Applying PSS, mLE, SALI and GALI Methods forExploring Order and Chaos in Two and ThreeDimensional Hamiltonian Systems with QuarticCouplingMohammed El ghamari ( mohammed.elghamari@usmba.ac.ma )

Université Sidi Mohamed Ben Abdellah Faculté des Sciences Dhar El Mahraz-Feshttps://orcid.org/0000-0002-8261-3219

Walid Chatar Université Sidi Mohamed Ben Abdellah Faculté des Sciences Dhar El Mahraz-Fes

Jouad Kharbach Université Sidi Mohamed Ben Abdellah Faculté des Sciences Dhar El Mahraz-Fes

Mohamed Benkhali Université Sidi Mohamed Ben Abdellah Faculté des Sciences Dhar El Mahraz-Fes

Rachid Masrour Université Sidi Mohamed Ben Abdellah Faculté des Sciences Dhar El Mahraz-Fes

Abdellah Rezzouk Université Sidi Mohamed Ben Abdellah Faculté des Sciences Dhar El Mahraz-Fes

Mohammed Ouazzani-Jamil Université Sidi Mohamed Ben Abdellah Faculté des Sciences Dhar El Mahraz-Fes

Research Article

Keywords: oincar´e Surface of Section, Maximum Lyapunov Exponent, Smaller Alignment Index,Generalized Alignment Index

Posted Date: October 20th, 2021

DOI: https://doi.org/10.21203/rs.3.rs-993970/v1

License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License

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Applying PSS, mLE, SALI and GALI

methods for exploring order and chaos in two

and three dimensional Hamiltonian systems

with quartic coupling

EL Ghamari Mohammed1*, Chatar Walid1, Kharbach

Jaouad1, Benkhali Mohamed1, Masrour Rachid1, Rezzouk

Abdellah1 and Ouazzani-Jamil Mohammed2

1Laboratoire de Physique du Solide, Faculte des Sciences Dhar ElMahraz, Universite Sidi Mohamed Ben Abdellah, B.P. 1796,

30000 Fez-Atlas, Morocco.2Laboratoire Systemes et Environnements Durables, Universite

Privee de Fes, Lot. Quaraouiyine Route Ain Chkef, Fez, Morocco.

*Corresponding author(s). E-mail(s):mohammed.elghamari@usmba.ac.ma;

Abstract

In this article, the complex nonlinear dynamics and chaos control havebeen examined in Hamiltonians systems with quartic coupling throughthe generalized three-dimensional (3D) Yang-Mills Hamiltonian systemwith four control parameters. We provide sufficient conditions on thefour control parameters of the system which guarantee the 3D inte-grability in the Liouvillian sense. Therefore, we get a classification ofthe 3D Yang-Mills Hamiltonian system in sense of integrability andnon-integrability. The integrable cases are identified and the detailedcalculations of their associated first integrals of motion are given. Thenature of the behavior orbits could be distinguished in a fast and effi-cient way by using a set of reliable methods based on the so-calledthe evolution of deviation vectors related to the studied orbit. This setof methods includes the Poincare surface of section (PSS), the maxi-mum Lyapunov exponent (mLE), the Smaller Alignment Index (SALI),the Generalized Alignment Index (GALI). In this view, the chaotic

1

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behavior will be explored and the order-chaos transition could be eval-uated both in 2D and 3D, when any control parameters on whichthe system depends vary. Finally, the efficiency and rapidity of theseproposed methods are proven by using several numerical illustrativeparadigms for identifying whether the system is in chaos or order state.

Keywords: Poincare Surface of Section, Maximum Lyapunov Exponent,Smaller Alignment Index, Generalized Alignment Index

1 Introduction

The theory of dynamical systems is an important branch of mathematicsintroduced by Newton around 1665. It provides mathematical models, forsystems evolving over time and following rules, generally expressed in ana-lytical form as a system of ordinary differential equations. The purpose ofdynamical system theory is to understand, or at least to describe the changesthat happen in a physical and artificial systems over time, especially at thelevel of Hamiltonian systems with several degrees of freedom (dof) that wereused to study and describe physical phenomena, in particular, energy trans-port and equipartition phenomena [1–4]. In the last decades, a new scientificconcept has been produced in an often-multidisciplinary way, under the termof ”chaos theory”. This concept has been an object of active research inthe fields of physics, mathematics and other scientific fields in recent years.Chaos is not as ”chaotic” as its name suggests; its disorder is only apparent.A chaotic system is unpredictable, but it is perfectly described by simple anddeterministic equations. A system is called deterministic when it is possibleto predict (calculate) its evolution over time. The exact knowledge of thestate of the system at a given moment (the initial moment) allows the precisecalculation of the state of the system at any time. The link between these twoparadoxical notions, deterministic and unpredictable, is manifested by thesensitivity to the initial conditions.

