Intermittency route to chaos for the nuclear billiard - a qualitative study
D. Felea, I. V. Grossu, C. C. Bordeianu, C. Besliu, Al. Jipa, A. A. Radu, C. M. Mitu, E. Stan
aa r X i v : . [ nu c l - t h ] D ec Intermittency route to chaos for the nuclear billiard - a qualitative study
Daniel Felea ∗ Institute of Space Sciences, P.O.Box MG 23, RO 77125, Bucharest-M˘agurele, Romania
Ion Valeriu Grossu, Cristian Constantin Bordeianu, C˘alin Be¸sliu, and Alexandru Jipa
Faculty of Physics, University of Bucharest, P.O.Box MG 11, RO 77125, Bucharest-M˘agurele, Romania
Aurelian-Andrei Radu, Ciprian-Mihai Mitu, and Emil Stan
Institute of Space Sciences, P.O.Box MG 23, RO 77125, Bucharest-M˘agurele, Romania (Dated: November 17, 2018)We analyze on a simple classical billiard system the onset of chaotical behaviour in differentdynamical states. A classical version of the ”nuclear billiard” with a 2 D deep Woods-Saxon potentialis used. We take into account the coupling between the single-particle and the collective degreesof freedom in the presence of dissipation for several vibrational multipolarities. For the consideredoscillation modes an increasing divergence of the nucleonic trajectories from the adiabatic to theresonance regime was observed. Also, a peculiar case of intermittency is reached in the vicinity of theresonance, for the monopole case. We examine the order-to-chaos transition by performing severaltypes of qualitative analysis including sensitive dependence on the initial conditions, single-particlephase space maps, fractal dimensions of Poincare maps and autocorrelation functions. PACS numbers: 24.60.Lz, 05.45.-a, 05.45.Pq, 21.10.Re
I. INTRODUCTION
Deterministic chaos, is usually defined as irregular, un-predictable behaviour of the trajectories generated bynonlinear systems whose dynamical laws, involving norandomness or probabilities, predict a unique time evo-lution of a given system.Over the last two decades an increasing number of pa-pers have treated the study of the deterministic chaoticalbehaviour of Fermi nuclear systems [1–27]. The interestfor analyzing the order-to-chaos transitions on such sys-tems was linked to the problem of the onset of dissipationof collective systems through mainly one-body and two-body processes. Among these we mention the interactionof the nucleons with the potential well, the evaporationof individual nucleons in nuclear peripheral interactions,and the collisions between nucleons without taking intoaccount the Pauli blocking effect.These kinds of analyses were performed for the firsttime by Burgio, Baldo et al. [1–3] considering a systemof nucleons which move within a container modelled as aWoods-Saxon type potential and kick the container wallswith a specific frequency. They discuss the damping ofthe movement and the relation with order-chaos transi-tion in single-particle dynamics.On that classical model worked Papachristou and col-lab. [28], studying the decay width of the Isoscalar GiantMonopole Resonance for various spherical nuclei. Fol-lowing also that formalism, the beginning of the chaoticbehaviour for a number of nucleons in various dynamicalregions at several multipolarities was surveyed [29–33]. ∗ [email protected] In this paper, we investigated the chaotic behaviour ofa single-nucleon in a two-dimensional (2 D ) deep Woods-Saxon potential well for specific physical phases. A qual-itative picture of the achievement of deterministic chaoswas shown for a comparative study between the adiabaticand the resonance stage of the nuclear interaction.Close to resonance we obtained characteristics of theintermittency regime, i.e. sudden change to a laminarbehaviour (so-called intermission) of a specific signal be-tween two turbulent phases, which has been detected overand over again in a plethora of experiments regarding theRayleigh-Benard convection, the driven nonlinear semi-conductor oscillator, the Belousov-Zhabotinskii chemicalreaction, and the Josephson junctions (for e.g., [35–44]).Albeit intermittency is a well-known phenomenon forbilliards [45–48], in particular for Hamiltonian systemswith divided phase space (e.g., mushroom [49–53] andannular billiards [54]), and for connected Hamiltoniansystems [55], we showed that this property also holds fora Woods-Saxon billiard container with inelastic particle-wall interactions. II. BASIC FORMALISM
