Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 764108, 5 pages http://dx.doi.org/10.1155/2013/764108 Research Article Stability Problems for Chua System with One Linear Control Camelia Pop ArieGanu “Politehnica” University of Timis ¸oara, Department of Mathematics, 2 Victoriei Place, 300006 Timis ¸oara, Romania Correspondence should be addressed to Camelia Pop Aries ¸anu; [email protected] Received 16 January 2013; Revised 11 March 2013; Accepted 13 March 2013 Academic Editor: Chein-Shan Liu Copyright © 2013 Camelia Pop Aries ¸anu. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A Hamilton-Poisson realization and some stability problems for a dynamical system arisen from Chua system are presented. e stability and dynamics of a linearized smooth version of the Chua system are analyzed using the Hamilton-Poisson formalism. is geometrical approach allows to deduce the nonlinear stabilization near different equilibria. 1. Introduction e Chua system was proposed by Chua as a model of a nonlinear electrical circuit for the generation of chaotic oscil- lations. For some parameter values it has chaotic behavior. In this work we apply the geometric methods underlying the Hamilton-Poisson approach to analyze the characteristic features of the system. is geometrical approach makes it possible to find new properties that facilitate the dynamics description of the system, as well as its stability analysis. e work is divided as follows: the first part presents a short overview of the Chua’s system and its Hamilton-Poisson realization. e Casimir function can be found only for a specific relation between the system’s parameters. Finding the Casimir for the general case remains an open problem. e stability problem is discussed in the last section. Applying a control about axis we stabilized the only one equilib- rium state of our dynamics. e nonlinear stability of the equilibrium state is studied via energy-Casimir method. For a specific case of the system’s parameters, the equilibrium point admits periodic orbits, presented in the last paragraph, too. 2. The Poisson Geometry Associated to a Smooth Linear Version of Chua System e original Chua system of differential equations on 3 has the following form (see [1] for details): ̇ = ( − ℎ ()), ̇ = − + , ̇ = −, (1) where the characteristic function ℎ is a piecewise linear function, with , being real parameters. First of all let us take as a characteristic function in system (1) the function ℎ() = 0 so the system (1) becomes ̇ = , ̇ = − + , ̇ = −. (2) e goal of this section is to try to find a Hamilton- Poisson structure for the system (2). In order to do this, let us recall very briefly the definitions of general Poisson manifolds and the Hamilton-Poisson systems. Definition 1 (see [2, 3]). Let be a smooth manifold and let ∞ () denote the set of the smooth real functions on .A Poisson bracket on is a bilinear map from ∞ ()× ∞ () into ∞ (), denoted as (,) → {,} ∈ ∞ () , , ∈ ∞ () (3) which verifies the following properties: (i) skew-symmetry: {, } = − {, } ; (4)
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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 764108 5 pageshttpdxdoiorg1011552013764108
Research ArticleStability Problems for Chua System with One Linear Control
Camelia Pop ArieGanu
ldquoPolitehnicardquo University of Timisoara Department of Mathematics 2 Victoriei Place 300006 Timisoara Romania
Correspondence should be addressed to Camelia Pop Ariesanu cariesanuyahoocom
Received 16 January 2013 Revised 11 March 2013 Accepted 13 March 2013
Academic Editor Chein-Shan Liu
Copyright copy 2013 Camelia Pop Ariesanu This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
A Hamilton-Poisson realization and some stability problems for a dynamical system arisen from Chua system are presented Thestability and dynamics of a linearized smooth version of the Chua system are analyzed using the Hamilton-Poisson formalismThisgeometrical approach allows to deduce the nonlinear stabilization near different equilibria
1 Introduction
The Chua system was proposed by Chua as a model of anonlinear electrical circuit for the generation of chaotic oscil-lations For some parameter values it has chaotic behaviorIn this work we apply the geometric methods underlyingthe Hamilton-Poisson approach to analyze the characteristicfeatures of the system This geometrical approach makes itpossible to find new properties that facilitate the dynamicsdescription of the system as well as its stability analysis Thework is divided as follows the first part presents a shortoverview of the Chuarsquos system and its Hamilton-Poissonrealization The Casimir function can be found only for aspecific relation between the systemrsquos parameters Finding theCasimir for the general case remains an open problem Thestability problem is discussed in the last section Applyinga control about 119874119910 axis we stabilized the only one equilib-rium state of our dynamics The nonlinear stability of theequilibrium state is studied via energy-Casimirmethod For aspecific case of the systemrsquos parameters the equilibrium pointadmits periodic orbits presented in the last paragraph too
2 The Poisson Geometry Associated to aSmooth Linear Version of Chua System
The original Chua system of differential equations on 1198773 has
the following form (see [1] for details)
= 120572 (119910 minus ℎ (119909))
= 119909 minus 119910 + 119911
= minus120573119910
(1)
where the characteristic function ℎ is a piecewise linearfunction with 120572 120573 being real parameters First of all let ustake as a characteristic function in system (1) the functionℎ(119909) = 0 so the system (1) becomes
= 120572119910
= 119909 minus 119910 + 119911
= minus120573119910
(2)
The goal of this section is to try to find a Hamilton-Poisson structure for the system (2) In order to do this let usrecall very briefly the definitions of general Poissonmanifoldsand the Hamilton-Poisson systems
Definition 1 (see [2 3]) Let 119872 be a smooth manifold and let119862infin(119872) denote the set of the smooth real functions on 119872 A
Poisson bracket on119872 is a bilinearmap from119862infin(119872)times119862
infin(119872)
into 119862infin(119872) denoted as
(119865 119866) 997891997888rarr 119865 119866 isin 119862infin
(119872) 119865 119866 isin 119862infin
(119872) (3)
which verifies the following properties(i) skew-symmetry
119865 119866 = minus 119866 119865 (4)
2 Journal of Applied Mathematics
(ii) jacobi identity