Hamiltonian systems face difficulties in distinguishing the nature of orbits(ordered or chaotic), because the orbits that represent this motion are dis-tributed in an unpredictable way in phase space. For that, we need a fastand precise tool to inform us about the chaotic or ordered nature of an orbit.Dynamical systems are expressed in analytical form as a system of ordinarydifferential equations who can be solved by a variety of methods: analyticaland numerical.

During the last years, multitude of analytical and numerical methodshave been developed to determine rapidly and reliably the regular or chaoticnature of orbits (the solution of ordinary differential equations provides orderor chaotic behavior). Among the analytical methods used, for instance: The

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Ziglin theorem [5], the study of separability [6, 7], Lie algebra [8], the Liou-ville theorem [9] and the Painleve analysis [10, 11]. At the same time, severalwell-documented numerical methods there exist nowadays in order to studythe identification of regular or chaotic behavior of orbits and the distinctionregular and chaotic states in nonlinear dynamical systems, we can cite forexemple the Poincare Surface of Section [2, 12], the Mean Exponential Growthfactor of Nearby Orbits (MEGNO) method [13–15], the Largest LyapunovExponent [16–19], the Small Alignment Index (SALI) [20–23], the GeneralizedAlignment Index (GALI) methods [23–26], the latter is a generalization of thetechnique of detection of chaos by the Smaller Alignment Index (SALI), andthe Shannon entropy [27].

Our purpose in this contribution is to study the three-dimensional inte-grability in the Liouvillian sense of three-dimensional Hamiltonian systemswith quartic potantials through the generalized three-dimensional Yang-MillsHamiltonian system with four control parameters. We get a complete classifi-cation of the 3D Yang-Mills Hamiltonian system in sense of integrability andnon-integrability. We determine the integrable cases and the associated firstintegrals of motion for the system, which belongs to the family of dynami-cal Hamiltonian systems describing coupled anisotropic quartic anharmonicoscillators. To distinguish chaos from order and observe the chaos-order-chaostransition of states of system both in 2D and 3D, when one of the controlparameters is changed, a serie of numerical methods are performed like, thePoincare surface of section (PSS), the maximum Lyapunov exponent (mLE),the Small Alignment Index SALI and the Generalized Alignment Index GALImethods.

The structure of this paper is as follows. In section 2, we provide basicconcepts as necessary preliminaries which describe the link between threedimensional Hamiltonian systems with quartic coupling and the generalizedthree-dimensional Yang-Mills problem. In section 3, we investigate analyticallythe three dimensional integrability of the generalized Yang-Mills Hamiltoiansystem by constructing the three-dimensional first integrals of motion. Section4, contains the results and the discussion, we try to apply PSS, mLE, SALI andGALI methods for exploring order and chaos behavior of the proposed two andthree-dimensional systems and evaluating the chaos-order-chaos transition.Finally, the last section summarizes the main conclusions for this work.

2 Basic concepts

Let us consider as a three dimensional Hamiltonian systems with quarticcoupling the three coupled nonlinear anharmonic oscillators with quartic

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interaction governed by the following generic Hamiltonian.

H3D =1

2

(

p2x + p2y + p2z)

+1

2

[

α(x2 + y2) + βz2]

+1

4

(

x2 + y2 + γz2)2+δz2

(

x2 + y2)

(1)where α, β, γ and δ are arbitrary control parameters, and px, py and pz are,respectively, generalized momenta corresponding to the generalized coordi-nates x, y and z.To bring out the connection between the Hamiltonian 1 and the general-ized three dimensional Yang-Mills Hamiltoian system, we could introduce thefollowing cylindrical coordinates

x = ρ cos θ

y = ρ sin θ

z = z

px = cos θpρ −sin θ

ρpθ

py = sin θpρ +cos θ

ρpθ

pz = pz

(2)

The Hamiltonian 1 takes the following form

H3D =1

2

(

p2ρ + p2z)

+m2

2ρ2+

1

2

(

αρ2 + βz2)

+1

4

(

ρ2 + γz2)2

+ δρ2z2 (3)

Where pρ and pz are, respectively, generalized momenta corresponding to thegeneralized coordinates ρ and z; m represents the value of the cyclic integralassociated to the cyclic coordinate θ, pθ = m.

For m = 0, H3D is equivalent to a Hamiltonian with two degrees of freedom(2D). It is known in literature as Yang-Mills Hamiltonian system which alsoknown in several problems in quantum mechanics [28, 29], celestial mechanicsand cosmological models [30, 31], scalar field theory [32, 33], etc. For thispurpose, there are several important studies such as the non-integrabilityaccording to the theory of Morales-Ramis [34], the Hamiltonian of Yang-Millsin the sense of Liouville-Arnold [35], and the regular motion and chaotic ofthe Hamiltonian [15].

The integrability of the 2D Yang-Mills Hamiltonian has been studied byseveral authors, see for example [28, 31, 36–38]. We would like to emphasizethat research works in this direction are interested in the integrability of thissystem only in two dimensions.