We chose for the present analysis a simple dynamicalsystem as considered in several previous papers by Burgioand collab. [1–3]. This system contains a number of A spinless and chargeless nucleons, with no internal struc-ture. The nucleons move in a 2 D deep Woods-Saxonpotential well considered as a ”nuclear billiard” and hitperiodically the oscillating surface of the well with a cer-tain frequency. The Bohr Hamiltonian of such a systemin polar coordinates is considered as: H ( r j , θ j , α ) = A X j =1 " p r j m + p θ j mr j + V ( r j , R ( θ j )) + p α M + M Ω α , (1)where (cid:8) p r j , p θ j , p α (cid:9) are the conjugate momenta of theparticle and collective coordinates { r j , θ j , α } , the nucleonmass m is 938 MeV, Ω is the oscillating frequency of thecollective variable α , and M = mAR is the Inglis mass.The Woods-Saxon potential is constant inside the bil-liard and a very steeply rising function on the surface: V ( r j , R ( θ j )) = V h r j − R ( θ j ,α ) a i , (2)with V = − a has a very small value 0 .
01 fm. The vibrat-ing surface can be written as in [1–3], depending on thecollective variable and Legendre polynomials P L (cos θ j ): R j = R ( θ j , α ) = R [1 + αP L (cos θ j )] , (3)where R = 6 fm, and L the multipolarity vibrationdegree of the potential well is 0 for the monopole, 1 forthe dipole, and 2 for the quadrupole case.Once the hamiltonian is chosen, the numerical simula-tions are based on the solution of the Hamilton equations: · r j = p r j m , · p r j = p θ j mr j − ∂V∂r j , (4) · θ j = p θ j mr j , · p θ j = − ∂V∂R j · ∂R j ∂θ j , (5) · α = p α M , · p α = − M Ω α − A X j =1 (cid:18) ∂V∂R j · ∂R j ∂α (cid:19) . (6)The Hamilton equations were solved with a Runge-Kutta type algorithm (order 2-3) having an optimizedstep size and taking into account that the absolute errorfor each variable is less than 10 − . Total energy wasverified to be conserved with high accuracy to a relativeerror level of 10 − .The equilibrium deformation parameter − α , which is themean collective variable, can be calculated (for e.g., for L = 0) by equating the mechanical pressure of the wall, P wall : P wall = M Ω πR · − α − α . (7) The pressure exerted by the particles is P part , ρ de-notes the particle density, and T is the apparent temper-ature of the system that equals the 2D kinetic energy,using the natural system of units (¯ h = c = k B = 1): P part = ρT = ATπR (cid:16) − α (cid:17) . (8)Thus, one gets the equation for the equilibrium valueof the collective coordinate in 2D, in the monopole case: − α (cid:16) − α (cid:17) = 2 TmR Ω . (9)Then a small perturbation of this collective variablewas considered α ( t = 0 fm / c) = − α +0 .
15 [1–3] and theevolution of the physical system was thoroughly investi-gated.
III. THE QUALITATIVE ANALYSIS OF THEROUTE TO CHAOSA. On the resonance condition
One can choose the wall oscillation taking place close toadiabatic conditions, imposing a wall frequency smallerthan the single-particle one. Thus, the frequency of vi-bration Ω ad was chosen less than 0 . c/ fm, which corre-sponds to an oscillation period equal to: τ wall = 2 π Ω ad ≥ .
66 fm / c . (10)By introducing the maximum particle speed: v = r Tm , (11)and the parameters as in [1–3]: R = 6 fm and T =36 MeV, one can obtain the value for the single-particleperiod: τ part = 2 R v ≈ .
