---
abstract: 'We extended a previous qualitative study of the intermittent behaviour of a chaotical nucleonic system, by adding a few quantitative analyses: of the configuration and kinetic energy spaces, power spectra, Shannon entropies, and Lyapunov exponents. The system is regarded as a classical “nuclear billiard” with an oscillating surface of a 2D Woods-Saxon potential well. For the monopole and dipole vibrational modes we bring new arguments in favour of the idea that the degree of chaoticity increases when shifting the oscillation frequency from the adiabatic to the resonance stage of the interaction. The order-chaos-order-chaos sequence is also thoroughly investigated and we find that, for the monopole deformation case, an intermittency pattern is again found. Moreover, coupling between one-nucleon and collective degrees of freedom is proved to be essential in obtaining chaotic states.'
author:
- Daniel Felea
- Cristian Constantin Bordeianu
- Ion Valeriu Grossu
- Călin Beşliu
- Alexandru Jipa
- 'Aurelian-Andrei Radu'
- Emil Stan
title: 'Intermittency route to chaos for the nuclear billiard - a quantitative study'
---

\[intro\]Introduction
=====================

We begin by briefly reminding that a conjugated continuous effort has been made to relate the emergence of the collective energy dissipation through one and two-body nuclear processes with the chaotical behaviour of nuclear systems .

A few options in choosing the collective oscillation frequencies have come into focus in the past years, in connection with the onset of chaoticity for “nuclear billiards”. First of all, the issue of dissipation into thermal motion of the adiabatic collective vibrational energy of the potential well was treated for several multipolarities by Burgio, Baldo *et al.* . On the other hand, when trying to associate different vibration frequencies to various nuclear processes, the path to chaos was found to be changed with the order of multipole [@felea-01; @felea-02; @bordeianu-08a; @bordeianu-08b; @bordeianu-08c; @felea-09a].

This paper was intended to bring a quantitative argumentation, based on a systematic study of the configuration and kinetic energy spaces, power spectra, informational entropies, and largest Lyapunov exponents. The study was done in completion of a few qualitative types of analysis previously presented [@felea-09a]: sensitive dependence on the initial conditions, single-particle phase space maps, fractal dimensions of Poincare maps, and autocorrelation functions.

In short, we remind that, by studying the nucleonic dynamics in a Woods-Saxon potential, one can find an increase of the chaotical degree of the system behaviour as raising the frequency of 2D wall oscillation.

The main result of [@felea-09a] was reported in relation with an intermittent route to chaos for the monopole vibrations close to the resonance phase of a nuclear interaction. Still, we mention that the purpose of these two coupled etudes was only to emphasize the detection of such intermission for the “nuclear billiards” and not to establish its type according to [@pomeau-80], nor to compare it with other intermittency patterns from known experimental results .

\[sec:1\]Toy model
==================

We continue the study on a classical dynamical system proposed by Burgio, Baldo *et al.* , system composed of a number of $A$ nucleons with no charge, spin, or internal structure. A two-dimensional deep Woods-Saxon potential well, regarded as “nuclear billiard”, is periodically hit by the nucleons. The Bohr Hamiltonian in polar coordinates is a sum of two components: kinetic ($E_{kin.}$) and potential ($E_{pot.}$), the kinetic one decoupling into radial ($E_r$), centrifugal ($E_L$), and collective terms ($E_{coll.}$):

$$E_{kin.} = E_r + E_L + E_{coll.} = {\sum_{j=1}^{A}}\left( \frac{p_{r_{j}}^{2}}{2m}+\frac{p_{\theta _{j}}^{2}}{2mr_{j}^{2}} \right)%
+\frac{p_{\alpha }^{2}}{2M},$$

$$E_{pot.} = {\sum_{j=1}^{A}} V\left(r_{j},R\left( \theta _{j}\right) \right)+\frac{M\Omega ^{2}\alpha ^{2}}{2}.$$

The phase space is defined by particle and collective momenta and their conjugate coordinates: $\left( r,p_{r}\right)$, $\left( \theta,p_{\theta}\right)$ and $\left( \alpha,p_{\alpha}\right)$. The collective coordinate $\alpha $ oscillates with $\Omega $ frequency, the Inglis mass $M$ is equal to $mAR_{0}^{2}$, and the nucleon mass: $m=938\ \rm{MeV}$.

For the time being we are not interested in studying the nucleon dynamics beyond the Woods-Saxon barrier:

$$V\left( r_j,R\left( \theta _j\right) \right) =\frac{V_0}{1+\exp \left[ \frac{r_j-R\left( \theta _j,\alpha \right) }a\right] },$$

and therefore we choose a deep well: $V_{0}=-1500\ \rm{MeV}$ and accordingly, a low value for the diffusivity coefficient: $a=0.01\ \rm{fm}$.