The well know 2D Yang-Mills Hamiltonian has the following form:

H =1

2

(

p2x + p2y)

+1

2

(

a1x2 + a2y

2)

+1

4

(

x4 + a3y4)

+1

2a4x

2y2 (4)

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where a1 = α, a2 = β, a3 = γ2 and a4 = γ + 2δ

We summarize all integrable cases of generalized 2D Yang-Mills Hamilto-nian and the corresponding Hamiltonian functions and first integrals of motionas followingCase (A) a4 = 0,

H =1

2(p2x + p2y) +

1

2

(

a1x2 + a2y

2)

I2D =1

2p2x +

1

2a1x

2 +1

4x4 +

1

4

(

x4 + a3y4)

Case (B) a2 = a1 and a3 = a4 = 1,

H =1

2(p2x + p2y) +

a1

2

(

x2 + y2)

+1

4

(

x2 + y2)2

I2D = xpy − ypx

Case (C) a2 = a1, a3 = 1 and a4 = 3,

H =1

2(p2x + p2y) +

a1

2

(

x2 + y2)

+1

4

(

x4 + y4 + 6x2y2)

I2D = pxpy + xy(a1 + x2 + y2)

Case (D) a2 = 4a1, a3 = 8 and a4 = 3,

H =1

2(p2x + p2y) +

a1

2

(

x2 + 4y2)

+1

4

(

x4 + 8y4 + 6x2y2)

− 4x3ypxpy + x4p2y

I2D = p4x + x2(x2 + 6y2 + 2a1)p2x + x4

(

x4

4+ y4 + x2y2 + 2a1y

2 + a1x2 + a21

)

Case (E) a2 = 4a1, a3 = 16 and a4 = 6

H =1

2(p2x + p2y) +

a1

2

(

x2 + 4y2)

+1

4

(

x4 + 16y4 + 12x2y2)

I2D = px (xpy − ypx) + x2y(x2 + 2y2 + a1)

Case (F) a3 = a4 = 1 and ∀(a1, a2),

H =1

2(p2x + p2y) +

1

2

(

a1x2 + a2y

2)

+1

4

(

x2 + y2)2

I2D =1

2(xpy − ypx)

2− (a1 − a2)

(

p2x +x4

2+

x2y2

2+ a1x

2

)

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3 Three-dimensional first integrals of motion

To study the 3D integrability in the Liouvillian sense of the Hamiltonian 1, weneed to calculate two additional integrals, according to Liouville’s Theorem:”A Hamiltonian with n degrees of freedom is completely integrable, if thereexists n− 1 other integral of motion in involution H, Ij = 0, j = 1, ..., n− 1,which are functionally independent [39]”.We apply spherical transformation on the Hamiltonian 1:

x = r sin θ cosϕ

y = r sin θ sinϕ

z = r cos θ

px = sin θ cosϕpr +cos θ cosϕ

rpθ −

sinϕ

r sin θpϕ

py = sin θ sinϕpr +cos θ sinϕ

rpθ −

cosϕ

r sin θpϕ

pz = cosϕpr −sinϕ

rpϕ

(5)

So, the Hamiltonian 1 takes the following form:

H3D =1

2

(

p2r +p2θr2

)

+k2

2r2 sin2 θ+

r2

2(α sin2 θ + β cos2 θ)

+r4

4

[

(sin2 θ + γ cos2 θ)2 + 4δ sin2 θ cos2 θ]

(6)

where pr and pθ are generalized momenta corresponding to the generalizedcoordinates r and θ respectively; k represents the value of the cyclic integralassociated to the cyclic coordinate ϕ

J1 = pϕ = k (7)

Moreover, we can determine the cyclic variable ϕ from ϕ−ϕ0 =∫ ∂H

∂kdt, where

we notice that the Hamiltonian 6 distinguishes the reduced problem from the2D-YM Hamiltonian after ignoring the cyclic coordinate pϕ.The Hamilton equations corresponding to the Hamiltonian 6 are:

r = pr

θ = pθ

pr =p2θr3

+k2

r3 sin2 θ− r

(

α sin2 θ + β cos2 θ)

− r3[

(

sin2 θ + γ cos2 θ)2

+ 4δ sin2 θ cos2 θ]

(8)

pθ =k2 cos θ

r2 sin3 θ+ r2(β − α) sin θ cos θ

− r4 sin θ cos θ[

(1− γ)(

sin2 θ + γ cos2 θ)

+ 2δ(

1− 2 sin2 θ)]

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The Hamilton equations 8 admit the energy integral

J2 =1

2

(

p2r +p2θr2

)

+k2

2r2 sin2 θ+

r2

2(α sin2 θ + β cos2 θ)

+r4

4

[

(sin2 θ + γ cos2 θ)2 + 4δ sin2 θ cos2 θ]

= h (9)

where h is a constant representing the value of the energy integral.Hamilton’s equations 8 are completely integrable, according to Jacobi’s the-ory, if there exists a complementary integral of motion J3 independent of theintegral of the energy.

