Classical Chaos in the Collective Geometric Nuclear Model

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Classical Chaos in the Collective Geometric Nuclear Model

Chaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsClassical Chaos in the Collective GeometricNuclear ModelMatú² Kurian and Pavel CejnarSummary22/09/2005, Thesis Defense


OutlineChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaos1 Motivation and Goal2 Introduction to Classical ChaosOur SystemNumericalResultsSummary3 Introduction of the System4 Numerical Results


OutlineChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaos1 Motivation and Goal2 Introduction to Classical ChaosOur SystemNumericalResultsSummary3 Introduction of the System4 Numerical Results


OutlineChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaos1 Motivation and Goal2 Introduction to Classical ChaosOur SystemNumericalResultsSummary3 Introduction of the System4 Numerical Results


OutlineChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaos1 Motivation and Goal2 Introduction to Classical ChaosOur SystemNumericalResultsSummary3 Introduction of the System4 Numerical Results


Physical BackgroundChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummaryThe GCM has two (vibrational) degrees of freedom in casewe do not consider rotations.Additionally, if the system can rotate about a xed axis, wehave to add one degree of freedom.Goal bluntIs a there a signicant dierence between chaoticity of therotating and non-rotating case?Note: The non-rotating case was investigated in 2004 byP. Stránský in his thesis.


Integrable SystemsChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummaryDenition (Integrability)Let N be a system's number of degrees of freedom.The the system is integrable if there exist N integrals of motionthat are compatible and independent.A non-integrable system possess fewer such constants ofmotion.Fact (Non-integrability of the GCM)Unless B vanishes, the GCM is non-integrable.


Lyapunov ExponentChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummarySuppose we have a trajectory, z(t) = [q(t), p(t)]. Consideranother one, ˜z(t) initially very close to the rst one:˜z(t) − z(t) = δ(t), δ(0) = εˆδ 0 , |ˆδ 0 | = 1, ε ∈ R .Denition (Rate of Divergence)LetR(t) ≡ |δ(t)||δ(0)|so in our case:R(t) = |δ(t)|εThe time-evolution of R(t) is crucial. We usually studydependence of ln R(t), however. Let us dene the limit:Denition (Leading Lyapunov Exponent)ln R(t)σ ≡ lim .t→∞ t.


Regular and Chaotic TrajectoriesChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummaryDenition (Regular (Stable) Trajectory)A trajectory is regular, if its Lyapunov exponent is zero.CorollaryA trajectory with non-zero Lyapunov exponent is not regularand will be further termed as chaotic.Note: Both stable and regular are used interchangeably asare unstable and chaotic.


Trajectories and IntegrabilityChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummaryRegularity of trajectories is closely related to the (non-)integrability of a system.Fact (Chaotic Trajectories and (Non-) Integrable Systems)There are no chaotic trajectories in an integrable system.However, there are both regular and chaotic trajectories in achaotic system.Note: Consequently an integrable system is often calledregular and a non-integrable one chaotic.


Measure of RegularityChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummaryA function that serves as an indicator of a system's regularity isa measure of regularity.A point in the phase space can be considered as an initialcondition for an either regular or chaotic trajectory. Thereforethe measure could be dened as fraction of regular volume≡ Vreg. VFor practical reasons (numerics) this has to be slightly redened.f (V)regDenition (Measure of Regularity)The measure of regularity is dened as a (volume) density ofpoints that belong to regular trajectories:f (V)reg≡ N regN total.


All Pieces TogetherChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummaryNow our method essentially is:We generate random initial conditions for given values ofconstants of motion (E, j) and parameter B.We solve equations of motion and distinguish regular andchaotic trajectories by their distinctive Lyapunovexponents.We calculate measure of regularity f (V)reg (B, j).


Dynamical VariablesChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummaryOur system is essentially the (nuclear) Geometric CollectiveModel.As there is not enough time to introduce the model itself,only the relevant pieces of the GCM will be put forward.This means dynamical variables and The Lagrangian.The model features ve real dynamical variables:α 20, α I 21, α R 21, α I 22, α R 22.Additionally, there are four real constants A, B, C, and K.


The LagrangianChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummaryThe Lagrangian of the system is a polynomial expression:L = K (| ˙α20 |22 + 2| ˙α 21| 2 + 2| ˙α 22| 2)− A ( |α 20| 2 + 2|α 21| 2 + 2|α 22| 2)+ B { (α 20 −|α20 | 2 − 3|α 21| 2 + 6|α 22| 2)− 3 √ }6 [α22(α R 21α R R 21− α21α I 21) I + 2α22α I 21α R 21]I− C ( |α 20| 2 + 2|α 21| 2 + 2|α 22| 2) 2Note: The equations of motion can be derivedstraight-forwardly.


ScalingChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummaryFact (Scaling)It can be shown that the Lagrangian can be scaled(P. S., 2004).As a result only one constant will remain, we choose B.Henceforth, we will put −A = C = K = 1.


Angular MomentumChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummaryThe system alow us to construct angular momentum. Theexpressions are somewhat unorthodox, but it can be veriedthat the angular momentum is very solid indeed, as{J i , J j } = ε ijk J k holds.J 1 = √ 2 [ √2(αI21˙α R 22− ˙α21α I R 22+ ˙α21α R I 22− α R 21˙α 22)+I+ √ ]3( ˙α 20α I 21− α 20˙α21)IJ 2 = − √ 2 [ √2( ˙αR21α R 22− α R 21˙α R 22+ ˙α21α I I 22− α I 21˙α 22)+I+ √ ]3( ˙α 20α R 21− α 20˙α21)RJ 3 = 2 [ ˙α R 21α I 21− α R 21˙α I 21+ 2( ˙α R 22α I 22− α R 22˙α I 22)]


Reduction of the ProblemChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummaryLemma (Zero Angular momentum (P. S.))If there is no angular momentum, then α R 21, α R 2I, α I 22and thecorresponding velocities identically vanish.This means we end up with 2 dynamical variables that can berewritten into β, γ, as it has been already mentioned.Lemma (Some Angular momentum)If ⃗ J = (0, 0, J 3 ) then α R 21, α R 21and their velocities vanish.As a result, we obtain 3 variables and recast them into β, γ, δ.


Introduction of jChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummaryDenition (j)j is dened as the fraction of the angular momentum'smagnitude and the maximum magnitude possible for a givenenergy E and parameter B. Formally:j(E, B) ≡ J maxJ , J max = max{J | E(..) − E = 0} .Note that j cannot be determined easily, as the expressions forJ 3 and energy constraint are complicated.


j(E, B)Chaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummary


PlansChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummaryWhole PictureWe have sucient knowledge of the system and can easilyderive relevant equations of motion.We can use the algorithm that has been outlined tocalculate f (V)reg (B, j).


Correspondence with the Previous WorkChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummary


Region of Low jChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummary


Region of Mid jChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummary


Region of High jChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummary


SummaryChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummaryWe developed analytical and numerical means forinvestigating regularity of the rotating GCM.We computed measures of regularity for a wide range of j.A Remaining Question: Unideal correspondence with thenon-rotating case.


AcknowledgementsChaos inGeometricModelMatú²KurianMotivationand GoalIntro toChaosOur SystemNumericalResultsSummaryI would like to pay tribute and give thanks to the followingpeople:Dr. Pavel Cejnar (ÚƒJF)Martin Fraas (ÚTF), David Kofro¬ (ÚTF), Michal Macek(ÚƒJF), Old°ich Kepka (ÚƒJF), Ji°í Pe²ek (ÚTF)and all others

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