Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

IMPACT FACTOR 2018: 0.726
5-year IMPACT FACTOR: 0.869

CiteScore 2018: 0.90

SCImago Journal Rank (SJR) 2018: 0.323
Source Normalized Impact per Paper (SNIP) 2018: 0.821

Mathematical Citation Quotient (MCQ) 2017: 0.32

ICV 2017: 161.82

Open Access
See all formats and pricing
More options …
Volume 17, Issue 1


Volume 13 (2015)

Stability Problems and Analytical Integration for the Clebsch’s System

Camelia Pop / Remus-Daniel Ene
Published Online: 2019-04-09 | DOI: https://doi.org/10.1515/math-2019-0018


The nonlinear stability and the existence of periodic orbits of the equilibrium states of the Clebsch’s system are discussed.. Numerical integration using the Lie-Trotter integrator and the analytic approximate solutions using Multistage Optimal Homotopy Asymptotic Method are presented, too.

Keywords: optimal control; nonlinear ordinary differential systems; nonlinear stability; multistage optimal homotopy asymptotic method

MSC 2010: 34H15; 65Nxx; 65P40; 70H14; 74G10; 74H10

1 Introduction

The Clebsch’s system was proposed in 1870 (see [1] for details) and it represents a specific famous case of the Kirchoff equations which describes the motion of a rigid body in an ideal fluid. The Clebsch’s case was obtained from the equations:


by taking the quadratic Hamiltonian




The physical meaning of p is the total angular momentum, whereas x represents the total linear momentum of the system.

If we consider now the Hamiltonian


the equations (1) become:


where a1, a2, a3 are different and nonzero constants. It is well-known that its first integrals are:





During the time since its publication, a lot of problems about the Clebsch’s system have been studied like its almost Lie-Poisson structure ([2]), the Lax formulation ([3]) or its Hirota-Kimura type discretization ([4]).

The paper’s structure is as follows: first, the nonlinear stability of the equilibrium states of Clebsch’s dynamics is discussed. About this problem, only partial results were found in [2] due to the fact that the existence of a Hamilton-Poisson structure is still an open problem, and for the almost Hamilton-Poisson structure proposed only one Casimir function was found instead of two. We use here the Arnold’s method in order to obtain some new results, which does not require a Hamilton-Poisson structure. The existence of the periodic orbits around the nonlinear stable states is the subject of the second part. The last part is committed to the numerical and analytical integration. Two methods are proposed: the Lie-Trotter integrator for numerical integration and the Multistage Optimal Homotopy Asymptotic Method to find the analytic approximate solutions. Numerical simulations obtained via Mathematica 10 are presented, too.

2 Stability problems

The equilibrium states of the dynamics (3) are



Proposition 1

If a1 < a2 and a1 < a3 then the equilibrium states e1M,M are nonlinear stable.


We shall make the proof using Arnold’s method ([5]).

Let Fα,β,γC (R6, R) given by


Following Arnold’s method, we have successively:

Fα,β,γ( e1M,M ) = 0 iff α = 1 − a1, β = − 1, γ = 0.

Considering the space





for any vX, i.e. v=0ab0cdt , so


is positive definite if a1 < a2 and a1 < a3. □

Using the same arguments we obtain the following results:

Proposition 2

The equilibrium states e2M,M are nonlinearly stable if a2 < a1 and a2 < a3.

Proposition 3

The equilibrium states e3M,M are nonlinearly stable if a3 < a1 and a3 < a2.

Proposition 4

The equilibrium states e4M,N,P are nonlinearly stable for any M, N, PR.


For this case we consider the function Gα,βC (R6, R),


Following Arnold’s method, we have successively:

Gα,β( e4M,N,P ) = 0 iff α = 0, β = 0.

Let us consider the space





for any vY, i.e. v=000abct , so


is positive definite.□

3 Periodic orbits

Proposition 5

If a1 < a2 and a1 < a3 then, near to e1M,M = (M, 0, 0, M, 0, 0), the reduced dynamics have, for each sufficiently small value of the reduced energy, at least one periodic solution whose period is close to 2πλ , where



We use the Moser-Weinstein theorem with zero eigenvalue, see [6] for details:

  1. The restriction of our dynamics (3) to the coadjoint orbit:




    gives rise to a classical Hamiltonian system.

  2. The matrix of the linear part of the reduced dynamics is:



    and has purely imaginary roots. More exactly:


  3. span (∇H2( e1M,M ), ∇H3( e1M,M ), ∇H4( e1M,M )) = V0 = span100000,000100, , where


  4. The smooth function F1−a1,−1,0C (R6, R) given by


has the following properties:

  • It is a constant of motion of the dynamics (3).

  • F1−a1,−1,0( e1M,M ) = 0.