If k = 0 the Hamiltonian 9 describes the 2D-YM, which proves that the3D-YM and the 2D-YM are equivalent to the zero level of the cyclic integral7. We explain why the 3D-YM system can be integrated in the six cases givenin the preceding section. By absurdity, we presume that 3D-YM is integrableoutside these cases and that the equivalence between 3D-YM and 2D-YM ona zero level of the cyclic integral makes the 2D-YM integrable outside thesesix integrable cases. But 2D-YM is integrable except for these cases. This con-tradiction confirms that 3D-YM is integrable in the six cases given in thepreceding section.In order to confirm Liouville’s 3D integrability, we must construct a comple-mentary integral of motion J3. This is why we give two examples of calculationof these complementary integrals for two cases. In general, it is well known thatcomplementary integrals of motion are either quadratic or quartic in terms ofmomenta. For 3D-YM, if the complementary integral of motion is quadratic,it takes the form 10, but if it is quartic, it takes the form 11.

J3 = I2D + k(

Λ3p2r + Λ2prpθ + Λ1p

2θ + Λ0

)

(10)

J3 = I22D + k(

Λ3p2r + Λ2prpθ + Λ1p

2θ + Λ0

)

(11)

Where I2D is the first integral of motion of 2D Yang-Mills Hamiltonian indi-cated in section 2 in spherical coordinates (r, θ, ϕ) and Λi(i = 0...3) beingfunctions in the generalized coordinates r, θ.Notice that the order of the complementary integral 11 is doubled and thatwe can name doubled order. This situation is found in several problems[40, 41]. Using the derivative of J3 with respect to time t or using the Poissonbracket H3D, J3, by matching the coefficients of moments to zero and usingHamilton’s equations 8, we obtain a system of nonlinear partial differentialequations whose solution gives the expressions of Λi(r, θ). Therefore, theadditional integrals J3 become:

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Case (A’) a4 = 0,Three-dimensional Yang-Mills Hamiltonian in this case is written as

H3D =1

2(p2x + p2y + p2z) +

1

2

[

a1(

x2 + y2)

+ a2z2]

+1

4

(

x4 + y4 + a3z4)

= J ′

1

(12)

The Hamiltonian is separable, so we use the direct method to find the integralsof motion J2 and J3, which are given by

J ′

2 =1

4(p2x − p2y + p2z) +

1

4

[

a1(

x2 − y2)

+ a2z2]

+1

8

(

x4 − y4 + a3z4)

(13)

J ′

3 =1

4(p2x + p2y − p2z) +

1

4

[

a1(

x2 + y2)

− a2z2]

+1

8

(

x4 + y4 − a3z4)

(14)

For the below cases we use the method indicated above.Case (B’) a1 = a2, a3 = a4 = 1,The complementary integral J3 is quartic in terms of the momenta

J3 =

(

p2θ +k2

sin2 θ

)2

(15)

Case (C’) a1 = a2, a3 = 1, a4 = 3,The complementary integral J3 is quartic in terms of the momenta

J3 = I22D + k

(

k cos2 θ

r2 sin2 θp2r −

2k cos θ

r3 sin θprpθ +

k

r4p2θ + 2kr2 cos2 θ

)

(16)

Case (D’) a2 = 4a1, a3 = 8, a4 = 3,The complementary integral J3 is quartic in terms of the momenta

J3 = I2D + k

(

2k

r2p2r +

4k cos θ

r3 sin θprpθ +

2k cos2 θ

r4 sin2 θp2θ

)

+k

(

k3

r4 sin4 θ+ kr2

(

1 + cos2 θ)

)

(17)

Case (E’) a2 = 4a1, a3 = 16, a4 = 6,The complementary integral J3 is quartic in terms of the momenta

J3 = I22D + k

(

2kp2r +2k cos θ

r sin θprpθ +

2k

r2 sin2 θp2θ + Λ0

)

(18)

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with

Λ0 =k3

r2 sin4 θ

(

1 + sin2 θ)

+ 2a1kr2(

1 + 2 cos2 θ)

+ kr4(

1 + 3 cos4 θ + 8 cos2 θ)

Case( F’) a3 = a4 = 1, ∀(a1, a2),The complementary integral J3 is quadratic in terms of the momenta

J3 = I2D + k

(

1

2p2r +

1

2r2p2θ +

k [k − 2(a1 − a2)]

2r2 sin2 θ

)

+k

(

r2

2

(

a1 sin2 θ + a2 cos

2 θ)

+k

2 sin2 θ

)

(19)

It is clear that on a zero level of the cyclic integral pϕ = k = 0 , the comple-mentary integral of all cases except for case (A’) is reduced to the quadraticintegral of 2D-YM.