When considering the two-dimensional case, the frontier of the collective motion is described as a function of the collective variable $\alpha $ and of the Legendre polynomials $P_{L}\left( \cos \theta_{j}\right)$ [@burgio-95; @baldo-96; @baldo-98; @felea-09a]:

$$R_j = R\left( \theta _j,\alpha \right) =R_0\left[ 1+\alpha P_L\left( \cos \theta_j\right) \right].$$

The oscillation degree of the potential well $L$ is considered for the monopole $\left(0\right)$, dipole $\left(1\right)$ and quadrupole case $\left(2\right)$.

If the surface has a stationary behaviour, or whenever one takes into account the uncoupled Hamilton equations ([UCE]{}) for the particle:

$$\stackrel{\cdot }{r_j}=\frac{p_{r_j}}m,\ 
\stackrel{\cdot }{\theta _j}=\frac{p_{\theta _j}}{mr_j^2},\ 
\stackrel{\cdot }{p_{r_j}}=\frac{p_{\theta _j}^2}{mr_j^3}-\frac{\partial V}{\partial r_j},\ 
\stackrel{\cdot }{p_{\theta _j}}=-\frac{\partial V}{\partial R_j}\cdot \frac{\partial R_j}{\partial \theta _j},$$

and collective degrees of freedom (*d.o.f.*):

$$\stackrel{\cdot }{\alpha }=\frac{p_\alpha }M,\ 
\stackrel{\cdot }{p_\alpha}=-M\Omega ^2\alpha -\sum_{j=1}^{A}\left(\frac{\partial V}{\partial R_j} \cdot \frac{\partial R_j}{\partial \alpha }\right),$$

$R_{0}$ has a fix value, chosen for consistency with previous papers [@burgio-95; @baldo-96; @baldo-98; @felea-09a] as $6\ \rm{fm}$.

A Runge-Kutta type algorithm (order 2-3) with an optimized step size was used for solving the system of differential equations, while keeping the absolute errors for the phase space variables under $10^{-6}$ and conserving the total energy with relative error: $\Delta E/E \approx 10^{-8}$ (Fig. \[fig:1\]).

We imposed the equilibrium condition between the pressure exerted by the particles and the mechanical pressure of the wall [@burgio-95; @baldo-96; @baldo-98; @felea-09a] and thus obtained for the initial equilibrium value of the collective variable, perturbed with a small value:

$$\alpha_{0}=\frac{-1+\sqrt{1+8T/{mR_0^2\Omega ^2}}}{2}+0.15,$$

where $T=36\ \rm{MeV}$ [@burgio-95; @baldo-96; @baldo-98; @felea-09a] is the two-dimensional kinetic energy and also the temperature of the nuclear system, when considering the natural system of units ($\hbar=c=k_{B}=1$).

\[sec:2\]The quantitative analysis of the route to chaos
========================================================

We carry on the study begun in [@felea-09a] by gradually changing the degree of vibration of the potential wall from a slow motion (adiabatic state) to a rapid one (the so-called resonance state of the interaction). In the first case, the collective motion is described by a radian frequency smaller than $0.05\ c/\rm{fm}$. The latter is dominated by frequencies close to the one-nucleon collisional frequency:

$$\omega _{part}=\frac{\pi}{R_0}\cdot\sqrt{\frac{2T}{m}}\approx 0.145\ c/\rm{fm}.$$

The physical motivation for studying the one-nucleon chaotical dynamics in the “nuclear billiard”, ranging the frequencies from the adiabatic to the resonance regime of a nuclear interaction, is explained in some detail in [@felea-09a].

We briefly remind that the process of nuclear multifragmentation can be viewed as a resonance process and that for smaller excitation energies, nuclear evaporation or breakup of a projectile nucleus occurs when the energy is shared between the collective and one-nucleon degrees of freedom.

By using a few types of analyses: sensitive dependence on the initial conditions, single-particle phase space maps, fractal dimensions of Poincare maps and autocorrelation functions, we emphasized that an intermittent route to chaos is observed in the monopole case when increasing the vibrational frequency to $\Omega = 0.1\ c/\rm{fm}$ [@felea-09a]. In the resonance phase of the interaction the onset of chaotical behaviour was found to be earlier than at any other adiabatic oscillations of the Woods-Saxon potential well.

We present here other methods [@schuster-84] promoting the idea that the degree of chaoticity increases when moving from the adiabatic to the resonance regime: analyses of the configuration and kinetic energy spaces, power spectra, generalized informational entropies and Lyapunov exponents. Furthermore, we try to identify possible pathways to chaos, including the intermittent one, previously put in evidence [@felea-09a].

Configuration and kinetic energy spaces
---------------------------------------

In order to establish if a specific physical system presents a chaotic dynamics and to identify possible routes to chaos we analyzed the behaviour of a small bunch of trajectories in the configuration and in the kinetic energy space, respectively. For example, we took five trajectories separated by an $\epsilon = \Delta r=0.01\ \rm{fm}$ aperture, while keeping the rest of the initial phase space variables constant and let the system evolve over a given time $\left(\Delta t = 1,600\ \rm{fm}/c\right)$.