  • If a1 < a2 and a1 < a3 then





Then our assertion follows. □

Using similar arguments, the following results hold:

Proposition 6

If a2 < a1 and a2 < a3 then near to e2M,M = (0, M, 0, 0, M, 0), the reduced dynamics has, for each sufficiently small value of the reduced energy, at least one periodic solution whose period is close to 2πω , where


Proposition 7

If a3 < a1 and a3 < a2 then near to e3M,M = (0, 0, M, 0, 0, M), the reduced dynamics has, for each sufficiently small value of the reduced energy, at least one periodic solution whose period is close to 2πσ , where


Remark 1

The existence of periodic orbits around the equilibrium states e4M,N,P remains an open problem. Due to the fact that the states e4M,N,P are not regular points for the Hamiltonian H2, the method described above does not work.

4 Numerical integration

We shall discuss now the numerical integration of the equations (3) via the Lie-Trotter formula ([7]).

Figures 1 and 2 present numerical simulations obtained with MATHEMATICA 10.

 The Lie-Trotter integrator of the system (3), projection on (Ox1x2x3) plane (a1 = 10, a2 = 2, a3 = 3, x1(0) = x2(0) = x3(0) = p1(0) = p2(0) = p3(0) = 1).
Figure 1

The Lie-Trotter integrator of the system (3), projection on (Ox1x2x3) plane (a1 = 10, a2 = 2, a3 = 3, x1(0) = x2(0) = x3(0) = p1(0) = p2(0) = p3(0) = 1).

The Lie-Trotter integrator of the system (3), projection on (Op1p2p3) plane (a1 = 10, a2 = 2, a3 = 3, x1(0) = x2(0) = x3(0) = p1(0) = p2(0) = p3(0) = 1).
Figure 2

The Lie-Trotter integrator of the system (3), projection on (Op1p2p3) plane (a1 = 10, a2 = 2, a3 = 3, x1(0) = x2(0) = x3(0) = p1(0) = p2(0) = p3(0) = 1).

Following [2] or [7], the Lie-Trotter integrator can be written as:





Some of its properties are sketched in the following proposition:

Proposition 8

The numerical integrator (4) preserves the Hamiltonians H1, H2, H3 and H4 if




5 Analytic approximate solutions of the Clebsch System (3) using Multistage Optimal Homotopy Asymptotic Method

In order to find the analytical approximate solutions of the nonlinear differential system (3) with the boundary conditions


we will use the Multistage Optimal Homotopy Asymptotic Method (MOHAM) [8], as follows:

  1. we divide the theoretical interval [t0, T] into some subintervals as [t0, t1), …, [tj−1, tj), …, where tj = T;

  2. we apply the Optimal Homotopy Asymptotic Method (OHAM) [9, 10] to find the first-order approximate solutions using only one iteration.

The initial approximation in each interval [tj−1, tj), j ∈ ℕ* is provided by the solution from the previous interval, so the analytical approximate solutions can be obtained for equations of the general form


subject to the initial conditions (5), where 𝓛 is a linear operator (which is not unique) and 𝓝 is a nonlinear one.

Now, choosing the linear operators 𝓛 as:


and the nonlinear operators 𝓝[xi(t)] and 𝓝[pi(t)], i = 1, 3 as


b1, b2, b3R, and following [9, 10] we are able to construct the homotopy given by:


where p ∈ [0, 1] is the embedding parameter, and H(t, Ci) ≠ 0 is an auxiliary convergence-control function, depending of the variable t and the parameters C1, C2, …, Cs.

The following properties hold:




For the functions F of the form


the following relation is obtained:


Considering the homotopy 𝓗 given by:


and using the linear operator given by Eq. (7), the solutions of the equation


for the initial approximations xi0 and pi0, i = 1, 3 respectively, are







The secular terms must be equal to zero, i.e.:





Also, to compute F1(t, Ci) we solve the equation


by taking into consideration that the nonlinear operator N presents the general form:


where m is a positive integer and hi(η) and gi(η) are known functions depending both on F0(η) and 𝓝.

Substituting Eqs. (16) into Eqs. (8), we obtain


Now, we observe that the nonlinear operators 𝓝[xi0(t)] and 𝓝[pi0(t)], i = 1, 3 respectively, are linear combinations of the functions


Although the equation (17) is a nonhomogeneous linear one, in most cases its solution cannot be found. In order to compute the function F1(t, Ci) we will use the third modified version of OHAM (see [9] for details), consisting of the following steps:

  • -

    We choose the auxiliary convergence-control functions Hi such that Hi ⋅ 𝓝[F0(t)] and 𝓝[F0(t)] have the same form. So, the first approximation of xi1 or pi1, i = 1, 3, denoted F1, becomes:


    where B12=i=111Bi;

  • -

    Next, by taking into account the equation (12), the first-order analytical approximate solution of the equations (6) - (5) is:


    where can be i or i, i = 1, 3, F0 can be xi0 or pi0, i = 1, 3, and F1 can be xi1 or pi1, i = 1, 3, respectively;

  • -

    Finally, the convergence-control parameters ω0, ω1 - ω6, B1-B12, C1-C12, which determine the first-order approximate solution (21), can be optimally computed by means of various methods, such as: the least square method, the Galerkin method, the collocation method, the Kantorowich method or the weighted residual method.