4 Results and discussion

As an additional confirmation of the analytical results, the aim of this numeri-cal study, is to apply PSS, mLE, SALI and GALI of order k (GALIk) methodsfor exploring order and chaos behavior of orbits of the Yung-Mills Hamilto-nian systems, and evaluating chaos-order-chaos transition for different valuesof the control parameters. This study concerns the integrable cases (C) and(E) in two dimensions, and (C’) and (E’) in three dimensions. As for the othercases, they will be dealt in subsequent works.First of all, we have plotted the three-dimensional evolution of trajectories inphase space, the PSS, the mLE and the time evolution of the SALI and theGALI for each two and three dimensional cases. It is worth pointing out thatthe numerical simulations have been carried out via computing programs inthe Maple and Matlab environments. In order to get both chaotic and orderbehaviors, we fix the appropriate initial conditions to an orbit, and we selecta variety of control parameters to change the integrability conditions. For oursituations, we choose to vary the control parameter a4.

4.1 Behavior of PSS, mLE and SALI for regular andchaotic motion for 2D case (C)

In this case, the integrability conditions are a2 = a1, a3 = 1, a4 = 3 whichdefine the conditions for regular behavior. To explore order and chaos, andevaluate the chaos-order-chaos transition, we only change the parameter a4.We plote the PSS, mLE and the time evolution of SALI and GALI.Furthermore, for fixed value of appropriate initial conditions andvarying the control parameters around the 2D integrable case (C)(a4 = 2, 3, 4). We select an orbit given by this initial conditions

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(a) (b) (c)

(d) (e) (f)

Fig. 1 The PSS for case (C) with initial conditions x = 0.1895, y = −0.0645, px =0.6078432605, py = −0.2214537225, step size = 0.05 and a1 = −2, a3 = 1; (a) a4 = 2, h =0.169663; (b) a4 = 3, h = 0.169737; (c) a4 = 4, h = 0.169812; (d) The time evolution of theSALI for a4 = 2, 3, 4 (blue, black, orange); (e, f) The time evolution of mLE for a4 = 2, 3, 4(blue, black, orange).

x = 0.1895, y = −0.0645, px = 0.6078432605, py = −0.2214537225, to cal-culate the time evolution of SALI, then we choose two vectors of deviationrandomly v1(0) = (1, 0, 0, 0) and v2(0) = (0, 0, 1, 0). Finally, we calculate themLE and the time evolution of SALI for three values of parameter a4.

For a4 = 3, the PSS indicates that the behavior is regular as it is shown inFigure 1-b. The time evolution of SALI in log-log scale, as it is shown in Figure1-d black, shows that SALI varies around 0.1460 and 1.4047 and remains con-stant after 103 units of time, indicating that the behavior is regular. Figure1-e black represents the mLE, when T = 104, it equals to zero. As it is shownin Figure 1-f black, the log10(mLE) presents a large fluctuation, and beginsto decrease according to a power law obtaining the value −3 for log10(T ) = 4.

For a4 = 4, after a small transient time in particular T = 175, the SALIsuddenly drops to zero which indicates that the behavior becomes chaoticas it is shown in Figure 1-d orange. The mLE reaches a positive value, thenremains constant after T = 104, as it is shown in Figure 1-e orange, whichmeans that the behavior is also chaotic. Moreover, log10(mLE) begins todecrease according to a power law, reaching the value −3 for log10(T ) = 4 as

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it is shown in Figure 1-f orange.

For a4 = 2, the SALI becomes null after a time T = 656, as it is shownin Figure 1-d blue, which leads to chaotic behavior, and also the mLE gets aconstant positive value as it is shown in Figure 1-e Blue. Similarly, Figure 1-fblue represents that also log10(mLE) shows chaotic behavior.

4.2 Behavior of GALI for regular and chaotic motion for2D case (C)

(a) (b) (c)

Fig. 2 The time evolution of GALIk(t), k = 2, 3, 4 for case (C) with initial conditionsx = 0.1895, y = −0.0645, px = 0.6078432605, py = −0.2214537225 and a1 = −2, a3 = 1;(a) a4 = 2, (b) a4 = 3, (c) a4 = 4.

Figure 2 represents the time evolution of GALIk, where using the sameorbit and the same variation of the control parameter a4.For a4 = 3, the GALI2 reaches a non-zero constant value, and GALI3, GALI4tend towards zero, as it is shown in Figure 2-b, according to the theoreticallypredicted power laws, this shows that the behavior of the system is regular.

For a4 = 2, the motion is dominated by the chaotic behavior, the GALIkbecomes null after a small transition of time as it is shown in Figure 2-a,following the exponential decreases, and the plots in Figure 2-a correspond toproportional functions to eσ1t , e2σ1t and e4σ1t, with σ1 = 0.2180, which arethe mLE orbits. The x-axis is linear while the y-axis is on a logarithmic scale.