33 fm / c . (12)In addition to [1–3] we introduced a physical constraintto this elementary physical system and continued thattype of analysis necessary for the study of a nonintegrabledynamical system. At the beginning we considered aphysical situation and we chose instead a static vibrating”nuclear billiard”, a projectile nucleus having the sameproperties, colliding with a target nucleus. It is well-known that the nuclear interaction, at incident energiesranging from MeV to GeV, can result in a multitudeof processes from the nuclear evaporation to completefragmentation or multifragmentation, according to theimpact parameter.It was shown in [56, 57] that during this kind of pro-cesses even for peripheral events an unnegligeable amountof energy is transferred by nucleon-nucleon scattering tothe nucleons of the projectile and not only the trans-verse momentum distributions, but also the longitudinalmomentum distributions as measured in the projectilefragmentation rest frame can reveal the centrality statusof the interaction. It can also offer a hint on the apparenttemperature of a Fermi gas of nucleons which was foundto be [57] near the isotopic temperatures, i.e. severalMeV [58–61].It was therefore supposed that the target fragmenta-tion can be associated with a resonance process. In or-der to obtain such behaviour, the wall frequency wasgradually increased to the resonance frequency Ω res =0 . c/ fm. However, nuclear evaporation or plainbreakup of a projectile nucleus can take place long beforethis regime is achieved by redistributing energy betweenthe nucleons themselves and also between single-particledegrees of freedom and collective ones. Individual nucle-ons or clusters can thus have enough kinetic energy toescalade the wall barrier.We should also emphasize that we can either have thecase that can be put in correspondence with a nuclearcollision process, i.e. the variation of the nucleonic fre-quency oscillation as the apparent temperature of thenucleons in the nuclei increases (Eqs. 11 and 12), main-taining the potential well vibration constant, or respec-tively, the inverse situation in which the period betweentwo consecutive collisions of the nucleon with the self-consistent mean field is kept invariable, while modifyingthe oscillation modes of the nuclear surface. The latterregards our studied case and is the reversed physical casepreviously described. It was used because of the specificchoice of the ”toy model” parameters described in [1–3].But the most realistic evolution of the nucleons in achosen potential can assume a simultaneous variation ofboth angular frequencies. The resonance condition ofthe coupled classical oscillators should remain howeveran important condition for a rapid appearance of a de-terministic chaotical behaviour of the physical system instudy at different time scales. A proper analysis of asystem should provide the variation of the collision ra-dian frequency of the nucleons inside the ”billiard” asthe apparent temperature increases and the change inthe vibrating potential period, supposing that the multi-polarity increases when pumping energy in the ”nuclearreservoir” during interaction. We can for example use insimulations, for nuclei with a large number of nucleons,the Liquid Drop Model or the Collective Model, whichpredict a frequency of vibration as function of the multi-polarity deformation degree:Ω L = r C L B L , (13) with C L being the elasticity coefficient, and B L themass coefficient for the oscillator of L multipolarity.We will briefly discuss on how the resonance conditionmight look like and for which particular case(s) it can beapplied. By reverting to the set of differential nonlinearHamilton equations and combining the last two of them(see Eq. 6), it emerged: .. α +Ω α = − V R aM · A X j =1 P L (cos θ j ) e rj − Rja h e rj − Rja i . (14)One can now compare the resulted equation with aharmonic oscillator system in the presence of dissipativeprocesses (”damped oscillator”) and of a external har-monic driving force of F amplitude: .. α +Γ . α +Ω α = − F R M · cos ωt. (15)In a first approximation we can neglect the dampingconstant Γ, as noticed from the Figs. 1-4. Besides, dis-sipation will appear if