For the transient stages from adiabatic to resonance, the temporal evolution of an initially confined trajectory bundle was studied for the monopole and dipole oscillation modes of the potential wall and also for the limit situation, in which the individual and collective degrees of freedom remain uncoupled (Figs. \[fig:2\] and \[fig:3\]).

The configuration space revealed a high degree of symmetry in $\left(r,\theta\right)$ plane in both cases, $\left( 4+2\right)$ uncoupled nonlinear differential equations and monopole (left and middle panels). Also, that the central zone remained uncovered, reflecting the conservation of the nucleon angular momentum.

Another important conclusion was issued from the definition of the stability concept of a dynamical system. For the aforementioned cases the dispersed trajectory pack periodically regroups on the frontier that delimits the forbidden zone of the phase space. This type of behaviour corresponds with the definition of stability given by Poisson (for e.g., in [@holmes-96]). We will herewith remind that the Poisson stability defines as steady the movement of a particle system of which configuration comes close, from time to time, to the initial position.

At a first glance, on the simple [UCE]{} case one can distinguish two extremities of the radius of the particle periodic motion in the 2D potential well: $r_{min}(UCE) \approx 1.42\ \rm{fm}$, and $r_{Max}(UCE) \approx 5.96\ \rm{fm}$. These values correspond to the roots of the boundary equation:

$$E = \frac{p_{\theta}^2}{2mr^2}+V\left(r \right),$$

for a given one-nucleon energy $E$, when the radial component of velocity vanishes ($\stackrel{\cdot }{r}=0$).

The analysis of the particle motion in the configuration space is similar to that applied to any system with bound unclosed trajectories. The nucleonic motion takes place within a circular crown (the so-called *annulus*) determined by the concentric circles of $r_{min}$ and $r_{Max}$ radii. The trajectory is symmetric about any turning point.

For $\Delta t = 1,600\ \rm{fm}/c$ we descry in the [UCE]{} case as much as $27$ distinct apocenters (at $r=r_{Max}$). These are correlated with a number of $13$ complete and one incomplete revolutions about the center of the force field (*i.e.* $27$ straight lines before $r(t)$ changes its sense of variation).

The particle completely sweeps over twice the 2D configuration space after $1,537\ \rm{fm}/c$. However, we notice that the bound trajectories are open, which means that the orbits never pass twice through a given point (see Fig. \[fig:2\]), which is in concordance with Bertrand’s theorem. We briefly remind that for a bound orbit to be closed, the angle between two consecutive apocenters must be:

$$\Delta \theta = 2\pi \cdot \frac{n_{1}}{n_{2}},$$

*i.e.* after $n_{2}$ revolutions about the center, the radius vector should sweep out a multiple $n_{1}$ of $2\pi$ [radians]{}. For the [UCE]{} case above considered we consequently obtain: $\Delta \theta_{UCE} \approx 4\pi/13$ [radians]{} (Fig. \[fig:2\] - left column).

The kinetic energy points are displayed in right isosceles triangular shaped patterns (Fig. \[fig:3\]), whose hypotenuses are described by Eqs. (11) and (12), for the two specific non-chaotical situations: [UCE]{} and the intermittent monopolar “window” emerged at $\Omega = 0.1\ c/\rm{fm}$:

$$E_{L_{UCE}} = 18.05 - E_r,$$

$$E_{L_{\Omega=0.1\: c/\rm{fm}}} = 18.09 - E_r.$$

Moreover, for the intermittency frequency of monopole oscillations, one can notice a second smaller segment with the same negative slope:

$$E_{L_{\Omega=0.1\: c/\rm{fm}}} = 5.87 - E_r.$$

For the uncoupled differential equations there are a couple of extreme values for the centrifugal kinetic energy: $E_{L_{min}}(UCE) = 1.02\ \rm{MeV}$, and $E_{L_{Max}}(UCE) = 18.05\ \rm{MeV}$, associated with $r_{Max}(UCE)$ and respectively, with $r_{min}(UCE)$ (left column of Fig. \[fig:3\] and Eq. (1)).