6 Numerical Examples and Discussions

In this section, the accuracy and validity of the MOHAM technique is proved using a comparison of our approximate solutions with numerical results obtained via the fourth-order Runge-Kutta method in the following case: we consider the initial value problem given by (3) with initial conditions Ai = 1, i = 1, 6, a1 = 10, a2 = 2 and a3 = 3.

The convergence-control parameters b1, b2, b3, ω0, ω1 - ω6, Bi|i=1,11¯,Ci|i=1,12¯ are optimally determined by means of the least-square method.

  • For 1 : the convergence-control parameters on the interval [0, 2] are:


    and the convergence-control parameters on the interval [2, 5] are given by:


  • -

    for 2 : the convergence-control parameters on the interval [0, 2] are:


    and the convergence-control parameters on the interval [2, 5] are given by:


  • -

    for 3 : the convergence-control parameters on the interval [0, 2] are:


    and the convergence-control parameters on interval [2, 5] are:


  • -

    for 1 : the convergence-control parameters on the interval [0, 2] are:


    and the convergence-control parameters on the interval [2, 5] are:


  • -

    for 2 : the convergence-control parameters on the interval [0, 2] are:


    and the convergence-control parameters on the interval [2, 5] are:


  • -

    for 3 : the convergence-control parameters on the interval [0, 2] are:


    and the convergence-control parameters on the interval [2, 5] are:


Finally, Tables 1 - 4 emphasizes the accuracy of the MOHAM technique by comparing the approximate analytic solutions 1 and 1 respectively presented above with the corresponding numerical integration values (via the 4th-order Runge-Kutta method), and the Lie-Trotter integrator. Finally, the results obtained using MOHAM are much closer to the original solution in comparison to the results obtained using Lie-Trotter integrator. These comparisons show the effectiveness, reliability, applicability, efficiency and accuracy of the MOHAM against to the Lie-Trotter integrator.

Table 1

The comparison between the approximate solutions 1 given by Eq. (22) and the corresponding numerical solutions for a1 = 10, a2 = 2 and a3 = 3 (relative errors: ϵx1 = |x1numerical1MOHAM|)

Table 2

The comparison between the approximate solutions 1 given by Eq. (23) and the corresponding numerical solutions for a1 = 10, a2 = 2 and a3 = 3 (relative errors: ϵx1 = |x1numerical1MOHAM|)

Table 3

The comparison between the approximate solutions 1 given by Eq. (28) and the corresponding numerical solutions for a1 = 10, a2 = 2 and a3 = 3 (relative errors: ϵp1 = |p1numerical1MOHAM|)

Table 4

The comparison between the approximate solutions 1 given by Eq. (29) and the corresponding numerical solutions for a1 = 10, a2 = 2 and a3 = 3 (relative errors: ϵp1 = |p1numerical1MOHAM|)

Remark 2

Figures 3 and 4 present the comparisons between the analytical approximate solutions given by MOHAM and numerical results provided by Runge-Kutta 4th steps integrator. We can see that the analytical approximate solutions and Runge-Kutta 4th steps integrator’s results are quite the same.

Profiles of the functions x̄1, x̄2, and x̄3 respectively, given by Eqs. (22), (24), (26) on [0, 2] and Eqs. (23), (25), (27) on [2, 5] respectively: - - - - - numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ MOHAM solution.
Figure 3

Profiles of the functions 1, 2, and 3 respectively, given by Eqs. (22), (24), (26) on [0, 2] and Eqs. (23), (25), (27) on [2, 5] respectively: - - - - - numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ MOHAM solution.

Profiles of the functions p̄1, p̄2, and p̄3 respectively, given by Eq. (28), (30), (32) on [0, 2] and Eqs. (29), (31), (33) on on [2, 5] respectively: - - - - - numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ MOHAM solution.
Figure 4

Profiles of the functions 1, 2, and 3 respectively, given by Eq. (28), (30), (32) on [0, 2] and Eqs. (29), (31), (33) on on [2, 5] respectively: - - - - - numerical solution, ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ MOHAM solution.

7 Conclusion

The stability problem and the existence of the periodic orbits represent important issues for any differential equations system, so a lot of methods were developed ([11]) in order to obtain better results.

The Clebsch’s system arises from physics like a lot of other systems: the rigid body ([12]), the Maxwell-Bloch equations ([13]), the heavy top dynamics ([14]), the spacecraft dynamics ([15]), the Ishii’s equations ([16]), and the list could continue. For all these examples, the energy-methods provided us conclusive results, being a good reason to use it again.