only the collective coordinate α is averaged over a large number of events, for all mul-tipolarities considered [1, 3]. For the uncoupled equa-tions ( U CE ) case there is no damping either, as the so-lution of a homogeneous Eq. 14 is a purely harmonic one: α = α cos Ω t .Also, we remark that for the dipole collective oscil-lations: P (cos θ j ) = cos ω j t and that, for a statis-tical ensemble of A nucleons, the one-particle radianfrequency can be approximated with a constant, ω j ≈ ω . For the Woods-Saxon potential energy: V ( t ) = P Aj =1 V ( r j , θ j , α ), it can always be derived a conserva-tive force F : F ( t ) = − A X j =1 ∂V ( t ) ∂r j = V a · A X j =1 e rj − Rja h e rj − Rja i . (16)A similarity between Eq. 14 and 15 was eventuallyobtained, because once shifting towards collectives of nu-cleons, the total force becomes a sum over a number oftime dependent forces, each acting on a single nucleon.Therefore, the net force for a pack of nucleons tends tovary little with time: F = F ( t ), and the tendency in-creases with A .In conclusion, the resonance condition, as stated in Eq.14, can be definitely used for the multi-particle couplingswith a dipolar shape of the potential well. This was ver-ified by studying the variance of the largest Lyapunovexponent and of the Kolmogorov-Sinai entropy with theradian frequency of wall oscillation on a ten-nucleons sys-tem [33]. For the other multipole degrees taken into ac-count, and as well, for the dipole oscillations coupledwith a single-nucleon dynamics, the nonlinear characterof the differential Equation 14 offers a somewhat intri-cate perspective over obtaining an analytical generalizedresonance condition, and for the moment is left aside forfurther investigation.As we confine the following analysis on the single-particle chaotical dynamics, the frequency matching ofthe coupled oscillators: ω part = Ω res = 0 . c/ fm, willbe generically designated as the resonance stage of theinteraction. B. Dependence on the initial conditions
In order to detect chaos in simple systems, several cus-tomary methods are used [62]. The sensitive dependenceon the initial conditions, the so-called ”Butterfly effect”is the first type of analysis presented in this article. Fora given multipolarity degree we studied the time depen-dence of both, single-particle and collective variables,with small perturbations applied to a single parameter(for e.g., ∆ r = 0 .
01 fm). This type of analysis is pre-sented for four cases, ranging from
U CE to quadrupoledeformations of the two-dimensional wall surface, and forthe two physical regimes in study, adiabatic and of reso-nance, respectively (Figs. 1-4).At a first glance, the time dependence of the col-lective degrees of freedom ”looks chaotic” for all con-sidered situations with the exception of (4 + 2) uncou-pled nonlinear differential equations case (Fig. 1), asone would naturally expect. In this case, a periodicalstructure would definitely appear in sharp contrast withthe quasi-periodicity shown for different multipole col-lective modes. Also, the time evolution of the single-particle variables is clearly chaotic in all cases where thestrong coupling between the (4 + 2) nonlinear differentialequations describing the single-particle dynamics appears(Figs. 2-4).The sensitive dependence on the initial conditions, i.e. the decoupling of the trajectories at macroscopic scale, isfound to give the first hints on the behaviour of the nu-clear system, in its evolution from the quasi-stable states,which can hardly develop a chaotic motion in time (adi-abatic state), to the unstable ones, characterized by arapid divergence of the particle trajectories in the phasespace (resonance regime). Also, in the monopole casethe order-to-chaos transition clearly shows an intermit-tent pattern at Ω = 0 . c/ fm (Fig. 2). C. Maps of phase space
Another type of qualitative analysis, indicating differ-ent ways toward a chaotic behaviour is based on the one-dimensional maps of phase space.Thus, for a temporal scale of 3 ,