As for the monopolar intermittency, we can distinguish just five distinct values for the $E_{L}$: $1.32\ \rm{MeV}$, $2.13\ \rm{MeV}$, $3.01\ \rm{MeV}$, $5.87\ \rm{MeV}$, and $18.09\ \rm{MeV}$ (central plot of Fig. \[fig:3\]), correlated with stationary radii: $r_{1} \approx 5.26\ \rm{fm}$, $r_{2} %
\approx 4.14\ \rm{fm}$, $r_{3} \approx 3.49\ \rm{fm}$, $r_{4} \approx 2.51\ \rm{fm}$, and $r_{5} \approx %
1.43\ \rm{fm}$ (Eq. (1), Fig. 2 of [@felea-09a], and central plot of Fig. \[fig:2\]). Thus, the nucleonic motion for the intermittent case is composed of alternated revolutions about the force field centre, forming a cyclic symmetrical structure, for e.g., $r_{1}$, $r_{5}$, $r_{2}$, $r_{4}$, $r_{3}$, $r_{4}$, $r_{2}$, $r_{5}$, $r_{1}$, and so on (Fig. \[fig:2\]). This behaviour can be easily verified through the sensitivity dependence on the initial conditions analysis, previously presented (third column of Fig. 2 - [@felea-09a]). It should also be mentioned that, following this radius alternation, the nucleon covers in 2D configuration space $\approx 2\pi$ radians after $6$ full revolutions in almost $590\ \rm{fm}/c$.

Concluding, we highlight once more, that ordered, non-chaotical events, exhibit periodical symmetrical patterns in the configuration and kinetic energy spaces. This was shown to be a characteristic feature of the uncoupled nonlinear Hamilton equations case and also, of the steady, intermittent behaviour arisen in the monopole case at $0.1\ c/\rm{fm}$ vibrational radian frequency.

A tendency to compactly fill the kinetic energy space when increasing the monopolar vibrations (from $0.05\ c/\rm{fm}$ to $0.145\ c/\rm{fm}$) was observed, except for the intermittency situation above described. For the $L = 1$ oscillation mode of the potential well, it seems that at the same frequency ($\Omega = 0.1\ c/\rm{fm}$) a somewhat intermittent behaviour could also come out, but this was proved to be elusive, as verified when reverting to this issue with the help of informational entropies and Lyapunov exponents and analyzing the system on longer time periods.

Power spectra
-------------

In order to better distinguish between a multiple periodical behaviour that can also exhibit an erratic pattern and chaos we used the Fourier transform of the analyzed signals:

$$x\left( \omega \right) = \lim_{T\rightarrow \infty} \int^{T}_{0} e^{i\omega t} \cdot x\left( t\right) \ dt\ ,$$

$$x\left( \omega \right) = \lim_{N\rightarrow \infty} \sum^{N}_{n=0} e^{i\omega t_{n}} \cdot x\left( t_{n}\right).$$

For a multiple periodical movement the power spectrum:

$$P\left( \omega \right) = \left|x\left( \omega \right)\right|^{2},$$

will only contain a number of discrete lines: the fundamental frequencies of the system and their associated sets of harmonics, while the chaotical behaviour is completely aperiodical and is represented by a continuous or quasi-continuous broadband.

The obtained results are presented in Figures \[fig:4\]-\[fig:7\]. As a persistent feature of the physical system analyzed one should mention that for the monopole and dipole deformation degrees of the potential well (Figures \[fig:5\] and \[fig:6\]) the chaotic behaviour increases in time, thus confirming previous results.

The transition towards a chaotic regime was put again in evidence once passing from the adiabatic to the resonance stage of the interaction. The power spectra reveal, as expected, the intermittent feature of the transition in the monopolar case at $\Omega=0.1\ c/\rm{fm}$.

This can be detected for periods of time large enough ($\Delta t \geq 1,600\ \rm{fm}/c$) to positively identify chaotic patterns, by transition from a quasi-continuous spectrum of the one-nucleon radial coordinate ($\Omega _{ad}=0.02\ c/\rm{fm}$) to a discrete periodical one, containing fundamental frequencies of the system and its harmonics ($\Omega=0.1\ c/\rm{fm}$) and again to a continuous spectrum at the resonance vibrational frequency ($\Omega _{res}=0.145\ c/\rm{fm}$) (Fig. \[fig:5\] - right panels). The order-chaos-order-chaos sequence can be also spotted out for the monopole oscillations in the power spectra of the collective degree of freedom (Fig. \[fig:7\] - second column).

The temporal series of the radius variable show a symmetrical sawtooth waveform for the uncoupled situation at any chosen vibration frequency, and also for the monopole case at adiabatic collective oscillations. For the rest, in general an asymmetrical sawtooth form defines the series, but sometimes, more complicated patterns appear at higher multipole orders (Figures 1-4 - [@felea-09a]).

The difference between two successive maxima in the temporal series of the radius variable for the [UCE]{} case is: $T_{0_{UCE}} \approx 1,537/26 = 59.1\ \rm{fm}/c$ (left column of Fig. \[fig:2\]) and can be also obtained from the sensitive dependence on the initial conditions analysis (see Fig. 1 - [@felea-09a]). The fundamental frequency for the radial sawtooth temporal series: $\omega_{0_{UCE}}=2\pi/T_{0_{UCE}}=0.106\ c/\rm{fm}$ and its first three harmonics: $\omega_{1_{UCE}}=0.212\ c/\rm{fm}$, $\omega_{2_{UCE}}=0.318\ c/\rm{fm}$, and $\omega_{3_{UCE}}=0.424\ c/\rm{fm}$, can be easily traced down in Figure \[fig:4\].