119865 119866119867 + 119866 119867 119865 + 119867 119865 119866 = 0 (5)
(iii) leibniz rule
119865 119866 sdot 119867 = 119865 119866 sdot 119867 + 119866 sdot 119865119867 (6)
Proposition 2 (see [2 3]) Let sdot sdot be a Poisson structure on119877119899 Then for any 119891 119892 isin 119862
infin(119877119899 119877) the following relation holds
119891 119892 =
119899
sum
119894119895=1
119909119894 119909119895
120597119891
120597119909119894
120597119892
120597119909119895
(7)
Let the matrix given by
Π = [119909119894 119909119895] (8)
Proposition 3 (see [2 3]) Any Poisson structure sdot sdot on 119877119899 is
completely determined by the matrix Π via the relation
119891 119892 = (nabla119891)119905Π(nabla119892) (9)
Definition 4 (see [2 3]) A Hamilton-Poisson system on 119877119899
is the triple (119877119899 sdot sdot 119867) where sdot sdot is a Poisson bracket on
119877119899 and 119867 isin 119862
infin(119877119899 119877) is the energy (Hamiltonian) Its
dynamics is described by the following differential equationssystem
= Π sdot nabla119867 (10)
where 119909 = (1199091 1199092 119909
119899)119905
Definition 5 (see [2 3]) Let sdot sdot be a Poisson structure on119877119899
ACasimir of the configuration (119877119899 sdot sdot) is a smooth function
119862 isin 119862infin(119877119899 119877) which satisfies
119891 119862 = 0 forall119891 isin 119862infin
(119877119899 119877) (11)
Now following the above results we are able to prove thefollowing
Proposition 6 The system (2) has the Hamilton-Poissonrealization
(1198773 Π = [Π
119894119895] 119867) (12)
where
Π =
[[[[[[
[
01
120572 minus 120573(119909 minus 119910 + 119911) 119910
minus1
120572 minus 120573(119909 minus 119910 + 119911) 0
1
120572 minus 120573(119909 minus 119910 + 119911)
minus119910 minus1
120572 minus 120573(119909 minus 119910 + 119911) 0
]]]]]]
]
119867 (119909 119910 119911) = 120573119909 + 120572119911 120572 120573 isin 119877 120572 = 120573
(13)
The next step is to try to find the Casimir functionsof the configuration described by Proposition 1 Since thePoisson structure is degenerate we can try to obtain Casimirfunctions The defining equation for the Casimir functionsdenoted by 119862 is Π119894119895120597
119895119862 = 0 The determination of a Casimir
in a finite dimensional Hamilton-Poisson system could bedone via the algebraic method of Hernandez-Bermejo andFairen (see [4])
Let us observe that the rank of Π is constant and equalto 2 Then there exists only one functionally independentCasimir associated to our structure Following the methoddescribed in [4] the Casimir function is the solution of
119889119911 = minus119889119909 minus119910 (120573 minus 120572)
119909 minus 119910 + 119911119889119910 (14)
For the integrability of this Pfaifian system one needs anintegrant factor to transform it into an equivalent one suchthat the above form is exact The existence of such anintegrant factor is guaranteed by the Frobenius theorem If120573 minus 120572 = minus2 an integrant factor is 120593(119909 119910 119911) = 119909 + 119910 + 119911
and one obtains a Casimir of our configuration given by thefollowing expression
119862 (119909 119910 119911) =1
31199093minus
2
31199103
+1
31199113+ 1199092119911 minus 119909119910
2+ 1199091199112minus 1199102119911
(15)
Consequently we have derived the following result
Proposition 7 If 120573 minus 120572 = minus2 then the real smooth function119862 1198773
rarr 119877
119862 (119909 119910 119911) =1
31199093minus
2
31199103
+1
31199113+ 1199092119911 minus 119909119910
2+ 1199091199112minus 1199102119911
(16)
is the only one functionally independent Casimir of theHamilton-Poisson realization of the system (2)
Journal of Applied Mathematics 3
3 Stability and Stabilization by OneLinear Control
Let us pass now to discuss the stability problem (see [3]for details) of the system (2) It is not hard to see that theequilibrium states of our dynamics are
119890119872
= (119872 0 minus119872) 119872 isin 119877 (17)
Let 119860 be the matrix of the linear part of our system Thecharacteristic roots of 119860(119890
119872) are given by
1205821= 0 120582
23=
1
2(minus1 plusmn radic1 + 4 (120572 minus 120573)) (18)
If 120572 le 120573 then the equilibrium states 119890119872 119872 isin 119877 are spectrallystable Moreover the equilibrium states 119890
119872 119872 isin 119877 are
unstable
Let us consider now the case 120572 le 120573 We shall prove thatthe equilibrium states
119890119872
= (119872 0 minus119872) 119872 isin 119877 (19)
of the system (2) may be nonlinear stabilized by a particularlinear control applied to the axis 119874119910
The system (2) with one control about the axis 119874119910 can bewritten in the following form
= 120572119910
= 119909 minus 119910 + 119911 + 119906
= minus120573119910
(20)
where 119906 isin 119862infin(1198773 119877) In all that follows we shall employ the
feedback
119906 = 119896119910 119896 isin 119877 (21)
Proposition 8 The controlled system (20)-(21) is a Hamilton-Poisson mechanical system with the phase space 119875 = 119877
3 thePoisson structure
Π =
[[[[[[
[
01
120572 minus 120573(119909 + (119896 minus 1) 119910 + 119911) 119910
minus1
120572 minus 120573(119909 + (119896 minus 1) 119910 + 119911) 0
1
120572 minus 120573(119909 + (119896 minus 1) 119910 + 119911)
minus119910 minus1
120572 minus 120573(119909 + (119896 minus 1) 119910 + 119911) 0
]]]]]]
]
(22)
and the Hamiltonian 119867 given by
119867 = 120573119909 + 120572119911 (23)
where 120572 120573 isin 119877 120572 = 120573
Remark 9 If 119896 = 1 then the function119862 isin 119862infin(1198773 119877) given by
119862 (119909 119910 119911) = 1199092+ 2119909119911 minus (120572 minus 120573) 119910
2+ 1199112 (24)
is a Casimir of the configuration from Proposition 8If 119896 = 1 finding the Casimir of the structure remains an
open problem
The phase curves of the dynamics (20)-(21) are theintersections of the surfaces 119867 = const and 119862 = const seeFigure 1
Proposition 10 If 119896 = 1 then the system (20)-(21) may berealized as a Hamilton-Poisson system in an infinite number ofdifferent ways that is there exists infinitely many different (ingeneral nonisomorphic) Poisson structures on 119877
3 such that thesystem (20)-(21) is induced by an appropriate Hamiltonian
Proof The triples
(1198773sdot sdot119886119887 119867119888119889) (25)
where
119891 119892119886119887
= nabla119862 sdot (nabla119891 times nabla119892) forall119891 119892 isin 119862infin
