The dynamics of Hamiltonians with non-integrable normal form

Transcription

1 Chaotic Modeling and Simlation (CMSIM) 1: 3 8, 2016 The dynamic of Hamiltonian with non-integrable normal form Ferdinand Verhlt Mathematich Intitt, Univerity of Utrecht, The Netherland ( f.verhlt@.nl) Abtract. In general Hamiltonian ytem are non-integrable bt their dynamic varie coniderably depending on the qetion whether the correponding normal form i integrable or not. We will explore thi ie for two and three degree of freedom ytem; additional remark on Hamiltonian chain can be fond in [9]. A pecial device, the qadratic part of the Hamiltonian H 2(p, q) i ed to illtrate the relt. Keyword: Hamiltonian Chao, normal form, Hamiltonian time erie. 1 Integrability ver non-integrability We will conider time-independent Hamiltonian ytem, Hamiltonian H(p, q), p, q R n with n 2 degree of freedom (DOF). A more detailed tdy i fond in [9]. Regarding mechanic, or more generally dynamical ytem, Hamiltonian ytem are non-generic. In addition we have that the exitence of an extra independent integral beide the energy for two or more degree of freedom i again non-generic for Hamiltonian ytem (hown by Poincaré in 1892, [4] vol. 1). So the following qetion i relevant: why wold we bother abot the integrability of Hamiltonian ytem? We give a few reaon, leaving ot the ethetic argment: Symmetrie play a large part in mathematical phyic model. Symmetrie may ometime indce integrability bt more often integrability of the normal form. An example i dicrete (or mirror) ymmetry. Near-integrability play a part in many model of mathematical phyic where the integrability, althogh degenerate, can be a good tarting point to analyze the dynamic. Integral of normal form may help. Non-integrability i too crde a category, it take many different form. A firt crde characterization i to ditingih non-integrable Hamiltonian ytem with integrable or non-integrable normal form. Received: 22 Jly 2015 / Accepted: 30 October 2015 c 2016 CMSIM ISSN

2 4 Ferdinand Verhlt 2 How to pinpoint (non-)integrability? Looking for a moking gn indicating integrability there are a few approache: 1. Poincaré [4] vol. 1: A periodic oltion of a time-independent Hamiltonian ytem ha two characteritic exponent zero. A econd integral add two characteritic exponent zero except in inglar cae. Thi can be oberved (for an explicit Hamiltonian ytem) a a contino family of periodic oltion on the energy manifold. Finding ch a contino family can be either a pecial degeneration of the ytem or a ign of the exitence of an extra integral. 2. Symmetrie of core; trong ymmetrie like pherical or axial ymmetry indce extra integral. Weaker ymmetrie may or may not indce an integral. An example i tdied in [6] where dicrete ymmetry i explored in two degree of freedom ytem. It i hown for intance that the pring-pendlm diplay many degeneration depending on the reonance tdied. 3. Degeneration in variational eqation or bifrcation are degeneration that often gget the preence of integral. 3 Normal form There are many paper and book on normalization. A rather complete introdction i [5]. One conider k-jet of Hamiltonian: H(p, q) = H 2 + H H k, ally in the neighborhood of table eqilibrim (p, q) = (0, 0). The H m are homogeneo polynomial in the p, q variable. An important featre i that H 2 (p, q)(t) i an independent normal form integral, ee [5]; it phyical interpretation i that H 2 i the energy of the linearized eqation of motion. The implication of the exitence of thi integral i that near table eqilibrim, for two DOF, the normal form i for all reonance ratio integrable o that chao ha for two DOF near table eqilibrim generally a maller than algebraic meare. Thi explain a lot of analytic and nmerical relt in the literatre (ee again [5]). In general, for more than two DOF, integrability of the normal form can not be expected withot additional amption. If we find integrability, it retrict the amont of chao and alo of Arnold diffion. Example: Bran parameter family Two DOF normal form are integrable bt it i till intrctive to conider them. An exampe i Bran family of Hamiltonian: H(p, q) = 1 2 (p2 1 + p q ω 2 q 2 2) a 1 3 q3 1 a 2 q 1 q 2 2.