As for the “window” of intermittency at $L = 0$, we obtained: $T_{0_{int}} \approx 590/12 = 49.2\ \rm{fm}/c$ (Fig. 2 - [@felea-09a] and central plot of current Fig. \[fig:2\]). This gives the corresponding fundamental frequency: $\omega_{0_{int}}=0.128\ c/\rm{fm}$ and its associated harmonics: $\omega_{1_{int}}=0.256\ c/\rm{fm}$, $\omega_{2_{int}}=0.384\ c/\rm{fm}$, and $\omega_{3_{int}}=0.512\ c/\rm{fm}$ (Fig. \[fig:5\]).

Shannon entropies
-----------------

In order to further investigate route to chaos, we paid attention to the time evolution of the generalized informational entropy (or Shannon entropy), introduced as usually :

$$S_{Shannon}\left(t\right)=-{\sum^{N\left(t\right)}_{k=1}} p_k \cdot \ln p_k,$$

$N\left(t\right)$ being the number of gradually occupied cells until the time $t$.

This type of entropy is actually a number which quantifies the time rate of information production for a chaotic trajectory [@ott-93]. We consider in the first place the case of a particle that at every moment occupies a cell of the two-dimensional lattice phase space with a $p_k$ probability:

$$p_k=1/N_{total\ cells},$$

where:

$$N_{total\ cells}=N_{r} \cdot N_{p_{r}} \cdot N_{\theta},$$

$N_{r}$, $N_{p_{r}}$, and $N_{\theta}$ are the number of bins of the $\left( r,p_{r},\theta\right)$ lattice. For $p_{\theta}$ is a constant of motion for the monopole and the [UCE]{} cases, we use for comparisons only these three phase space variables.

As an alternative measure for the above defined entropy we also used the cumulative filling percentage of the one-nucleon phase space:

$$\eta\left(t\right)=\frac{N\left(t\right)}{N_{total\ cells}} \cdot 100\ \left(\%\right).$$

In the first place, for a given wall frequency of vibration and for a certain multipolarity (here, for $\Omega _{res}=0.145\ c/\rm{fm}$ and $L = 0$), we studied the dependence of the Shannon entropy with the number of bins. A clear tendency for smoothing the entropy curve was found when decreasing the bin. A reduced number of cells ($N_{b}=2^3$) is characterized by an entropy formed from a small number of high-amplitude Heaviside functions. As the number of bins increases (for e.g., here to $12^3$), the entropy gets a more realistic representation, being composed of a superior number of low-amplitude step functions (Fig. \[fig:8\]).

Moreover, the filling percentage $\eta$ of the one-nucleon phase space maps can drastically differ with the size of the bin. Thus, after the system evolved over $400\ \rm{fm}/c$, a phase space with $8$ bins is entirely covered, $64$ bins can be filled in with $0.8594$ probability, a $26.95$ filling percentage for $512$ cells can be found, and we counted only as much as $226$ bins occupied out of a total of $1,728$ (*i.e.* $\eta = 13.08\ \%$).

At a first glance one can identify a series of entropy plateaus, which could be put in correspondence with stationary or quasi-stationary thermodynamic values of the system if a large number of particles would be under study. Some of them will vanish when considering a large number of bins. However, those surviving for $N_{b} \rightarrow \infty$ could be associated with stationary nucleonic states in the chosen potential well in the limit of a large number of degrees of freedom.

For a given 2D phase space lattice formed of $N_{b}=4^3$ bins we present in Figures \[fig:9\]-\[fig:12\] a comparison between the informational entropies of the physical system in study, starting from the adiabatic stage of interaction and gradually increasing the vibrational wall frequency towards the dipole resonance value, $\Omega _{res}=0.145\ c/\rm{fm}$. The slopes for the resonance frequency case were found to be significantly higher than for the adiabatic one ($\Omega _{ad}=0.02\ c/\rm{fm}$) for all multipolarities involved.

Another comparison revealed significant differences between the onset times of the quasi-constant Shannon entropy values for all cases taken into consideration. Thus, for four vibrational radian frequencies and for four coupling modes of the Hamilton equations we show the informational entropy values after $800\: \rm{fm}/c$ (Table \[tab:1\]) and the associated phase space filling degrees (Table \[tab:2\]). Also, in Table \[tab:3\], are presented the periods of time after which the filling percentages $\eta$ equal unity.