(1198773 119877)
119862119886119887
= 119886119862 + 119887119867 119867119888119889
= 119888119862 + 119889119867
119886 119887 119888 119889 isin 119877 119886119889 minus 119887119888 = 1
119867 =1
120572 minus 120573(120573119909 + 120572119911)
119862 =1
2(1199092minus (120572 minus 120573) 119910
2+ 2119909119911 + 119911
2)
(26)
define Hamilton-Poisson realizations of the dynamics (20)-(21)
Using now the energy-Casimir method we can prove thefollowing
Proposition 11 If 120572 lt 120573 then the controlled system (20)-(21)may be nonlinear stabilized about the equilibrium states 119890119872 =
(119872 0 minus119872) 119872 isin 119877 for 119896 = 1
Proof Let
119867120593= 119862 + 120593 (119867) = 119909
2+ 2119909119911 minus (120572 minus 120573) 119910
2+ 1199112+ 120593 (120573119909 + 120572119911)
(27)
4 Journal of Applied Mathematics
minus10
minus5
0
5
10
119909
minus5
0
5119910
minus2minus1
01
119911
(a)
minus2
0
2
minus2minus1
01
2
minus2 minus1 0 1
minus2
0
2
minus2minus1
01
2
minus2 minus1 0 12
119909
119910
119911
(b)
Figure 1 The phase curves of the system (20)-(21) for 119886 = minus20 119887 = 2 and 119886 = 20 119887 = minus20 respectively
be the energy-Casimir function where 120593 119877 rarr 119877 is asmooth real valued function defined on 119877
Now the first variation of119867120593is given by
120575119867120593= 2 (119909 + 119911) 120575119909 minus 2 (120572 minus 120573) 119910120575119910
+ 2 (119909 + 119911) 120575119911 + (120573119909 + 120572119911) (120573120575119909 + 120572120575119911)
(28)
This equals zero at 119890119872 if and only if
(120573119872 minus 120572119872) = 0 (29)
The second variation of119867120593is given by
1205752119867120593= 2(120575119909)
2minus 2 (120572 minus 120573) (120575119910)
2
+ 2(120575119911)2+ 4120575119909120575119911 + sdot (120573120575119909 + 120572120575119911)
2
(30)
If 120572 lt 120573 and having chosen 120593 such that
(120573119872 minus 120572119872) = 0
(120573119872 minus 120572119872) gt 0
(31)
we can conclude that the second variation of 119867120593(119890119872) is
positively defined and thus 119890119872 is nonlinearly stable
Proposition 12 If
120573 gt 0 120572 lt 0 120573 = |120572| (32)
then near to 119890119872
= (119872 0 minus119872) 119872 isin 119877lowast the reduced dynamics
has for each sufficiently small value of the reduced energy atleast 1 periodic solution whose period is close to 120587radicminus2120572
Proof We will use Moserrsquos theorem for zero eigenvalue see[5] for details
(i) The restriction of our dynamics (20)-(21) to thecoadjoint orbit
119909 minus 119911 = 2119872 (33)
gives rise to a classical Hamiltonian system(ii) Consider span
119877(nabla119867(119890
119872)) = 119881
120582=0= span
119877([1
0
minus1])
where
119881120582=0
=
[
[
119909
119910
119911
]
]
isin 1198773| 119860 (119890
119872)[
[
119909
119910
119911
]
]
= [
[
0
0
0
]
]
(34)
with 119860(119890119872) being the matrix of the linear part of the system
(20)-(21) at the equilibrium of interest 119890119872(iii)Thematrix of the linear part of our reduced dynamics
to (20)-(21) has purely imaginary roots at the equilibrium ofinterest
1205821= 0 120582
23= plusmn119894radicminus2120572 (35)
(iv) The smooth function 119862 isin 119862infin(1198773 119877) given by
119862 (119909 119910 119911) = 1199092+ 2119909119911 minus 2120572119910
2+ 1199112 (36)
has the following properties(i) It is a constant of motion for the dynamics (20)-(21)(ii) nabla119862(119890
119872) = 0
(iii) nabla2119862(119890119872)|119882times119882
gt 0where
119882= ker 119889119867(119890119872) = span
119877([
[
1
0
1
]
]
) (37)
Journal of Applied Mathematics 5
The assertion follows via Moserrsquos theorem for zero eigenval-ues
4 Conclusion
The paper presents Hamilton-Poisson realizations of adynamical system which represents a smooth linear versionof Chua system [1] As in many other examples (Toda Lattice[6] Lu system [7] Kowalevski top dynamics [8] and batterymodel [9]) the Poisson geometry offers us a different point ofview unlike other old approaches and specific tools to studythe dynamics Due to its chaotic behavior finding its exactsolutions is an open problemThis problem can be solved forthe parameter values for which it admits a Hamilton-Poissonrealization In this case the solution is given as an intersectionbetween two surfaces119867 = const and119862 = const In additionwe can apply energy-Casimir method to study the stability ofthe equilibria and Moserrsquos theorem to find the period orbitsaround this equilibria Unfortunately these methods cannotbe used for any parameters values but only for some specificones
Acknowledgments
The author would like to thank the referees very muchfor their valuable comments and suggestions This paperwas supported by the Development and Support of Mul-tidisciplinary Postdoctoral Programmes in Major Techni-cal Areas of National Strategy for Research-Development-Innovation project 4D-POSTDOC Contract no POS-DRU8915S52603 cofunded by the European SocialFund through Sectorial Operational Programme HumanResources Development 2007ndash2013
References
[1] L O Chua ldquoNonlinear circuitsrdquo IEEE Transactions on Circuitsand Systems vol 31 no 1 pp 69ndash87 1984 Centennial specialissue
[2] J-M Ginoux and B Rossetto ldquoDifferential geometry andmechanics applications to chaotic dynamical systemsrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 4 pp 887ndash910 2006
[3] M W Hirsch S Smale and R L Devaney DifferentialEquations Dynamical Systems and an Introduction to ChaosElsevier Academic Press New York NY USA 2003
[4] B Hernandez-Bermejo and V Fairen ldquoSimple evaluation ofCasimir invariants in finite-dimensional Poisson systemsrdquoPhysics Letters A vol 241 no 3 pp 148ndash154 1998
[5] P Birtea M Puta and R M Tudoran ldquoPeriodic orbits inthe case of a zero eigenvaluerdquo Comptes Rendus MathematiqueAcademie des Sciences Paris vol 344 no 12 pp 779ndash784 2007
[6] P A Damianou ldquoMultiple Hamiltonian structures for Todasystems of type119860-119861-119862rdquo Regular amp Chaotic Dynamics vol 5 no1 pp 17ndash32 2000
[7] C Pop C Petrisor and D Bala ldquoHamilton-Poisson realizationsfor the Lu systemrdquo Mathematical Problems in Engineering vol2011 Article ID 842325 13 pages 2011
[8] A Aron P Birtea M Puta P Susoi and R Tudoran ldquoStabilityperiodic solutions and numerical integration in the Kowalevski
top dynamicsrdquo International Journal of Geometric Methods inModern Physics vol 3 no 7 pp 1323ndash1330 2006