200 fm / c we repre-sented the evolution of the multipolarity deformation de-gree of the potential well in the phase space of the vi-brational variables ( α ↔ p α ), as well as in the 1 D phase space of a single-particle ( r ↔ p r ). This type of analy-sis was performed for the U CE , monopole, dipole andquadrupole cases, using the same physical conditions,from the adiabatic to the resonance phase (Figs. 5-12).A remark that can be formulated from the study ofthese phase space maps is related to the form of the tra-jectories specific to those described by a harmonic oscil-lator, especially for the ( α ↔ p α ) maps. It can be easilyput to the test (Figs. 5, 7, 9, and 11) that the ellipsearea S ho is proportionally inverse with the wall vibrationfrequency: S ho = 2 π · E coll Ω , (17)where E coll is the sum of the last two terms from (1),being the energy of the collective nucleonic motion.The phase space filling degree raises as the vibrationwall frequency moves towards the resonance value for allmultipolarities took into account (Figs. 7-12), with theexception of the uncoupled nonlinear differential equa-tions case (Figs. 5 and 6). This confirms that the char-acteristic time of the macroscopic decoupling of the nu-cleon trajectories in the Woods-Saxon potential evolvestowards smaller values, once Ω is increased and alsothat the coupling between the collective variable motionand the particle dynamics is essential in amplifying thechaotic behaviour.An intriguing aspect is revealed by the comparison ofthe filling degrees of the phase space as a function ofthe nucleon oscillation frequency in the chosen potential.One would expect that the trajectories degenerate from asimple orbit towards a compact filling of the ( α ↔ p α ) or( r ↔ p r ) plane. Moreover, we observe several attractionbasins around a few standard orbits, preeminently in themonopole case at Ω = 0 . c/ fm, characteristic feature ofan intermittent behaviour (Figs. 7 and 8). D. Fractal dimensions of Poincare maps
The one-dimensional phase space maps offer a hint ontheir filling degree in time. In the same manner one canuse the Poincare maps for the ”nuclear billiard” regardedas a deterministic dynamical system. When analyzingthe behaviour of close trajectories in the phase space,starting from a periodical orbit, Poincare showed [63, 64]that there are only three distinct possibilities: the curvecan be either a closed one (with a fix distance to a pe-riodical orbit), or it can be a spiral that asymptoticallywraps/unfolds to/from the periodical trajectory.In order to simplify the temporal analysis for peculiarorbits, Poincare suggested a simple and effective method.By choosing a transversal section with ( N − D thatintersects a N D geometrical variety, the sequence of theintersection points is easier to be studied than the whole
N D curve. Therefore, the cases previously described forthe spiral generate a series of points that draw near in or
FIG. 1. The sensitive dependence on the initial small perturbation of the radius parameter (0.01 fm/c) when adiabatic andresonance conditions are imposed (uncoupled one-nucleon and collective degrees of freedom case). move away from the intersection point of the periodicalorbit with the transversal section.Applying the Poincare maps type analysis to the sim-ple chosen physical system, one would expect that thecompact filling up of a 1 D map of phase space for vibra-tion frequencies of the Woods-Saxon well that match theone-particle oscillation frequencies inside the ”billiard”(Figs. 5-12), to reverberate by a large density of chaoticdistributed intersection points.In order to estimate the degree of fractality of suchdensities, we computed the fractal dimensions d f of thePoincare maps with one-nucleon radial degrees of free-dom ( r ↔ p r ), when choosing for the transversal sectionthe polar pair ( θ ↔ p θ ) as a constant. As only for themonopole and U CE cases the orbital kinetic momentumis a constant of motion, we plotted the one-particle ra-dius as a function of radial momentum when the polarcoordinate is not kept constant, but quasi-constant in or-der to increase the probability of intersecting the sectionfor a given time evolution of the system: θ = θ ± ǫ θ . (18)We then chose a very small value for the ( θ ↔ p θ )thickness: ǫ θ = 10 − radians and let the system evolvefor a period of ∆ t = 5 · fm / c.The values were calculated using the