  1). Finally, each IC was integrated inside an ensemble of ten other −3 "ghost-systems" surrounding the central IC with infinitesimally close displacements (∼ 10 around both variables ai and ei). The clear advantage of taking ensembles of systems have already been thoroughly discussed in [5, 13]. Fig. 6 shows a comparison between the escape time estimated via the Shannon approach τesc(S) (red squares) and the values outcoming from the crude numerical integration of the equations of motion (as the ones in Fig. 5) (black dots). The value of τesc(S) corresponding to a given IC was obtained as follows. For each planet, a coefficient DS,i and an escape time τesc,i are derived in the fashion 2 2 σ(Li,Gi) (∆Li) + (∆Gi) DS,i = q0,i(t)S˙i(t); τesc,i = K , (20) q DS,i

where σ(Li,Gi) = (Lmax,i −Lmin,i)(Gmax,i −Gmin,i) is the maximum area reached out by the phase variables (Li,Gi) of each trajectory during the elapsed time, while the numerator in the expression of τesc,i is given by a quarter of the extent of the partition box in the action-variables (Li,Gi) centered in the IC. Ultimately, the final estimate for the global escape time of the system was acquired as the minimum of the individual escape times, τesc = min{τesc,1,τesc,2}. We tested two different values for the K factor, whose magnitude may be attached to the dynamics of the system, more precisely to the direction in which diffusion proceeds. In Fig. 6, we observe that K = 1 shows a very reasonable agreement with the results coming from the long term integrations of the Ncorp code. Notwithstanding, it is noticeable the sharply structures outlined by the red squares in both panels, coincident with the nominal positions of the MMRs highlighted in Fig. 5.

3This values is completely arbitrary. The entropy evaluates the variables (L,G) along time and to be sure that the system would run for a sufficient time so that it would make sense to compute the derivative. A sufficiently long integration time T is directly related to the ergodicity hypothesis. 12

E. Dynamical maps: MEGNO vs τesc

As a second experiment, we proposed the comparison between dynamical maps of HD 181433 obtained by different approaches.

Fig. 7 displays two dynamical maps constructed in the (a2,e2) domain of the HD 181433 system’s phase space. The left-hand panel shows a map parameterized by the MEGNO indicator, hY i, computed for each trajectory through 4 integrations along a 5 × 10 time-span. For a grid of 400 × 400 initial values of (a2,e2), with 4 AU ≤ a2 ≤ 10 AU and e2 ∈ [0.0,0.8]), both the numerical integration and the computation of the MEGNO value were performed by the Ncorp routine [5], applying a Bulirsh-Stoer integrator with precision ll = 13 and a sampling rate of h = 1 year.

Figure 7. A MEGNO map (left) and a τesc-map (right) in the (a2,e2) plane corresponding to the same ICs for HD 181433. The black symbol indicates the nominal position of the system, while the black lines highlight other MMRs present in the phase space.

The right-hand panel of Fig. 7 shows a dynamical map constructed from diffusion estimations in the same region 5 of the phase plane (a2,e2), i.e. a τesc-map. We used a grid of 100 × 100 ICs that were integrated for 10 years, with h = 10−1 yr and using an ensemble of five "ghost-systems". We took, as in the first experiment a partition of q = 1600 × 1600, cells onto the boxes in the (ai,ei)-planes, with ∆ai = ∆h and ∆ei = 0.5 (i = 1,2). Therefrom, it is 6 N = (ne + 1)T/h = 6 × 10 , where ne is the number of "ghost-systems" inside the ensemble, and hence N/q ∼ 2 (we recall that the mandatory relation of the Shannon theory lies upon the relation N/q0  1, which clearly is respected 4 in this experiment). From the estimations of the diffusion coefficients DS,i;i = 1,2 for each one of the 10 ICs, we obtained a corresponding (global) escape time τesc that serves as a predictive measure for the instability time-scales for the HD 181433 system, through which we parameterized the τesc-map (see the respective color-bar). At first, let us notice the qualitative good agreement between both maps. Indeed, in a general way, the Shannon estimates of the system’s lifetime shows a good correspondence with the indications of regularity/irregularity of solutions coming from the MEGNO-map. Also, the τesc-map shows that prominent regions of stability (high lifetimes) are coincident with the presence of several MMRs: besides the 4/1, 5/1 and 6/1 already indicated before, for values of a2 > 7 AU notice also the 7/1, 8/1, 9/1, 10/1 and 12/1 commensurabilities. The nominal position of the system 10 (considering the solutions proposed in [21]) lies in a region with escape times τesc ∼ 10 years, very close to unstable solutions of high eccentricities (e2 > 0.5), and more stable solutions for e2 < 0.4 (corroborating the results given in [21]).

Furthermore, we should highlight the quantitative information revealed by the τesc-map against a more qualitative information provided by the MEGNO map. In particular, we notice the much more prominent gradient of the system life-time observed in the right-hand panel in the transient region between unstable ICs and long-term stable solutions 7 9 for which the τesc values range in between 10 -10 yrs, in comparison with the MEGNO map where such a region is revealed just as irregular corresponding mostly to values hY i  2. 13

IV. CONCLUSIONS

The Shannon entropy proves to be a very efficient tool to display the global and local dynamics of a multidimensional system as well as to provide accurate estimates of the diffusion rate. Its computation is rather simple, it just requires a counting box scheme after solving the equations of motion of the system for a given ensemble of initial conditions and the computation of the mean time derivative of the entropy evolution. Herein an improvement of the best choice of the partition is given, the size of the unit cell depends on the character of the diffusion, i.e. rather confined or extended in action space, that leads to a larger or smaller surface density of iterates on the adopted section. The application to a 4D map reveals its efficiency to estimate time-scales for chaotic instabilities in relatively short motion times in comparison with the ones derived from the diffusion coefficient obtained from the variance evolution. Particularly interesting is the implementation to the 3BP. Its computation is quite simple since the solution of the first variational equations is not required to get DS. This fact allows to get the escape times for each planet involved in the system, while in general this cannot be done when the variational equations are required. An escape-time map constructed by recourse to the Shannon entropy estimates provides much more dynamical information in comparison with a MEGNO-map, namely, the MMR resonance structure together with the actual time-scale of stability of the system.

ACKNOWLEDGEMENTS

This work was supported by grants from Consejo Nacional de Investigaciones Científicas y Técnicas de la República Argentina (CONICET), the Universidad Nacional de La Plata and the Universidad Nacional de Córdoba, IAG- Universidade de São Paulo and CAPES.

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The authors declare that they have no conflict of interest. P. M. Cincotta, C. Giordano, R. Alves Silva, C. Beaug´e: Conceptualization, Methodology, Software, Formal analysis, Writing-Original Draft, Writing-Review & Editing

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