  ----------------------- ------------------------------------------------------------------------------------- -- -- --
   Oscillation frequency                            [UCE]{} & $L=0$ & $L=1$ & $L=2$\                                  
                            $\Omega \:_{ad} = 0.020\: c/\rm{fm}$ & $3.6889$ & $3.7136$ & $3.7842$ & $3.4340$\         
                            $\Omega \:_{ad} = 0.050\: c/\rm{fm}$ & $3.6889$ & $4.0775$ & $4.0775$ & $3.2958$\         
                           $\Omega \:\;\ \   = 0.100\: c/\rm{fm}$ & $3.6889$ & $3.6889$ & $3.8501$ & $4.0073$\        
                            $\Omega _{res} = 0.145\: c/\rm{fm}$ & $3.6889$ & $4.1589$ & $4.0431$ & $4.0254$\          
  ----------------------- ------------------------------------------------------------------------------------- -- -- --

  : \[tab:1\]The computed $S_{Shannon}\left(t=800\: \rm{fm}/c\right)$ of the $\left( r\leftrightarrow p_{r}\leftrightarrow\theta\right)$ one-particle phase space maps at several multipolarities and frequencies of wall vibration

  ----------------------- --------------------------------------------------------------------------------- -- -- --
   Oscillation frequency                          [UCE]{} & $L=0$ & $L=1$ & $L=2$\                                
                            $\Omega \:_{ad} = 0.020\: c/\rm{fm}$ & $62.50$ & $64.06$ & $68.75$ & $48.44$\         
                            $\Omega \:_{ad} = 0.050\: c/\rm{fm}$ & $62.50$ & $92.19$ & $92.19$ & $42.19$\         
                           $\Omega \:\;\ \   = 0.100\: c/\rm{fm}$ & $62.50$ & $62.50$ & $73.44$ & $85.94$\        
                            $\Omega _{res} = 0.145\: c/\rm{fm}$ & $62.50$ & $100.00$ & $89.06$ & $87.50$\         
  ----------------------- --------------------------------------------------------------------------------- -- -- --

  : \[tab:2\]The filling percentage $\eta$ of the $\left( r\leftrightarrow p_{r}\leftrightarrow\theta\right)$ one-particle phase space maps at several multipolarities and frequencies of wall vibration

  ----------------------- ------------------------------------------------------------------------------------ -- -- --
   Oscillation frequency                            [UCE]{} & $L=0$ & $L=1$ & $L=2$\                                 
                            $\Omega \:_{ad} = 0.020\: c/\rm{fm}$ & $>10^{5}$ & $6,023$ & $6,359$ & $5,356$\          
                           $\Omega \:_{ad} = 0.050\: c/\rm{fm}$ & $>10^{5}$ & $1,618$ & $4,223$ & $>10^{5}$\         
                           $\Omega \:\;\ \   = 0.100\: c/\rm{fm}$ & $>10^{5}$ & $11,442$ & $3,241$ & $2,758$\        
                             $\Omega _{res} = 0.145\: c/\rm{fm}$ & $>10^{5}$ & $729$ & $1,887$ & $10,571$\           
  ----------------------- ------------------------------------------------------------------------------------ -- -- --

  : \[tab:3\]The time (in $\rm{fm}/c$) at which the informational entropies of the $\left( r\leftrightarrow p_{r}\leftrightarrow\theta\right)$ one-particle phase space maps at several multipolarities and frequencies of wall vibration have the maximum value (*i.e.* $\eta  = 100\ \%$)

We continue the analysis by further defining the Shannon entropy for a group of $w$ nearby orbits:

$$S_{traject.\ pack}\left(t\right)=ln\ N_{w}\left(t\right),$$

so that the number of occupied cells is:

$$1 \leq N_w(t) \leq w,$$

thus describing the spread of the trajectories at each moment of time $t$ (Figs. \[fig:13\]-\[fig:16\]). When reaching the maximum divergence, the entropy for five distinct phase space paths gets its highest value (Table \[tab:4\]).

  ----------------------- ------------------------------------------------------------------------------- -- -- --
   Oscillation frequency                         [UCE]{} & $L=0$ & $L=1$ & $L=2$\                               
                            $\Omega \:_{ad} = 0.020\: c/\rm{fm}$ & $>10^{4}$ & $1,095$ & $555$ & $688$\         
                             $\Omega \:_{ad} = 0.050\: c/\rm{fm}$ & $>10^{4}$ & $855$ & $476$ & $122$\          
                           $\Omega \:\;\ \   = 0.100\: c/\rm{fm}$ & $>10^{4}$ & $4,133$ & $333$ & $395$\        
                             $\Omega _{res} = 0.145\: c/\rm{fm}$ & $>10^{4}$ & $279$ & $327$ & $469$\           
  ----------------------- ------------------------------------------------------------------------------- -- -- --

  : \[tab:4\]The time (in $\rm{fm}/c$) at which the one-particle Shannon entropies of a pack of $w=5$ close orbits begin having the maximum value (*i.e.* $S_{traject.\ pack} = 1.60944$) for various coupling degrees between the one-nucleon and the collective *d.o.f.* and for the standard chosen wall frequencies