box counting al-gorithm in a specific Visual Basic 6 application [65, 66]on Poincare maps. The fractal dimension of a map cov-ered by N boxes of length r is described by the followingexpression (see, for e.g., [67, 68]): d f = lim r → log N ( r )log (1 /r ) , (19)where we took: r = 1 / n , 0 ≤ n ≤ n max . The maxi-mum value n max denotes the highest chosen resolution.We calculated d f as slopes of the linear fits of log N ( r )2 versus n = log /r points, where we took the inferior limitof log (1 /r ) to be 5, and the superior threshold according FIG. 2. The sensitive dependence on the initial small perturbation of the radius parameter (0.01 fm/c) when adiabatic andresonance conditions are imposed (monopole case). to the highest resolutions for which a saturation plateauappears: N = N ( r ) (Fig. 13). At resolutions abovethreshold, no additional information on the points distri-bution can be gained.The points from the Poincare maps are distributed insuch a way that up to a certain resolution (32 ×
32 pixels)the algorithm does not take into account the individualpixels. In this first region analyzed the alignment is quitegood, denoting a correlation between points. The infor-mation is not uniformly distributed in the phase spacebut represents a ”cloud” of points with d f greater than1. This large-scale region thus offers a global image ofthe Poincare maps.Above 32 ×
32 resolution appear the effects linked withthe contribution of the individual dots of the Poincaremaps, reflected by significant variations of the fractal di-mensions. By choosing the points for the fitting proce-dure in such a way, we are also in agreement with thefollowing criterion: the fitted points should be singledout so their associated resolutions are to be pertainedto at least three successive decimal logarithmic intervals
TABLE I. The d f of the ( r ↔ p r ) Poincare single-particlemapsOscillation frequency Uncoupled eqs. Monopole caseΩ ad = 0 . c/ fm 0 . . ad = 0 . c/ fm 0 . . . c/ fm 0 . . res = 0 . c/ fm 0 . . [69]. In Figure 13 we selected two points with resolutionsbelonging to the first interval (cid:2) ÷ (cid:3) , three points to (cid:2) ÷ (cid:3) , and we took one point in the (cid:2) ÷ (cid:3) range, just until constant values N are reached.The small-scale d f thus computed (Table I) representthe filling degrees of a fine detailed phase space and canbe correlated with the Shannon entropies calculated in[70].It can be remarked that the fractal dimensions gener- FIG. 3. The sensitive dependence on the initial small perturbation of the radius parameter (0.01 fm/c) when adiabatic andresonance conditions are imposed (dipole case). ally increase with the specific vibration frequencies, fromthe quasi-stationary equilibrium regime at adiabatic os-cillations to the unstable chaotical states of the nucle-onic system in the resonance domain. At the same time,it should be mentioned that for the monopole case thefractal dimension has an intermittent deportment closeto the resonance phase of interaction, at Ω = 0 . c/ fm. E. Autocorrelation functions
The transition toward a quasi-periodical, aperiodicalor chaotic behaviour can be also analyzed with the au-tocorrelation function of the variables characteristic forthe temporal evolution of the studied physical system.When the ”nuclear billiard” evolves to chaos, essentialchanges can be noticed in the shape of the autocorrela-tion function of a specific variable (continuous or discretedistributed). This type of function measures the correla-tion between a sequence of signals, and is usually definedas: C ( τ ) = lim T →∞ T Z T ξ ( t ) · ξ ( t + τ ) dt ; (20) ξ ( t ) = x ( t ) − lim T →∞ T Z T x ( t ) dt ; (21) C ( τ ) = lim N →∞ N N X n =0 ξ ( t n ) · ξ ( t n +1 ) ; (22) ξ ( t n ) = x ( t n ) − lim N →∞ N N X n =0 ξ ( t n ) . (23)For a laminar regular regime this function either has aconstant value or presents decreasing oscillations in time.During a chaotic phase, it falls in, having an exponen-tially decreasing behaviour for uncorrelated signals, asshown in [71]. FIG. 4. The sensitive dependence on the initial small perturbation of the radius parameter (0.01 fm/c) when adiabatic andresonance conditions are imposed (quadrupole case).