[9] A Aron G Girban and S Kilyeni ldquoA geometric approach ofa battery mathematical model for on-line energy monitoringrdquoin Proceedings of the International Conference on Computer as aTool (EUROCON rsquo11) pp 1ndash4 Lisbon Portugal April 2011
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2 Journal of Applied Mathematics
(ii) jacobi identity
119865 119866119867 + 119866 119867 119865 + 119867 119865 119866 = 0 (5)
(iii) leibniz rule
119865 119866 sdot 119867 = 119865 119866 sdot 119867 + 119866 sdot 119865119867 (6)
Proposition 2 (see [2 3]) Let sdot sdot be a Poisson structure on119877119899 Then for any 119891 119892 isin 119862
infin(119877119899 119877) the following relation holds
119891 119892 =
119899
sum
119894119895=1
119909119894 119909119895
120597119891
120597119909119894
120597119892
120597119909119895
(7)
Let the matrix given by
Π = [119909119894 119909119895] (8)
Proposition 3 (see [2 3]) Any Poisson structure sdot sdot on 119877119899 is
completely determined by the matrix Π via the relation
119891 119892 = (nabla119891)119905Π(nabla119892) (9)
Definition 4 (see [2 3]) A Hamilton-Poisson system on 119877119899
is the triple (119877119899 sdot sdot 119867) where sdot sdot is a Poisson bracket on
119877119899 and 119867 isin 119862
infin(119877119899 119877) is the energy (Hamiltonian) Its
dynamics is described by the following differential equationssystem
= Π sdot nabla119867 (10)
where 119909 = (1199091 1199092 119909
119899)119905
Definition 5 (see [2 3]) Let sdot sdot be a Poisson structure on119877119899
ACasimir of the configuration (119877119899 sdot sdot) is a smooth function
119862 isin 119862infin(119877119899 119877) which satisfies
119891 119862 = 0 forall119891 isin 119862infin
(119877119899 119877) (11)
Now following the above results we are able to prove thefollowing
Proposition 6 The system (2) has the Hamilton-Poissonrealization
(1198773 Π = [Π
119894119895] 119867) (12)
where
Π =
[[[[[[
[
01
120572 minus 120573(119909 minus 119910 + 119911) 119910
minus1
120572 minus 120573(119909 minus 119910 + 119911) 0
1
120572 minus 120573(119909 minus 119910 + 119911)
minus119910 minus1
120572 minus 120573(119909 minus 119910 + 119911) 0
]]]]]]
]
119867 (119909 119910 119911) = 120573119909 + 120572119911 120572 120573 isin 119877 120572 = 120573
(13)
The next step is to try to find the Casimir functionsof the configuration described by Proposition 1 Since thePoisson structure is degenerate we can try to obtain Casimirfunctions The defining equation for the Casimir functionsdenoted by 119862 is Π119894119895120597
119895119862 = 0 The determination of a Casimir
in a finite dimensional Hamilton-Poisson system could bedone via the algebraic method of Hernandez-Bermejo andFairen (see [4])
Let us observe that the rank of Π is constant and equalto 2 Then there exists only one functionally independentCasimir associated to our structure Following the methoddescribed in [4] the Casimir function is the solution of
119889119911 = minus119889119909 minus119910 (120573 minus 120572)
119909 minus 119910 + 119911119889119910 (14)
For the integrability of this Pfaifian system one needs anintegrant factor to transform it into an equivalent one suchthat the above form is exact The existence of such anintegrant factor is guaranteed by the Frobenius theorem If120573 minus 120572 = minus2 an integrant factor is 120593(119909 119910 119911) = 119909 + 119910 + 119911
and one obtains a Casimir of our configuration given by thefollowing expression
119862 (119909 119910 119911) =1
31199093minus
2
31199103
+1
31199113+ 1199092119911 minus 119909119910
2+ 1199091199112minus 1199102119911
(15)
Consequently we have derived the following result
Proposition 7 If 120573 minus 120572 = minus2 then the real smooth function119862 1198773
rarr 119877
119862 (119909 119910 119911) =1
31199093minus
2
31199103
+1
31199113+ 1199092119911 minus 119909119910
2+ 1199091199112minus 1199102119911
(16)
is the only one functionally independent Casimir of theHamilton-Poisson realization of the system (2)
Journal of Applied Mathematics 3
3 Stability and Stabilization by OneLinear Control
Let us pass now to discuss the stability problem (see [3]for details) of the system (2) It is not hard to see that theequilibrium states of our dynamics are
119890119872
= (119872 0 minus119872) 119872 isin 119877 (17)
Let 119860 be the matrix of the linear part of our system Thecharacteristic roots of 119860(119890
119872) are given by
1205821= 0 120582
23=
1
2(minus1 plusmn radic1 + 4 (120572 minus 120573)) (18)
If 120572 le 120573 then the equilibrium states 119890119872 119872 isin 119877 are spectrallystable Moreover the equilibrium states 119890
119872 119872 isin 119877 are
unstable
Let us consider now the case 120572 le 120573 We shall prove thatthe equilibrium states
119890119872
= (119872 0 minus119872) 119872 isin 119877 (19)
of the system (2) may be nonlinear stabilized by a particularlinear control applied to the axis 119874119910
The system (2) with one control about the axis 119874119910 can bewritten in the following form
= 120572119910
= 119909 minus 119910 + 119911 + 119906
= minus120573119910
(20)
where 119906 isin 119862infin(1198773 119877) In all that follows we shall employ the
feedback
119906 = 119896119910 119896 isin 119877 (21)
Proposition 8 The controlled system (20)-(21) is a Hamilton-Poisson mechanical system with the phase space 119875 = 119877
3 thePoisson structure
Π =
[[[[[[
[
01
120572 minus 120573(119909 + (119896 minus 1) 119910 + 119911) 119910
minus1
120572 minus 120573(119909 + (119896 minus 1) 119910 + 119911) 0
1
120572 minus 120573(119909 + (119896 minus 1) 119910 + 119911)
minus119910 minus1
120572 minus 120573(119909 + (119896 minus 1) 119910 + 119911) 0
]]]]]]
]
(22)
and the Hamiltonian 119867 given by
119867 = 120573119909 + 120572119911 (23)
where 120572 120573 isin 119877 120572 = 120573
Remark 9 If 119896 = 1 then the function119862 isin 119862infin(1198773 119877) given by
119862 (119909 119910 119911) = 1199092+ 2119909119911 minus (120572 minus 120573) 119910
2+ 1199112 (24)
is a Casimir of the configuration from Proposition 8If 119896 = 1 finding the Casimir of the structure remains an
open problem
The phase curves of the dynamics (20)-(21) are theintersections of the surfaces 119867 = const and 119862 = const seeFigure 1