For a4 = 4, where the time evolution of GALIk for k = 2, 3, 4, as it is shownin Figure 2-c, all the values of GALI tend rapidly towards zero, according tothe exponential decreases, with σ1 = 0.2337.

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4.3 Behavior of 3D projection, mLE and SALI forregular and chaotic motion for 3D case (C’)

For fixed value of appropriate initial conditions x = −0.005, y = −0.42,z = 0.0, px = 0.5239885194, py = 0.0,pz = 0.0844906298 and varying thecontrol parameters around the 3D integrable case (C’) (a4 = 2, 3, 4), witha4 = 3 corresponds to 3D integrability conditions.

For the case a4 = 3 and according to Figure 3-b, the PSS is dominated bythe regular states of the trajectories. The time evolution of SALI remains non-zero as it is shown in Figure 3-d blue, and varies around [0.7561, 1.24345], italso shows that the behavior is regular. The mLE tends to zero after T = 104

as it is shown in Figure 3-e bleu, on the other hand the log10(mLE) decreasesin time T according to a power law, and becomes log10(mLE) = −3 whenT = 104, as it is shown in Figure 3-f blue, which also indicates that the systempresents a regular behavior, so the 3D integrability of the system is justified.

For the cases a4 = 2 and a4 = 4, as it is shown, respectively, in Figures3-a and 3-c, the PSS is dominated by the random distribution states of thetrajectories which means that the system manifests a chaotic behavior. Thetime evolution of SALI for a4 = 4 abruptly tends to zero after T = 199, asit is shown in Figure 3-d black. For a4 = 2 the SALI reaches the limit of theprecision of the computer 10−18 to T = 800, as it is shown in Figure 3-d red,while the time evolution of SALI shows again that the behavior is chaotic. ThemLE reaches a positive value, constant for both cases a4 = 2 (red) and a4 = 4(black) as it is shown in Figure 3-e, but with a very long time which alsoindicates chaotic behavior. The log10(mLE), for a4 = 2 and a4 = 4, appearsto stabilize after a time of T = 104 as it is shown, respectively, in Figure 3-fred and Figure 3-f black, so the system confirms his chaotic behavior.

4.4 Behavior of GALI for regular and chaotic motion for3D case (C’)

For the case a4 = 3, Figure 4-b represents the time evolution of GALIk fork = 2, 3, 4, 5, 6, the GALI2 and GALI3 reach non-zero constant values. How-ever, GALI4, GALI5, and GALI6 tend to zero according to the decrease ofthe asymptotic power law t−2, t−4, and t−6, this shows that the behavior isregular.

For the cases a4 = 2 and a4 = 4 as it is shown, respectively, in Figure 4-aand Figure 4-c, the time evolution of GALIk describes the chaotic behavior,where the values of GALIk abruptly tend to zero, following the exponentialdecreases with σ1 = 0.2305,σ2 = 0.0007 for a4 = 2, and σ1 = 0.2547, σ2 =0.0018 for a4 = 4.

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(a) (b) (c)

(d) (e) (f)

Fig. 3 The 3D evolution of trajectories for case (C’) with initial conditions x = −0.005,y = −0.42, z = 0.0, px = 0.5239885194, py = 0.0,pz = 0.0844906298 step size= 0.1 anda1 = −2, a3 = 1; (a) a4 = 2, h = −0.027794442; (b) a4 = 3, h = −0.027792237; (c)a4 = 4, h = −0.027792237; (d) The time evolution of the SALI for a4 = 2, 3, 4 (red, blue,black); (e, f) The time evolution of mLE for a4 = 2, 3, 4 (red, blue, black).

4.5 Behavior of PSS, mLE and SALI for regular andchaotic motion for 2D case (E)

In order to explore order and chaos and evaluate the chaos-order-chaos transi-tion, for fixed value of appropriate initial conditions, we vary only the controlparameter a4 around the 2D integrable case (E), i.e. a4 = 5, 6, 7, with a4 = 6corresponds to 2D integrability conditions.

First, let’s discuss the integrable case where a4 = 6, according to Figure5-b the PSS indicates regular behavior. The SALI varies around 0.2034 and1.4036, and seems constant after 103-time units, as it is shown in Figure 5-dblack, which indicates that the system is integrable (regular behavior). ThemLE tends to zero after 104 units of time, as it is shown in Figure 5-e black,therefore the behavior of the system is regular. Moreover, the log10(mLE) rep-resents a large variation and starts to decrease according to a power law untilit reaches the value log10(mLE) = −3 for T = 104, as it is shown in Figure5-f black, which also shows that the system confirmes his regular behavior.

However, we observe the motion in the chaos state, when we vary theparameter a4 especially while a4 = 5 and a4 = 7. The SALI abruptly decreases

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(a) (b) (c)

Fig. 4 The time evolution of GALIk(t), k = 2, 3, 4, 5, 6 for case (C’) with initial conditionsx = −0.005, y = −0.42, z = 0.0, px = 0.5239885194, py = 0.0,pz = 0.0844906298 anda1 = −2, a3 = 1; (a) a4 = 2, (b) a4 = 3, (c) a4 = 4.