3 Chaotic Modeling and Simlation (CMSIM) 1: 3 8, :1 reonance normal mode in phae ot phae a 1 3a 2 1/3 2/15 1/3 2/3 normal mode Fig. 1. Periodic oltion obtained from the normal form to cbic term of Bran family of dicrete-ymmetric Hamiltonian. The horizontal line correpond with iolated periodic oltion on the energy manifold, dot at 1/3 (the Hénon-Heile cae) and at 1/3 correpond with contino familie of periodic oltion. The analyi i given in [8] and mmarized in [5]; conider for intance ω = 1, a 1 and a 2 0 are parameter. The normal form to cbic term prodce two normal mode and, depending on the parameter, two familie of in-phae periodic oltion, two familie of ot-phae periodic oltion and for pecific parameter vale two contino familie of periodic oltion on the energy manifold; ee fig. 1. Normalizing to qartic term the contino family at a 1 /(3a 2 ) = 1/3 (the Hénon-Heile Hamiltonian) break p into eparate periodic oltion; the contino family at a 1 /(3a 2 ) = 1/3 perit, thi Hamiltonian before normalization i already integrable. 4 Three degree of freedom Genine firt-order reonance are characterized by it normal form. Apart from the three action, thi contain at leat two independent combination angle. We have for three DOF: 1 : 2 : 1 reonance 1 : 2 : 2 reonance 1 : 2 : 3 reonance 1 : 2 : 4 reonance A baic analyi of the normal form to cbic order H = H 2 + H 3 yield hort-periodic oltion and integral. The e of integral give inight in the geometry of the flow, enable poible application of the KAM-theorem and may prodce meare-theoretic retriction on chao. 5 Integrability of normal form The normal form ha two integral, H 2 and H (or H 3 ). I there a third integral? To etablih (non-)integrability we have:

4 6 Ferdinand Verhlt Ingenio inpection of the normal form or obvio ign of integrability, ee van der Aa and F.V. [7]. Extenion into the complex domain and analyi of inglaritie, ee Ditermaat [2]. Applying Shilnikov-Devaney theory to etablih the exitence of a tranvere homoclinic orbit on the energy manifold, ee Hoveijn and F.V. [3]. Uing Ziglin-Morale-Rami theory to tdy the monodromy grop of a particlar nontrivial oltion; thi tdy may lead to non-integrability. Thi involve the variational eqation and the characteritic exponent in the pirit of Poincaré. In an extenion one introdce the differential Galoi grop aociated with a particlar oltion; if it i non-commtative, the ytem i non-integrable. See Chritov [1]. 5.1 The genine firt-order reonance A remarkable relt i that the normal form to cbic term of the 1 : 2 : 2 reonance i integrable with qadratic third integral, ee [7]. We have that p 1 = q 1 = 0 correpond with an invariant manifold of the normal form; the manifold conit of a contino et of periodic oltion and i a degeneration according to Poincaré with 4 characteritic exponent zero. The calclation of the normal form to qartic term prodce a break-p of thi contino et into ix periodic oltion on the energy manifold. It wa hown in [2] that the normal form to cbic term of the 1 : 2 : 1 reonance i non-integrable. Thi wa hown by inglarity analyi in the complex domain. A different approach wa ed in [1] where it wa hown that for a particlar oltion the monodromy grop i not Abelian; thi preclde that the normal form i integrable by meromorphic integral. Non-integrability wa hown in [1] for the 1 : 2 : 4 reonance in a imilar way. One identifie a particlar oltion in the (p 1, q 1 ) = (0, 0) bmanifold; the local monodromy grop i not Abelian which preclde integrability. The cae of the 1 : 2 : 3 reonance i different. The analyi in [3] how that a complex ntable normal mode (p 2, q 2 ) i preent. The normal form contain an invariant manifold N defined by H 2 = E 0, H 3 = 0. N contain an invariant ellipoid, alo homoclinic and heteroclinic oltion. They are forming an organizing center prodcing a horehoe map and chao in H 2 + H 3 + H 4. So, the normal form contain only two integral. Later, Chritov [1] howed by algebraic method that H 2 + H 3 i already nonintegrable, bt the coneqence for the dynamic are not yet clear. Technically, thi i hi mot complicated cae. 6 Dicion and coneqence In two DOF the Hamiltonian normal form i integrable to any order; thi retrict the chao near table eqilibrim to exponentially mall et between the invariant tori.