Proposition 10 If 119896 = 1 then the system (20)-(21) may berealized as a Hamilton-Poisson system in an infinite number ofdifferent ways that is there exists infinitely many different (ingeneral nonisomorphic) Poisson structures on 119877
3 such that thesystem (20)-(21) is induced by an appropriate Hamiltonian
Proof The triples
(1198773sdot sdot119886119887 119867119888119889) (25)
where
119891 119892119886119887
= nabla119862 sdot (nabla119891 times nabla119892) forall119891 119892 isin 119862infin
(1198773 119877)
119862119886119887
= 119886119862 + 119887119867 119867119888119889
= 119888119862 + 119889119867
119886 119887 119888 119889 isin 119877 119886119889 minus 119887119888 = 1
119867 =1
120572 minus 120573(120573119909 + 120572119911)
119862 =1
2(1199092minus (120572 minus 120573) 119910
2+ 2119909119911 + 119911
2)
(26)
define Hamilton-Poisson realizations of the dynamics (20)-(21)
Using now the energy-Casimir method we can prove thefollowing
Proposition 11 If 120572 lt 120573 then the controlled system (20)-(21)may be nonlinear stabilized about the equilibrium states 119890119872 =
(119872 0 minus119872) 119872 isin 119877 for 119896 = 1
Proof Let
119867120593= 119862 + 120593 (119867) = 119909
2+ 2119909119911 minus (120572 minus 120573) 119910
2+ 1199112+ 120593 (120573119909 + 120572119911)
(27)
4 Journal of Applied Mathematics
minus10
minus5
0
5
10
119909
minus5
0
5119910
minus2minus1
01
119911
(a)
minus2
0
2
minus2minus1
01
2
minus2 minus1 0 1
minus2
0
2
minus2minus1
01
2
minus2 minus1 0 12
119909
119910
119911
(b)
Figure 1 The phase curves of the system (20)-(21) for 119886 = minus20 119887 = 2 and 119886 = 20 119887 = minus20 respectively
be the energy-Casimir function where 120593 119877 rarr 119877 is asmooth real valued function defined on 119877
Now the first variation of119867120593is given by
120575119867120593= 2 (119909 + 119911) 120575119909 minus 2 (120572 minus 120573) 119910120575119910
+ 2 (119909 + 119911) 120575119911 + (120573119909 + 120572119911) (120573120575119909 + 120572120575119911)
(28)
This equals zero at 119890119872 if and only if
(120573119872 minus 120572119872) = 0 (29)
The second variation of119867120593is given by
1205752119867120593= 2(120575119909)
2minus 2 (120572 minus 120573) (120575119910)
2
+ 2(120575119911)2+ 4120575119909120575119911 + sdot (120573120575119909 + 120572120575119911)
2
(30)
If 120572 lt 120573 and having chosen 120593 such that
(120573119872 minus 120572119872) = 0
(120573119872 minus 120572119872) gt 0
(31)
we can conclude that the second variation of 119867120593(119890119872) is
positively defined and thus 119890119872 is nonlinearly stable
Proposition 12 If
120573 gt 0 120572 lt 0 120573 = |120572| (32)
then near to 119890119872
= (119872 0 minus119872) 119872 isin 119877lowast the reduced dynamics
has for each sufficiently small value of the reduced energy atleast 1 periodic solution whose period is close to 120587radicminus2120572
Proof We will use Moserrsquos theorem for zero eigenvalue see[5] for details
(i) The restriction of our dynamics (20)-(21) to thecoadjoint orbit
119909 minus 119911 = 2119872 (33)
gives rise to a classical Hamiltonian system(ii) Consider span
119877(nabla119867(119890
119872)) = 119881
120582=0= span
119877([1
0
minus1])
where
119881120582=0
=
[
[
119909
119910
119911
]
]
isin 1198773| 119860 (119890
119872)[
[
119909
119910
119911
]
]
= [
[
0
0
0
]
]
(34)
with 119860(119890119872) being the matrix of the linear part of the system
(20)-(21) at the equilibrium of interest 119890119872(iii)Thematrix of the linear part of our reduced dynamics
to (20)-(21) has purely imaginary roots at the equilibrium ofinterest
1205821= 0 120582
23= plusmn119894radicminus2120572 (35)
(iv) The smooth function 119862 isin 119862infin(1198773 119877) given by
119862 (119909 119910 119911) = 1199092+ 2119909119911 minus 2120572119910
2+ 1199112 (36)
has the following properties(i) It is a constant of motion for the dynamics (20)-(21)(ii) nabla119862(119890
119872) = 0
(iii) nabla2119862(119890119872)|119882times119882
gt 0where
119882= ker 119889119867(119890119872) = span
119877([
[
1
0
1
]
]
) (37)
Journal of Applied Mathematics 5
The assertion follows via Moserrsquos theorem for zero eigenval-ues
4 Conclusion
The paper presents Hamilton-Poisson realizations of adynamical system which represents a smooth linear versionof Chua system [1] As in many other examples (Toda Lattice[6] Lu system [7] Kowalevski top dynamics [8] and batterymodel [9]) the Poisson geometry offers us a different point ofview unlike other old approaches and specific tools to studythe dynamics Due to its chaotic behavior finding its exactsolutions is an open problemThis problem can be solved forthe parameter values for which it admits a Hamilton-Poissonrealization In this case the solution is given as an intersectionbetween two surfaces119867 = const and119862 = const In additionwe can apply energy-Casimir method to study the stability ofthe equilibria and Moserrsquos theorem to find the period orbitsaround this equilibria Unfortunately these methods cannotbe used for any parameters values but only for some specificones
Acknowledgments
The author would like to thank the referees very muchfor their valuable comments and suggestions This paperwas supported by the Development and Support of Mul-tidisciplinary Postdoctoral Programmes in Major Techni-cal Areas of National Strategy for Research-Development-Innovation project 4D-POSTDOC Contract no POS-DRU8915S52603 cofunded by the European SocialFund through Sectorial Operational Programme HumanResources Development 2007ndash2013
References
[1] L O Chua ldquoNonlinear circuitsrdquo IEEE Transactions on Circuitsand Systems vol 31 no 1 pp 69ndash87 1984 Centennial specialissue
[2] J-M Ginoux and B Rossetto ldquoDifferential geometry andmechanics applications to chaotic dynamical systemsrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 4 pp 887ndash910 2006
[3] M W Hirsch S Smale and R L Devaney DifferentialEquations Dynamical Systems and an Introduction to ChaosElsevier Academic Press New York NY USA 2003
[4] B Hernandez-Bermejo and V Fairen ldquoSimple evaluation ofCasimir invariants in finite-dimensional Poisson systemsrdquoPhysics Letters A vol 241 no 3 pp 148ndash154 1998
[5] P Birtea M Puta and R M Tudoran ldquoPeriodic orbits inthe case of a zero eigenvaluerdquo Comptes Rendus MathematiqueAcademie des Sciences Paris vol 344 no 12 pp 779ndash784 2007
[6] P A Damianou ldquoMultiple Hamiltonian structures for Todasystems of type119860-119861-119862rdquo Regular amp Chaotic Dynamics vol 5 no1 pp 17ndash32 2000