We represented the autocorrelation function for thestudy of the single-particle (Fig. 14) and collective dy-namics (Fig. 15). The analysis was thus done auto-correlating the radial momentum p r of the particle andrespectively the collective variable α for the U CE caseand for all multipolarities, from all stages of interactiontaken into account. One conclusion, that can be easilydrawn from these figures, confirmed the previous resultsobtained with the other two methods. Thus, it can benoticed the increase of the chaotic bearing once the os-cillation frequency of the potential well is varied fromthe adiabatic to the resonance regime, for all degrees ofmultipole.When studying the case with single-particle degrees offreedom uncoupled from the collective ones, the domi-nant behaviour is aperiodical oscillation in time charac-teristic for a steady adiabatic phase of the interaction,even though the vibration frequency is varied. Only atresonance the shape is somewhat changed, indicating agreater degree of chaos.As for the monopole to quadrupole deformations the decreasing of the autocorrelation function is indeed ofthe exponential type, steeper as the radian frequency ofoscillation is raised to Ω res = 0 . c/ fm.The exception is again found in the monopole case atΩ = 0 . c/ fm frequency of oscillation, when the inter-mittent phase of the interaction, reflected by aperiodicaloscillations, points out a steady behaviour prior to theresonance regime. IV. CONCLUSIONS
A comparative study was done between the interest-ing physical regimes of nuclear interaction: adiabatic andresonance, giving at this level only a qualitative pictureof the possible scenarios towards a pure deterministic be-haviour of chaotic type of the studied nucleonic system.We envisaged the single-nucleon dynamics in a Woods-Saxon potential. The coupling between individual andcollective degrees of freedom was shown to generate dif-ferent paths to chaos, according to the order of multi- −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−200002000 p α ( f m M e V / c ) −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−200002000 p α ( f m M e V / c ) −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−200002000 p α ( f m M e V / c ) −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−200002000 p α ( f m M e V / c ) ( α − p α ) phase−space map; N nucleons = 1; ∆ t = 3200 fm/c; Uncoupled Eqs. α (rad) Ω ad = . c /f m Ω ad = . c /f m Ω = . c /f m Ω r e s = . c /f m FIG. 5. The phase space of the collective degrees of freedom( α − p α ) for different wall frequencies (uncoupled single andcollective degrees of freedom case). p r ( M e V / c ) p r ( M e V / c ) p r ( M e V / c ) p r ( M e V / c ) (r − p r ) phase−space map; N nucleons = 1; ∆ t = 3200 fm/c; Uncoupled Eqs. r (fm) Ω ad = . c /f m Ω ad = . c /f m Ω = . c /f m Ω r e s = . c /f m FIG. 6. The phase space of the single-particle degrees of free-dom ( r − p r ) for different wall frequencies (uncoupled singleand collective degrees of freedom case). polarity. In general, for all vibrational modes at theresonance frequency of oscillation, the onset of chaoti-cal behaviour was found to be earlier than at any otheradiabatic vibrations of the 2D potential well.Also, a phase alternation of periodical and chaotic dy-namics was found in the monopole case of nuclear walloscillation at Ω = 0 . c/ fm, revealing a laminar dynamicsprior to the resonance stage of interaction.In order to verify the aforementioned results, the studywas completed with the inclusion of several quantitativeanalyses, inter alia we mention: power spectra, Shannoninformational entropies, and Lyapunov exponents [70]. −0.5 0 0.5 1 1.5 2 