(a) (b) (c)

(d) (e) (f)

Fig. 5 The PSS for case (E) with initial conditions x = −0.736320, y = 0.0455005, px =−0.0849987979, py = 0.3598360147, step size= 0.05 and a1 = −1, a3 = 16; (a) a4 = 5, h =−0.13056122; (b) a4 = 6, h = −0.13000000; (c) a4 = 7, h = −0.12943878; (d) The timeevolution of the SALI for a4 = 5, 6, 7 (blue, black, red); (e, f) The time evolution of mLEfor a4 = 5, 6, 7 (blue, black, red).

towards zero after a small transition of time, at T = 233 as it is shown inFigure 5-d blue, indicating that the behavior is chaotic. For a4 = 7, the SALIbecomes zero after 369-time units (figure 5-d red). On the other hand, themLE ends up obtaining constant positive values as it is shown Figure 5-e bluefor a4 = 5 and Figure 5-e red for a4 = 7. The log10(mLE) shows a fluctuationup to 200-time units and ends up stabilizing after this value, as it is shown

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Figure 5-f blue for a4 = 5 and Figure 5-f red for a4 = 7, which leads to chaoticbehavior.

4.6 Behavior of GALI for regular and chaotic motion for2D case (E)

(a) (b) (c)

Fig. 6 The time evolution of GALIk(t), k = 2, 3, 4 for case (E) with initial conditions x =−0.736320, y = 0.0455005, px = −0.0849987979, py = 0.3598360147 and a1 = −1, a3 = 16;(a) a4 = 5, (b) a4 = 6, (c) a4 = 7.

Figure 6 illustrates the time evolution of the GALIk for a regular andchaotic behavior. According to Figure 6-a where a4 = 5, all the values ofthe GALI suddenly tend towards zero after the exponential decreases withσ1 = 0.2421, this situation corresponds to chaotic behavior.

For a4 = 6 represents the time evolution of the GALIk for k = 2, 3, 4,the GALI2 remains constant not null, however, the GALI3 and the GALI4decrease to zero according to the asymptotic power laws, indicated by lines inthe Figure 6-b, in this case the system behaves regularly.

For a4 = 7, we notice that the behavior is chaotic, the time evolution of theGALIk confirms this nature as it is shown in Figure 6-c, because the values ofthe GALIk suddenly become zero following exponential decreases, indicatedby straight lines, with σ1 = 0.2016.

4.7 Behavior of 3D projection, mLE and SALI forregular and chaotic motion for 3D case (E’)

In the same way, we select an isolated orbit corresponding to the fixed value ofinitial conditions x = −0.004760, y = 0.0, z = −0.4531395, px = 0.1252732365,py = 0.08858155502, pZ = 0.4477669702. To test regular and chaotic behaviorand observe chaos-order-chaos transition around the 3D integrable case (E’),we choose to vary only the control parameter (a4 = 5, 6, 11), with a4 = 6

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corresponds to 3D integrability conditions.

(a) (b) (c)

(d) (e) (f)

Fig. 7 The 3D evolution of trajectories for case (E’) with initial conditions x = −0.004760,y = 0.0, z = −0.4531395, px = 0.1252732365, py = 0.08858155502, pZ = 0.4477669702 stepsize= 0.1 and a1 = −1, a3 = 16; (a) a4 = 5, h = −0.12998837; (b) a4 = 6, h = −0.130; (c)a4 = 11, h = −0.13000233; (d) The time evolution of the SALI for a4 = 5, 6, 11 (black, blue,red); (e, f) The time evolution of mLE for a4 = 5, 6, 11 (black, blue, red).

When the control parameter a4 = 6, the evolution of trajectories is char-acterized by regular states which proves the integrability of the system. Whena4 takes the value 5 and 11, we observe a dramatic change in the states ofthe trajectories indicating the completely chaotic behavior. For a4 = 6, theSALI remains different from zero and varies around 0.2223 and 1.0518 as it isshown in Figure 7-d blue, which confirms the regular state. As for the mLE,it remains at zero for T = 104 as it is shown in Figure 7-e blue, which meansthat the behavior is regular. The log10(mLE) decreases over time followinga power law, it reaches the value −3.5 as it is shown in Figure 7-f blue.Thisindicates that the motion is regular, Thereby the 3D integrability of thesystem is verified.

For a4 = 5, Figure 7-d black shows that the SALI tends towards zeroafter T = 412, which correspond to the chaotic behavior. The mLE reaches apositive and constant value as it is shown in Figure 7-e black, so the motionis chaotic. Finally, the log10(mLE) reaches a constant value after some fluc-tuation, it is thus a chaotic motion as it is shown in Figure 7-f black.