5 Chaotic Modeling and Simlation (CMSIM) 1: 3 8, For three and more DOF, the itation i more complicated. If the normal form i integrable, chao i retricted to et that are algebraically mall with repect to the mall parameter that cale the energy with repect to table eqilibrim. We wold like to ditingih between vario kind of nonintegrability near eqilibrim. The chao i ally localized near homoclinic interection of table and ntable manifold. The phenomenon i mot triking after a Hamiltonian-Hopf bifrcation of a periodic oltion ha taken place, ee for the bifrcation diagram fig. 2. doble eigenvale EE EE Hamiltonian Hopf bifrcation C Fig. 2. A a parameter varie, eigenvale on the imaginary axi become coincident and then move into the complex plane. Conider the following explicit example of the 1 : 2 : 3 reonance: H(p, q) = 1 2 (p2 1 + q 2 1) + (p q 2 2) (p2 3 + q 2 3) + H 3 (p, q), H 3 (p, q) = q 2 1(a 2 q 2 + a 3 q 3 ) q 2 2(c 1 q 1 + c 3 q 3 ) bq 1 q 2 q 3. We will conider two cae. If a 2 > b, analyi of the normal form how that the (p 2, q 2 ) normal mode i ntable of type HH (hyperbolic-hyperbolic or 4 real eigenvale). If a 2 < b the (p 2, q 2 ) normal mode i ntable of type C (complex eigenvale). The H 2 (p, q)(t) time erie i hown in fig. 3 and fig 4. Both time erie diplay chaotic behavior, bt the cae of intability C involve the Devaney-Shilnikov bifrcation prodcing trong chaotic behavior; for more information ee [3]. Reference 1. O. Chritov, Non-integrability of firt order reonance in Hamiltonian ytem in three degree of freedom, Celet. Mech. Dyn. Atron. 112 pp , (2012). 2. J.J. Ditermaat, Non-integrability of the 1 : 2 : 1 reoannce, Ergod. Theory Dyn. Syt. 4 pp , (1984). 3. I. Hoveijn and F. Verhlt, Chao in the 1 : 2 : 3 Hamiltonian normal form, Phyica D 44 pp , (1990). 4. Henri Poincaré, Le Méthode Novelle de la Mécaniqe Céète, 3 vol., Gathier- Villar, Pari (1892, 1893, 1899). 5. j.a. Sander, F.Verhlt and J.Mrdock, Averaging Method in Nonlinear Dynamical Sytem, Springer (2007), 2nd ed.

6 8 Ferdinand Verhlt :2:3 reonance, chao H H :2:3 reonance, a 2 >b t t Fig. 3. The H 2(p, q)(t) time erie in the cae of normal mode intability HH. Fig. 4. The H 2(p, q)(t) time erie in the cae of normal mode intability C. 6. J.M. Twankotta and F. Verhlt, Symmetry and reonance in Hamiltonian ytem, SIAM J. Appl. Math. 61 pp , (2000). 7. E. van der Aa and F. Verhlt, Aymptotic integrability and periodic oltion of a Hamiltonian ytem in 1 : 2 : 2 reonance, SIAM J. Math. Anal. 15 pp , (1984). 8. Ferdinand Verhlt, Dicrete ymmetric dynamical ytem at the main reonance with application to axi-ymmetric galaxie, Tran. roy. Soc. London 290, pp (1979). 9. Ferdinand Verhlt, Integrability and non-integrability of Hamiltonian normal form, Acta Applicandae Mathematicae 137, pp (2015), DOI /