[7] C Pop C Petrisor and D Bala ldquoHamilton-Poisson realizationsfor the Lu systemrdquo Mathematical Problems in Engineering vol2011 Article ID 842325 13 pages 2011
[8] A Aron P Birtea M Puta P Susoi and R Tudoran ldquoStabilityperiodic solutions and numerical integration in the Kowalevski
top dynamicsrdquo International Journal of Geometric Methods inModern Physics vol 3 no 7 pp 1323ndash1330 2006
[9] A Aron G Girban and S Kilyeni ldquoA geometric approach ofa battery mathematical model for on-line energy monitoringrdquoin Proceedings of the International Conference on Computer as aTool (EUROCON rsquo11) pp 1ndash4 Lisbon Portugal April 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
3 Stability and Stabilization by OneLinear Control
Let us pass now to discuss the stability problem (see [3]for details) of the system (2) It is not hard to see that theequilibrium states of our dynamics are
119890119872
= (119872 0 minus119872) 119872 isin 119877 (17)
Let 119860 be the matrix of the linear part of our system Thecharacteristic roots of 119860(119890
119872) are given by
1205821= 0 120582
23=
1
2(minus1 plusmn radic1 + 4 (120572 minus 120573)) (18)
If 120572 le 120573 then the equilibrium states 119890119872 119872 isin 119877 are spectrallystable Moreover the equilibrium states 119890
119872 119872 isin 119877 are
unstable
Let us consider now the case 120572 le 120573 We shall prove thatthe equilibrium states
119890119872
= (119872 0 minus119872) 119872 isin 119877 (19)
of the system (2) may be nonlinear stabilized by a particularlinear control applied to the axis 119874119910
The system (2) with one control about the axis 119874119910 can bewritten in the following form
= 120572119910
= 119909 minus 119910 + 119911 + 119906
= minus120573119910
(20)
where 119906 isin 119862infin(1198773 119877) In all that follows we shall employ the
feedback
119906 = 119896119910 119896 isin 119877 (21)
Proposition 8 The controlled system (20)-(21) is a Hamilton-Poisson mechanical system with the phase space 119875 = 119877
3 thePoisson structure
Π =
[[[[[[
[
01
120572 minus 120573(119909 + (119896 minus 1) 119910 + 119911) 119910
minus1
120572 minus 120573(119909 + (119896 minus 1) 119910 + 119911) 0
1
120572 minus 120573(119909 + (119896 minus 1) 119910 + 119911)
minus119910 minus1
120572 minus 120573(119909 + (119896 minus 1) 119910 + 119911) 0
]]]]]]
]
(22)
and the Hamiltonian 119867 given by
119867 = 120573119909 + 120572119911 (23)
where 120572 120573 isin 119877 120572 = 120573
Remark 9 If 119896 = 1 then the function119862 isin 119862infin(1198773 119877) given by
119862 (119909 119910 119911) = 1199092+ 2119909119911 minus (120572 minus 120573) 119910
2+ 1199112 (24)
is a Casimir of the configuration from Proposition 8If 119896 = 1 finding the Casimir of the structure remains an
open problem
The phase curves of the dynamics (20)-(21) are theintersections of the surfaces 119867 = const and 119862 = const seeFigure 1
Proposition 10 If 119896 = 1 then the system (20)-(21) may berealized as a Hamilton-Poisson system in an infinite number ofdifferent ways that is there exists infinitely many different (ingeneral nonisomorphic) Poisson structures on 119877
3 such that thesystem (20)-(21) is induced by an appropriate Hamiltonian
Proof The triples
(1198773sdot sdot119886119887 119867119888119889) (25)
where
119891 119892119886119887
= nabla119862 sdot (nabla119891 times nabla119892) forall119891 119892 isin 119862infin
(1198773 119877)
119862119886119887
= 119886119862 + 119887119867 119867119888119889
= 119888119862 + 119889119867
119886 119887 119888 119889 isin 119877 119886119889 minus 119887119888 = 1
119867 =1
120572 minus 120573(120573119909 + 120572119911)
119862 =1
2(1199092minus (120572 minus 120573) 119910
2+ 2119909119911 + 119911
2)
(26)
define Hamilton-Poisson realizations of the dynamics (20)-(21)
Using now the energy-Casimir method we can prove thefollowing
Proposition 11 If 120572 lt 120573 then the controlled system (20)-(21)may be nonlinear stabilized about the equilibrium states 119890119872 =
(119872 0 minus119872) 119872 isin 119877 for 119896 = 1
Proof Let
119867120593= 119862 + 120593 (119867) = 119909
2+ 2119909119911 minus (120572 minus 120573) 119910
2+ 1199112+ 120593 (120573119909 + 120572119911)
(27)
4 Journal of Applied Mathematics
minus10
minus5
0
5
10
119909
minus5
0
5119910
minus2minus1
01
119911
(a)
minus2
0
2
minus2minus1
01
2
minus2 minus1 0 1
minus2
0
2
minus2minus1
01
2
minus2 minus1 0 12
119909
119910
119911
(b)
Figure 1 The phase curves of the system (20)-(21) for 119886 = minus20 119887 = 2 and 119886 = 20 119887 = minus20 respectively
be the energy-Casimir function where 120593 119877 rarr 119877 is asmooth real valued function defined on 119877
Now the first variation of119867120593is given by
120575119867120593= 2 (119909 + 119911) 120575119909 minus 2 (120572 minus 120573) 119910120575119910
+ 2 (119909 + 119911) 120575119911 + (120573119909 + 120572119911) (120573120575119909 + 120572120575119911)
(28)
This equals zero at 119890119872 if and only if
(120573119872 minus 120572119872) = 0 (29)
The second variation of119867120593is given by
1205752119867120593= 2(120575119909)
2minus 2 (120572 minus 120573) (120575119910)
2
+ 2(120575119911)2+ 4120575119909120575119911 + sdot (120573120575119909 + 120572120575119911)
2
(30)
If 120572 lt 120573 and having chosen 120593 such that
(120573119872 minus 120572119872) = 0
(120573119872 minus 120572119872) gt 0
(31)
we can conclude that the second variation of 119867120593(119890119872) is
positively defined and thus 119890119872 is nonlinearly stable
Proposition 12 If
120573 gt 0 120572 lt 0 120573 = |120572| (32)
then near to 119890119872
= (119872 0 minus119872) 119872 isin 119877lowast the reduced dynamics
has for each sufficiently small value of the reduced energy atleast 1 periodic solution whose period is close to 120587radicminus2120572
Proof We will use Moserrsquos theorem for zero eigenvalue see[5] for details
(i) The restriction of our dynamics (20)-(21) to thecoadjoint orbit
119909 minus 119911 = 2119872 (33)
gives rise to a classical Hamiltonian system(ii) Consider span
119877(nabla119867(119890
119872)) = 119881
120582=0= span
119877([1
0
minus1])
where
119881120582=0
=
[
[
119909
119910
119911
]
]
isin 1198773| 119860 (119890
119872)[
[
119909
119910
119911
]
]
= [
[
0
0
0
]
]
(34)
with 119860(119890119872) being the matrix of the linear part of the system