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For a4 = 11, according to Figure 7-d red, the SALI becomes zero after1200-time units which indicates that the motion is chaotic. Figure 7-e redshows that the mLE reaches a constant positive value, so proves that thebehavior is chaotic. Finally, the log10(mLE) as it is shown in Figure 7-f redalso indicates that the behavior is chaotic.

4.8 Behavior of GALI for regular and chaotic motion for3D case (E’)

We study the chaos-order-chaos transition for the Hamiltonian 3D. For itsverification, we plot the time evolution of GALI (Figure 8).For a4 = 5 the time evolution of the GALIk, k = 2, 3, 4, 5, 6 is represented in

(a) (b) (c)

Fig. 8 The time evolution of GALIk(t), k = 2, 3, 4, 5, 6 for case (E’) with initial conditionsx = −0.004760, y = 0.0, z = −0.4531395, px = 0.1252732365, py = 0.08858155502, pZ =0.4477669702 and a1 = −1, a3 = 16; (a) a4 = −5, (b) a4 = 6, (c) a4 = 11.

Figure 8-a, all values of the GALIk tend towards zero rapidly, following theexponential decreases with σ1 = 0.2547, σ2 = 0.0018, which shows that themotion is chaotic.

Figure 8-b shows the regular behavior for a4 = 6 by the time evolution ofGALIk for k = 2, 3, 4, 5, 6. We notice that the GALI2 and GALI3 end up hav-ing a constant non-zero value, as for GALI4, GALI5, and GALI6 tend towardszero according to the decrease of the asymptotic power law t−2, t−4, and t−6.Finally, for a4 = 5, all the values of GALI are cancelled after a small transi-tion of time as it is shown in Figure 8-c, following the exponential decreasesthat are shown in the same Figure with σ1 = 0.0627, σ2 = 0.0052.However, it should be mentioned that GALI4, GALI5, and GALI6 tendtowards zero for both regular and chaotic orbits, but with different time rates.

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5 Conclusion

To conclude, we have established the complete integrability of Hamiltonianssystems with quartic coupling through the generalized three-dimensionalYang-Mills Hamiltonian system with four control parameters, by explicitlyconstructing the time-independent, sufficient number of involutive first inte-grals of motion. The major advantage of this approach is that it determinesa sufficient and necessary condition for integrability of a Hamiltonian systemwith nonlinear ordinary differentiel equations. We have showed that 2D and3D Yang-Mills Hamiltonians are equivalent on a zero level of the cyclic inte-gral, and the conditions on the three-dimensional integrability parameters arepreserved. However, the possibility of the existence of other cases integrablein 2D and 3D far from those known should not be excluded until proven oth-erwise. We would like to mention that the dynamical system presented hereis an interesting problem, it represents first corrections beyond the harmonicapproximation in several physical situations with different physical interpre-tations.

To highlight the results presented here, we have applied and analysedsuccessfully the behavior of several numerical methods with PSS, mLE, SALIand GALI as a tools for exploring order and chaos, and evaluating chaos-order-chaos transition in 2D and 3D autonomous Hamiltonian systems withquartic coupling.We have verified numerically that for regular behavior, thePSS and 3D projection are easily visualized, and dominated by the regularstates of the trajectories, the time evolution of SALI exhibits small fluctu-ations around non-zero values and remain constant, the mLE tends to zeroafter a long time and log10(mLE) decreases in time according to a particularpower law, and the time evolution GALIk is constant, fluctuates aroundnon-zero values and tend to zero following a particular power laws. However,for chaotic behavior, the PSS and 3D projection are not easily visualized, itspresent a dramatic change of states of the trajectories indicating the chaoticbehavior, the time evolution of SALI converges exponentially to zero, themLE reaches a positive value, then remains constant after a long time andlog10(mLE) begins to decrease following a particular power law, and the timeevolition GALIk tends exponentially to zero.

Therefore, we have demonstrated that for a dependent set of a particularcontrol parameters, the system provides a regular behavior which confirmshis integrability. Also, we have confirmed that nearly the integrable cases, themotion becomes progressively chaotic, far from regular behavior, the systemfalls completely into chaos. So, the system is also known to exhibit the phe-nomenon of the chaos-order-chaos in which the system transits from a chaoticstate to a predominantly regular state and then back to a predominantlychaotic state.

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Finally, we believe that the results presented here show that SALI andGALI methods are powerful, faster and efficient computational techniquesthat can distinguish chaotic behavior from regular. In particular GALIkmethod with higher order k is the fastest and most efficient method to detectchaos, and very useful for many interesting problems with a dimensionalitygreater than two, it is defined as a more general method than the SALI.Otherwise, to compare with SALI and GALI, computing mLE takes a longtime to disciminate chaotic behavior from regular.

Data availability All data generated or analyzed during this study areincluded in this published article.

Declarations

Conflict of interest The authors have no conflict of interest to declare thatare relevant to the content of this article.

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