In this paper we analyze the nonlinear stability of the equilibrium states of Clebsch’s system. Some results obtained in [2] -like the stability of the equilibrium states e1M,N,e2M,N,e3M,N - are improved and some other new, like the stability of the equilibrium states e4M,N,P which are developed. Then, using Moser-Weinstein theorem with zero eigenvalue, we were able to prove the existence of the periodic orbits around the nonlinear stable equilibria.

In the last part, a comparison of the results obtained using numerical integrator, Lie-Trotter and Multistage Optimal Homotopy Asymptotic Method are analyzed. We summarize that the MOHAM’s analytic solutions are proved to be the best.


  • [1]

    Clebsch A., Über die Bewegung eines Köorpers in einer Flüssigkeit, Math. Ann., 1870, 3, 238–262. CrossrefGoogle Scholar

  • [2]

    Birtea P., Hogea C., Puta M., Some Remarks on the Clebsch’s System, Bull. Sci. Math., 2004, 128, 871–882. CrossrefGoogle Scholar

  • [3]

    Zhivkov A., Christovy O., Effective solutions of Clebsch and C. Neumann systems, Preprints aus dem Institut für Mathematik 3, 1998, 23 pages. Google Scholar

  • [4]

    Petrera M., Pfadler A., Suris Y., On integrability of Hirota-Kimura Type Discretizations, Regul. Chaotic Dyn., 2011, 16(3-4), 245–289. Web of ScienceCrossrefGoogle Scholar

  • [5]

    Arnold V., Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid, Doklady, 1965, tome 162, no 5, 773–777. Google Scholar

  • [6]

    Birtea P., Puta M., Tudoran R. M., Periodic orbits in the case of a zero eigenvalue, C. R. Acad. Sci. Paris, 2007, 344(12), 779–784. CrossrefGoogle Scholar

  • [7]

    Trotter H.F., On the product of semigroups of operators, Proc. Amer. Math. Soc., 1959, 10, 545–551. CrossrefGoogle Scholar

  • [8]

    Anakira N. Rabit, Alomari A. K., Jameel Ali, Hashim I., Multistage optimal homotopy asymptotic method for solving initial-value problems, J. Nonlinear Sci. App., 2016, 9(4), 1826–1843. CrossrefGoogle Scholar

  • [9]

    Marinca V., Herişanu N., Bota C., Marinca B., An optimal homotopy asymptotic method applied to the steady flow of a fourth grade fluid past a porous plate, Appl. Math. Lett., 2009, 22, 245–251. CrossrefWeb of ScienceGoogle Scholar

  • [10]

    Marinca V., Herişanu N., The Optimal Homotopy Asymptotic Method: Engineering Applications, Springer Verlag, Heidelberg, 2015. Google Scholar

  • [11]

    Hirsch M., Smale S., Devaney R., Differential equations, dynamical systems and an introduction to chaos, Elsevier, Academic Press, New York, 2003. Google Scholar

  • [12]

    Bloch A. M., Marsden J. E., Stabilization of Rigid Body Dynamics by the Energy-Casimir Method, Syst. Control Lett., 1990, 14, 341–346. CrossrefGoogle Scholar

  • [13]

    David D., Holm D., Multiple Lie-Poisson structures, reduction and geometric phases for the Maxwell-Bloch traveling wave equations, J. Nonlinear Sci., 1992, 2, 241–262. CrossrefGoogle Scholar

  • [14]

    Caşu I., Căruntu M., Puta M., The stability problem and the existence of periodic orbits in the heavy top dynamics, In: I. M. Mladenov and G.L. Naber (eds.), The Fourth International Conference on Geometry, Integrability and Quantization, (Varna, Bulgaria), 2002, June 6-15, Sofia: Coral Press, 248–256. Google Scholar

  • [15]

    Wang Y., Xu S., On the nonlinear stability of relative equilibria of the full spacecraft dynamics around an asteroid, Nonlinear Dynam., 2014, 78, 1–13. Web of ScienceCrossrefGoogle Scholar

  • [16]

    Tudoran R., On the dynamics and the geometry of the Ishii system, International Journal of Geometric Methods in Modern Physics, 2013, 10(01), Article ID: 1220017, https://doi.org/10.1142/S0219887812200174Web of Science

About the article

Received: 2018-01-31

Accepted: 2019-01-18

Published Online: 2019-04-09

Conflict of InterestConflict of Interests: The authors declare that there is no conflict of interests regarding the publication of this paper.

Citation Information: Open Mathematics, Volume 17, Issue 1, Pages 242–259, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2019-0018.

Export Citation

© 2019 Pop and Ene, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

Comments (0)

Please log in or register to comment.
Log in