(20)-(21) at the equilibrium of interest 119890119872(iii)Thematrix of the linear part of our reduced dynamics
to (20)-(21) has purely imaginary roots at the equilibrium ofinterest
1205821= 0 120582
23= plusmn119894radicminus2120572 (35)
(iv) The smooth function 119862 isin 119862infin(1198773 119877) given by
119862 (119909 119910 119911) = 1199092+ 2119909119911 minus 2120572119910
2+ 1199112 (36)
has the following properties(i) It is a constant of motion for the dynamics (20)-(21)(ii) nabla119862(119890
119872) = 0
(iii) nabla2119862(119890119872)|119882times119882
gt 0where
119882= ker 119889119867(119890119872) = span
119877([
[
1
0
1
]
]
) (37)
Journal of Applied Mathematics 5
The assertion follows via Moserrsquos theorem for zero eigenval-ues
4 Conclusion
The paper presents Hamilton-Poisson realizations of adynamical system which represents a smooth linear versionof Chua system [1] As in many other examples (Toda Lattice[6] Lu system [7] Kowalevski top dynamics [8] and batterymodel [9]) the Poisson geometry offers us a different point ofview unlike other old approaches and specific tools to studythe dynamics Due to its chaotic behavior finding its exactsolutions is an open problemThis problem can be solved forthe parameter values for which it admits a Hamilton-Poissonrealization In this case the solution is given as an intersectionbetween two surfaces119867 = const and119862 = const In additionwe can apply energy-Casimir method to study the stability ofthe equilibria and Moserrsquos theorem to find the period orbitsaround this equilibria Unfortunately these methods cannotbe used for any parameters values but only for some specificones
Acknowledgments
The author would like to thank the referees very muchfor their valuable comments and suggestions This paperwas supported by the Development and Support of Mul-tidisciplinary Postdoctoral Programmes in Major Techni-cal Areas of National Strategy for Research-Development-Innovation project 4D-POSTDOC Contract no POS-DRU8915S52603 cofunded by the European SocialFund through Sectorial Operational Programme HumanResources Development 2007ndash2013
References
[1] L O Chua ldquoNonlinear circuitsrdquo IEEE Transactions on Circuitsand Systems vol 31 no 1 pp 69ndash87 1984 Centennial specialissue
[2] J-M Ginoux and B Rossetto ldquoDifferential geometry andmechanics applications to chaotic dynamical systemsrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 4 pp 887ndash910 2006
[3] M W Hirsch S Smale and R L Devaney DifferentialEquations Dynamical Systems and an Introduction to ChaosElsevier Academic Press New York NY USA 2003
[4] B Hernandez-Bermejo and V Fairen ldquoSimple evaluation ofCasimir invariants in finite-dimensional Poisson systemsrdquoPhysics Letters A vol 241 no 3 pp 148ndash154 1998
[5] P Birtea M Puta and R M Tudoran ldquoPeriodic orbits inthe case of a zero eigenvaluerdquo Comptes Rendus MathematiqueAcademie des Sciences Paris vol 344 no 12 pp 779ndash784 2007
[6] P A Damianou ldquoMultiple Hamiltonian structures for Todasystems of type119860-119861-119862rdquo Regular amp Chaotic Dynamics vol 5 no1 pp 17ndash32 2000
[7] C Pop C Petrisor and D Bala ldquoHamilton-Poisson realizationsfor the Lu systemrdquo Mathematical Problems in Engineering vol2011 Article ID 842325 13 pages 2011
[8] A Aron P Birtea M Puta P Susoi and R Tudoran ldquoStabilityperiodic solutions and numerical integration in the Kowalevski
top dynamicsrdquo International Journal of Geometric Methods inModern Physics vol 3 no 7 pp 1323ndash1330 2006
[9] A Aron G Girban and S Kilyeni ldquoA geometric approach ofa battery mathematical model for on-line energy monitoringrdquoin Proceedings of the International Conference on Computer as aTool (EUROCON rsquo11) pp 1ndash4 Lisbon Portugal April 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
minus10
minus5
0
5
10
119909
minus5
0
5119910
minus2minus1
01
119911
(a)
minus2
0
2
minus2minus1
01
2
minus2 minus1 0 1
minus2
0
2
minus2minus1
01
2
minus2 minus1 0 12
119909
119910
119911
(b)
Figure 1 The phase curves of the system (20)-(21) for 119886 = minus20 119887 = 2 and 119886 = 20 119887 = minus20 respectively
be the energy-Casimir function where 120593 119877 rarr 119877 is asmooth real valued function defined on 119877
Now the first variation of119867120593is given by
120575119867120593= 2 (119909 + 119911) 120575119909 minus 2 (120572 minus 120573) 119910120575119910
+ 2 (119909 + 119911) 120575119911 + (120573119909 + 120572119911) (120573120575119909 + 120572120575119911)
(28)
This equals zero at 119890119872 if and only if
(120573119872 minus 120572119872) = 0 (29)
The second variation of119867120593is given by
1205752119867120593= 2(120575119909)
2minus 2 (120572 minus 120573) (120575119910)
2
+ 2(120575119911)2+ 4120575119909120575119911 + sdot (120573120575119909 + 120572120575119911)
2
(30)
If 120572 lt 120573 and having chosen 120593 such that
(120573119872 minus 120572119872) = 0
(120573119872 minus 120572119872) gt 0
(31)
we can conclude that the second variation of 119867120593(119890119872) is
positively defined and thus 119890119872 is nonlinearly stable
Proposition 12 If
120573 gt 0 120572 lt 0 120573 = |120572| (32)
then near to 119890119872
= (119872 0 minus119872) 119872 isin 119877lowast the reduced dynamics
has for each sufficiently small value of the reduced energy atleast 1 periodic solution whose period is close to 120587radicminus2120572
Proof We will use Moserrsquos theorem for zero eigenvalue see[5] for details
(i) The restriction of our dynamics (20)-(21) to thecoadjoint orbit
119909 minus 119911 = 2119872 (33)
gives rise to a classical Hamiltonian system(ii) Consider span
119877(nabla119867(119890
119872)) = 119881
120582=0= span
119877([1
0
minus1])
where
119881120582=0
=
[
[
119909
119910
119911
]
]
isin 1198773| 119860 (119890
119872)[
[
119909
119910
119911
]
]
= [
[
0
0
0
]
]
(34)
with 119860(119890119872) being the matrix of the linear part of the system
(20)-(21) at the equilibrium of interest 119890119872(iii)Thematrix of the linear part of our reduced dynamics
to (20)-(21) has purely imaginary roots at the equilibrium ofinterest
1205821= 0 120582
23= plusmn119894radicminus2120572 (35)
(iv) The smooth function 119862 isin 119862infin(1198773 119877) given by
119862 (119909 119910 119911) = 1199092+ 2119909119911 minus 2120572119910
2+ 1199112 (36)
has the following properties(i) It is a constant of motion for the dynamics (20)-(21)(ii) nabla119862(119890
119872) = 0