We begin the analysis with the [UCE]{} case. The single and collective uncoupled *d.o.f.* give birth to a quasi-laminar behaviour with a weak development of chaotic states. The one-particle informational entropy shows an identical evolution, no matter the frequency chosen. The orbit covers, after $800\ \rm{fm}/c$, only $62.50\ \%$ of the entire lattice (Table \[tab:2\] and Figure \[fig:9\]) and does not reach $100\ \%$, even after $\Delta t = 100,000\ \rm{fm}/c$ (Table \[tab:3\]). Also, the phase space is not covered up by all five trajectories for the whole range of $10,000\ \rm{fm}/c$ considered, when analyzing $S_{traject.\ pack}$ (Fig. \[fig:13\] and Table \[tab:4\]).

For the dipole oscillations mode, at $\Omega_{ad} = 0.05\ c/\rm{fm}$, it appears that, after only $800\ \rm{fm}/c$, the entropy closes in upon its maximum value: $S_{Max} = ln\ N_{total\ cells} = 4.1589$ (Fig. \[fig:11\] and Table \[tab:1\]). However, on long periods of time, the real tendency is towards filling up the nucleonic phase space as rapid as the vibrational frequency is increased (Table \[tab:3\]). The exact pattern is repeated when studying the Shannon entropy for closeby nucleonic trajectories (Fig. \[fig:15\] and Table \[tab:4\]).

We found quite the same feature for the monopole case, with exception for the intermittent “window” at $\Omega=0.1\ c/\rm{fm}$ (Tables \[tab:1\], \[tab:2\] and Fig. \[fig:10\]). The occupying rate is so small in the intermittent zone, that just at $11,442\ \rm{fm}/c$, the particle would have covered the whole phase space (see Table \[tab:3\]). A similar conclusion can be drawn from Table \[tab:4\] and Fig. \[fig:14\] (with a double temporal scale scanned for the intermittent frequency). The trajectory pack informational entropy reaches its highest value after the longest one-particle evolution time of all: $4,133\ \rm{fm}/c$.

The quadrupole oscillation also reveals an apparent intermittent pattern, this time at $\Omega_{ad} = %
0.05\ c/\rm{fm}$. We call it intermittent because after $800\ \rm{fm}/c$ the nucleon fills in only $42.19\ \%$ of the total number of bins (Figure \[fig:12\] and Table \[tab:2\]), and a longer time than $100,000\ \rm{fm}/c$ is required to get to $\eta  = 100\ \%$ (Table \[tab:3\]). However, this behaviour can be a misleading one, the Shannon entropy for a trajectory bunch showing exactly the opposite (see Fig. \[fig:16\] and Table \[tab:4\]), after $122\ \rm{fm}/c$ the orbits being completely dispersed.

Lyapunov exponents
------------------

We furthermore presented another quantitative analysis: the temporal evolution of the Lyapunov exponents, $\lambda\left(t\right)$. As previously shown, initial adjacent points in the phase space $\Delta x_{0} (t=0)$, can generate in time separated trajectories $\Delta x\left(t\right)$. When studying the evolution of a single phase space parameter, the one-dimensional Lyapunov exponent takes the form:

$$\lambda\left(t\right)=\lim_{\left|\Delta x_{0}\right|\rightarrow 0} ln\left|\frac{\Delta x\left(t\right)}{\Delta x_{0}}\right|.$$

The generalization for obtaining the multi-dimensional Lyapunov exponent is then straightforward:

$$\lambda\left(t\right)=ln\frac{\left(\sum_{k=1}^{m}\left[x_{k}\left(t\right)-x_{k0}\right]^{2}\right)^{\frac{1}{2}}}{0.01},$$

where the sum is taken over all $m = 4$ squared differences between final $x_{k}\left(t\right)$ and initial $x_{k0}$ one-nucleon phase space variables. Integration times of the order of $10^{3}$ [fm]{}/c exclude errors when computing the Lyapunov exponents.

In short, we here remind that the trajectories can be classified as function of the Lyapunov exponents. Thus, one can distinguish periodical behaviours, for $\lambda=0$, dissipative movements with a fixed point or a basin of attraction $\left(\lambda<0\right)$, and aperiodical chaotic states $\left(\lambda>0\right)$, when the iterative discrete evolution of the solution series (Eqs. 5 and 6) leads to a chaotic pattern.