2.5−200002000 p α ( f m M e V / c ) −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−200002000 p α ( f m M e V / c ) −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−200002000 p α ( f m M e V / c ) −0.3 −0.2 −0.1 0 0.1 0.2 0.3−200002000 p α ( f m M e V / c ) ( α − p α ) phase−space map; N nucleons = 1; ∆ t = 3200 fm/c; L = 0 (Monopole) Ω ad = . c /f m Ω ad = . c /f m Ω = . c /f m Ω r e s = . c /f m α (rad) FIG. 7. The phase space of the collective degrees of freedom( α − p α ) for different wall frequencies (L = 0). p r ( M e V / c ) p r ( M e V / c ) p r ( M e V / c ) p r ( M e V / c ) (r − p r ) phase−space map; N nucleons = 1; ∆ t = 3200 fm/c; L = 0 (Monopole) Ω ad = . c /f m Ω ad = . c /f m Ω = . c /f m Ω r e s = . c /f m r (fm) FIG. 8. The phase space of the single-particle degrees of free-dom ( r − p r ) for different wall frequencies (L = 0). −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−200002000 p α ( f m M e V / c ) −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−200002000 p α ( f m M e V / c ) −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−200002000 p α ( f m M e V / c ) −0.3 −0.2 −0.1 0 0.1 0.2 0.3−200002000 p α ( f m M e V / c ) Ω ad = . c /f m Ω ad = . c /f m Ω = . c /f m Ω r e s = . c /f m ( α − p α ) phase−space map; N nucleons = 1; ∆ t = 3200 fm/c; L = 1 (Dipole) α (rad) FIG. 9. The phase space of the collective degrees of freedom( α − p α ) for different wall frequencies (L = 1). p r ( M e V / c ) p r ( M e V / c ) p r ( M e V / c ) p r ( M e V / c ) Ω ad = . c /f m Ω ad = . c /f m Ω = . c /f m Ω r e s = . c /f m (r − p r ) phase−space map; N nucleons = 1; ∆ t = 3200 fm/c; L = 1 (Dipole) r (fm)r (fm) FIG. 10. The phase space of the single-particle degrees offreedom ( r − p r ) for different wall frequencies (L = 1). −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−200002000 p α ( f m M e V / c ) −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−200002000 p α ( f m M e V / c ) −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−200002000 p α ( f m M e V / c ) −0.3 −0.2 −0.1 0 0.1 0.2 0.3−200002000 p α ( f m M e V / c ) Ω ad = . c /f m Ω ad = . c /f m Ω = . c /f m Ω r e s = . c /f m ( α − p α ) phase−space map; N nucleons = 1; ∆ t = 3200 fm/c; L = 2 (Quadrupole) α (rad) FIG. 11. The phase space of the collective degrees of freedom( α − p α ) for different wall frequencies (L = 2). p r ( M e V / c ) p r ( M e V / c ) p r ( M e V / c ) p r ( M e V / c ) (r − p r ) phase−space map; N nucleons = 1; ∆ t = 3200 fm/c; L = 2 (Quadrupole) Ω ad = . c /f m Ω ad = . c /f m Ω = . c /f m Ω r e s = . c /f m r (fm)r (fm) FIG. 12. The phase space of the single-particle degrees offreedom ( r − p r ) for different wall frequencies (L = 2). FIG. 13. The fractal dimensions of the (radius ↔ radial mo-mentum) Poincare one-nucleon maps as linear slopes. τ (fm/c) τ (fm/c) τ (fm/c) C ( τ ) Ω ad = . c /f m Ω ad = . c /f m Ω = . c /f m Ω r e s = . c /f m C ( τ ) C ( τ ) τ (fm/c) C ( τ ) L = 0 L = 1 L = 2Uncoupled Eqs.
FIG. 14. The autocorrelation function which correlates theradial momentum of a single-particle in time for various stagesof nuclear interactions (from adiabatic to resonance regimes)and for several nuclear multipole deformations. C ( τ ) C ( τ ) C ( τ ) Ω = . c /f m Ω ad = . c /f m Ω ad = . c /f m Ω r e s = . c /f m τ (fm/c) C ( τ ) L = 0 L = 1 L = 2Uncoupled Eqs. τ (fm/c) τ (fm/c) τ (fm/c) FIG. 15. The autocorrelation function which correlates thecollective variable in time for various stages of nuclear inter-actions (from adiabatic to resonance regimes) and for severalnuclear multipole deformations.
ACKNOWLEDGMENTS
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