(iii) nabla2119862(119890119872)|119882times119882
gt 0where
119882= ker 119889119867(119890119872) = span
119877([
[
1
0
1
]
]
) (37)
Journal of Applied Mathematics 5
The assertion follows via Moserrsquos theorem for zero eigenval-ues
4 Conclusion
The paper presents Hamilton-Poisson realizations of adynamical system which represents a smooth linear versionof Chua system [1] As in many other examples (Toda Lattice[6] Lu system [7] Kowalevski top dynamics [8] and batterymodel [9]) the Poisson geometry offers us a different point ofview unlike other old approaches and specific tools to studythe dynamics Due to its chaotic behavior finding its exactsolutions is an open problemThis problem can be solved forthe parameter values for which it admits a Hamilton-Poissonrealization In this case the solution is given as an intersectionbetween two surfaces119867 = const and119862 = const In additionwe can apply energy-Casimir method to study the stability ofthe equilibria and Moserrsquos theorem to find the period orbitsaround this equilibria Unfortunately these methods cannotbe used for any parameters values but only for some specificones
Acknowledgments
The author would like to thank the referees very muchfor their valuable comments and suggestions This paperwas supported by the Development and Support of Mul-tidisciplinary Postdoctoral Programmes in Major Techni-cal Areas of National Strategy for Research-Development-Innovation project 4D-POSTDOC Contract no POS-DRU8915S52603 cofunded by the European SocialFund through Sectorial Operational Programme HumanResources Development 2007ndash2013
References
[1] L O Chua ldquoNonlinear circuitsrdquo IEEE Transactions on Circuitsand Systems vol 31 no 1 pp 69ndash87 1984 Centennial specialissue
[2] J-M Ginoux and B Rossetto ldquoDifferential geometry andmechanics applications to chaotic dynamical systemsrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 4 pp 887ndash910 2006
[3] M W Hirsch S Smale and R L Devaney DifferentialEquations Dynamical Systems and an Introduction to ChaosElsevier Academic Press New York NY USA 2003
[4] B Hernandez-Bermejo and V Fairen ldquoSimple evaluation ofCasimir invariants in finite-dimensional Poisson systemsrdquoPhysics Letters A vol 241 no 3 pp 148ndash154 1998
[5] P Birtea M Puta and R M Tudoran ldquoPeriodic orbits inthe case of a zero eigenvaluerdquo Comptes Rendus MathematiqueAcademie des Sciences Paris vol 344 no 12 pp 779ndash784 2007
[6] P A Damianou ldquoMultiple Hamiltonian structures for Todasystems of type119860-119861-119862rdquo Regular amp Chaotic Dynamics vol 5 no1 pp 17ndash32 2000
[7] C Pop C Petrisor and D Bala ldquoHamilton-Poisson realizationsfor the Lu systemrdquo Mathematical Problems in Engineering vol2011 Article ID 842325 13 pages 2011
[8] A Aron P Birtea M Puta P Susoi and R Tudoran ldquoStabilityperiodic solutions and numerical integration in the Kowalevski
top dynamicsrdquo International Journal of Geometric Methods inModern Physics vol 3 no 7 pp 1323ndash1330 2006
[9] A Aron G Girban and S Kilyeni ldquoA geometric approach ofa battery mathematical model for on-line energy monitoringrdquoin Proceedings of the International Conference on Computer as aTool (EUROCON rsquo11) pp 1ndash4 Lisbon Portugal April 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
The assertion follows via Moserrsquos theorem for zero eigenval-ues
4 Conclusion
The paper presents Hamilton-Poisson realizations of adynamical system which represents a smooth linear versionof Chua system [1] As in many other examples (Toda Lattice[6] Lu system [7] Kowalevski top dynamics [8] and batterymodel [9]) the Poisson geometry offers us a different point ofview unlike other old approaches and specific tools to studythe dynamics Due to its chaotic behavior finding its exactsolutions is an open problemThis problem can be solved forthe parameter values for which it admits a Hamilton-Poissonrealization In this case the solution is given as an intersectionbetween two surfaces119867 = const and119862 = const In additionwe can apply energy-Casimir method to study the stability ofthe equilibria and Moserrsquos theorem to find the period orbitsaround this equilibria Unfortunately these methods cannotbe used for any parameters values but only for some specificones
Acknowledgments
The author would like to thank the referees very muchfor their valuable comments and suggestions This paperwas supported by the Development and Support of Mul-tidisciplinary Postdoctoral Programmes in Major Techni-cal Areas of National Strategy for Research-Development-Innovation project 4D-POSTDOC Contract no POS-DRU8915S52603 cofunded by the European SocialFund through Sectorial Operational Programme HumanResources Development 2007ndash2013
References
[1] L O Chua ldquoNonlinear circuitsrdquo IEEE Transactions on Circuitsand Systems vol 31 no 1 pp 69ndash87 1984 Centennial specialissue
[2] J-M Ginoux and B Rossetto ldquoDifferential geometry andmechanics applications to chaotic dynamical systemsrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 16 no 4 pp 887ndash910 2006
[3] M W Hirsch S Smale and R L Devaney DifferentialEquations Dynamical Systems and an Introduction to ChaosElsevier Academic Press New York NY USA 2003
[4] B Hernandez-Bermejo and V Fairen ldquoSimple evaluation ofCasimir invariants in finite-dimensional Poisson systemsrdquoPhysics Letters A vol 241 no 3 pp 148ndash154 1998
[5] P Birtea M Puta and R M Tudoran ldquoPeriodic orbits inthe case of a zero eigenvaluerdquo Comptes Rendus MathematiqueAcademie des Sciences Paris vol 344 no 12 pp 779ndash784 2007
[6] P A Damianou ldquoMultiple Hamiltonian structures for Todasystems of type119860-119861-119862rdquo Regular amp Chaotic Dynamics vol 5 no1 pp 17ndash32 2000
[7] C Pop C Petrisor and D Bala ldquoHamilton-Poisson realizationsfor the Lu systemrdquo Mathematical Problems in Engineering vol2011 Article ID 842325 13 pages 2011
[8] A Aron P Birtea M Puta P Susoi and R Tudoran ldquoStabilityperiodic solutions and numerical integration in the Kowalevski
top dynamicsrdquo International Journal of Geometric Methods inModern Physics vol 3 no 7 pp 1323ndash1330 2006
[9] A Aron G Girban and S Kilyeni ldquoA geometric approach ofa battery mathematical model for on-line energy monitoringrdquoin Proceedings of the International Conference on Computer as aTool (EUROCON rsquo11) pp 1ndash4 Lisbon Portugal April 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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