Another way of measuring the system sensitivity to initial conditions is to compute the largest Lyapunov exponent ([LLE]{}). Usually a couple of methods can be employed, one based on the time dependence of the multi-dimensional Lyapunov exponent, the other on Wolf’s standard method that uses a Gram-Schmidt Reorthonormalization of the tangent vectors [@wolf-85]. In the latter, the [LLE]{} is obtained by taking the asymptotic value of the multi-dimensional Lyapunov exponent:

$$\lambda_{Max}=\lim_{t \rightarrow \infty} \frac{\lambda\left(t\right)}{t}.$$

Still, this method has the disadvantage that the integration times have to be at least an order of magnitude larger than those here considered. Other methods are slightly less efficacious, being more CPU time-consuming when simulating strong chaotic systems [@ramasubramanian-00].

We consequently used the first method and noticed the saturation behaviour, *i.e.* the arising of a plateau after a certain time $t_{c}$ (Fig. \[fig:17\]). The straight lines represent fits whose slopes match the [LLE]{} (Table \[tab:5\]). They are in inverse proportion with the onset times of chaoticity $(\tau = 1/\lambda_{Max})$, being a measure of the trajectory decoupling at a microscopic level.

[cccc]{} Oscillation frequency & $L=0$ & $L=1$ & $L=2$\
$\Omega \:_{ad} = 0.020\: c/\rm{fm}$ & $0.003939$ & $0.008689$ & $0.004306$\
$\Omega \:_{ad} = 0.050\: c/\rm{fm}$ & $0.004432$ & $0.009829$ & $0.023203$\
$\Omega \:\;\ \   = 0.100\: c/\rm{fm}$ & $0.000761$ & $0.015454$ & $0.014402$\
$\Omega _{res} = 0.145\: c/\rm{fm}$ & $0.008739$ & $0.016662$ & $0.010086$\

When the single-particle and collective *d.o.f.* remain uncoupled, the Lyapunov exponents basically oscillate between two quasi-stationary regimes. This happens for all vibrational frequencies involved, reflecting a periodical regrouping of orbits in two basins of attraction. The phase space not being covered, even after a hundred of thousand of $\rm{fm}/c$, computing the [LLE]{} becomes futile for this case.

One can remark for dipole oscillations (Fig. \[fig:17\] - middle column) a faster evolution towards reaching saturation states of the 4-dimensional Lypaunov exponents, once passing from the adiabatic $(\tau _{ad}=115\ \rm{fm}/c)$ to the resonance phase of the interaction $(\tau _{res}=60\ \rm{fm}/c)$.

In the monopolar case the intermittency can be easily traced at $0.1\ c/\rm{fm}$ vibrational frequency (Fig. \[fig:17\] - left panels). During the intermittent stage, independent nearby orbits microscopically diverge with the slowest rate of all: $\tau =1,314\ \rm{fm}/c$ (Table \[tab:5\]). In order to catch the heaving in sight of the stationary plateau at $\approx 4,917\ \rm{fm}/c$, the temporal scale was scanned over $6,400\ \rm{fm}/c$.

The study of the quadrupole collective oscillation case confirms the results obtained with all previous analyses. Namely, the neighbouring trajectories deviate one from each other after just $43\ \rm{fm}/c$ at an adiabatic frequency: $0.05\ c/\rm{fm}$. Also, when increasing $\Omega$, the [LLE]{} evolution pattern exactly matches that found with informational entropy measured for a group of orbits (Tables \[tab:4\] and \[tab:5\]).

\[sec:3\]Conclusions
====================

We investigated the chaotic nucleonic behaviour in a two-dimensional deep Woods-Saxon potential well for specific phases of the nuclear interaction. By comparing the order-to-chaos transition for these cases of interest, from adiabatic to resonance regime, it was shown that the couplings between the one-particle dynamics and high multipole vibrational modes significantly decrease the onset of the chaotic nucleonic motion towards realistic nuclear interaction time scale.

The quantitative study enfolded a plethora of analyses, pointing out that the paths to chaos for the “nuclear billiard” are dissimilar for the studied multipolarities. For the first two multipole degrees we noticed a more rapid emergence of chaotic states as moving on towards higher radian frequencies of oscillation. When analyzing the system with quadrupole collective deformations of the potential well, an order-strong chaos-weak chaos-order sequence is revealed. Still, as emphasized in the “Shannon entropies” subsection, the quadrupole case represents an intricate one, and further analysis would be required before concluding it.

Every type of quantitative analysis strengthened previous results regarding the monopolar intermittency route to chaos for the “nuclear billiard”. The collective oscillation frequency for the intermittent behaviour was located prior to the resonance state of interaction (at $\Omega =0.1\ c/\rm{fm}$).

Further studies along the above issues are currently in progress. The used formalism can be improved by adding spin and charge to the nucleons. A semi-quantal treatment of this problem, including Pauli blocking effect, is hoped to shed more light on the discussed issue in the near future.

We wish to thank to R.I. Nanciu, I.S. Zgură, A.Ş. Cârstea, G. Păvălaş, S. Zaharia, A. Gheaţă, M. Rujoiu, A. Mitruţ, and R. Mărginean for fruitful discussions on this paper.

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