Im + α α. β + I 1 I 1< 0 I 1= 0 I 1 > 0


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1 ON THE HAMILTONIAN ANDRONOVHOPF BIFURCATION M. Olle, J. Villanueva 2 and J. R. Pacha Departament de Matematica Aplicada I (UPC), Barcelona, Spain In this contribution, we consider an specic type of Hamiltonian H 0 and two of its basic facts are analyzed: the existence of a oneparameter family of periodic orbits with a transition from stability to complex instability and the appearance of a twoparameter family of bifurcating tori. The importance of this specic Hamiltonian is that it constitutes a versal deformation of such quasiperiodic unfolding: under some quite generic conditions, any Hamiltonian with three degrees of freedom and having a family of periodic orbits with a transition stablecomplex unstable can be reduced {through a canonical change{, to the sum of H 0 plus some remainder H, containing higher order terms. At the end, we give some ideas about the proof of the persistence of the invariant tori when H is added and the whole transformed Hamiltonian is taken into account. Complex instability Let H 0 be a real three degree of freedom Hamiltonian given by, H 0 ( I x y)=! I! 2 y x 2 jyj2 2 (jxj 2 2 =2 I y x) () where x T =(x x 2 ), y T =(y y 2 ), jxj 2 2 = x 2 x 2 2 jyj 2 2 = y 2 y 2 2 y x = y x 2 ; y 2 x and (u v w) is a polynomial of degree r beginning with terms of order two, so we can write, (u v w) = 2 (au2 bv 2 cw 2 ) duv euw fvw X 3lmnr l m n u l v m w n : (2) Note that H 0 depends on an action, I, and on the normal positions (x x 2 ) and momenta (y y 2 ), but it does not depend on the angle conjugate to the action,,sowe shall denote H 0 (I x y).
2 Im α Ι αι iω 2 α α I I Im αi α I i ω2 α I α I Re Re βι β Ι iω 2 β β I < 0 I = 0 I > 0 I I βi β I i ω2 I < 0 I = 0 I > 0 βi βi (a) d<0 (b) d>0 Figure : We note that when I =0,then ; 0 = 0 = i! 2 and ; 0 = 0 = ;i! 2 (collision of characteristics exponents). Therefore, the family changes its linear character from stable to complexunstable (when d < 0, g. (a)), or viceversa (when d>0, g. (b)). It is straightforward to check that the Hamiltonian () has a one parameter family of periodic orbits which can be expressed as, M I : 8 < : 2 (0 I 0))t 0 I = const. x = x 2 = y = y 2 =0 being the action I the parameter of the family. We can compute the characteristic exponents in the normal directions to the periodic orbit, which turn out to be I = i(! 2 fi ) p ;di O 2 O 2 (3) I = ;i(! 2 fi ) p ;di O 2 O 2 (4) where O 2 = O 2 (I ), and the coecients d and f which appear in the formulas are those of the development of, (2). Thus, if I is taken to be small enough, the character of the exponents depend mainly on the sign of the product ;di inside the square roots, so the evolution of the stability of the family is as shown in gure. The interest of Hamiltonian () relies on the following result. Theorem Let H, be a 3degree of freedom Hamiltonian and fm g 2R a nondegenerate oneparameter family of periodic orbits of the corresponding Hamiltonian system. Suppose, in addition, that this family undergoes a transition stablecomplex unstable as the one described in gure, for some value
3 of the parameter (say =0). Let! and! 2 be the angular frequency and the absolute value of the characteristic exponent form 0,thecritical (degenerate, resonant) periodic orbit. Then, if! =! 2 62 Q, it is always possible, to reduce formally the initial Hamiltonian H to a normal form given by (), with as in (2), but a power series instead of a polynomial. For a proof, see [3]. Nevertheless, though it is expressed formally (as an innite process), in the practice, the nonlinear reduction of the initial Hamiltonian is carried on only up to some suitable degree r, so the transformed Hamiltonian, H, can be casted into, H(I x y)=h 0 (I x y)h ( I x y) (5) where H is analytic, 2periodic on the angle, and holds the terms of degree higher than r, soitcanbethought of as a perturbation of H 0. The unfolding of 2D invariant tori We are interested in the quasiperiodic bifurcation phenomena linked with the transition stablecomplex unstable. We will describe such phenomena using the normal form H 0 (and we shall skip the remainder H o). The existence of the quasiperiodic solutions we are looking for is more easily shown if we rstchange to (canonical) polar coordinates, p x = 2r cos 2 y = ; I p 2 p sin 2 p r 2r cos 2 2r x 2 = ; p 2r sin 2 y 2 = ; I p 2 p cos 2 ; p r 2r sin 2 : 2r This introduces a second action I 2, together with its conjugate angle, 2 while r and p r are the new normal coordinates and momenta. With these polar coordinates, the Hamiltonian H 0 takes the form, H 0 (r I I 2 p r )=! I! 2 I 2 rp 2 r I 2 2 4r (r I I 2 ) (6) (we use the same notation for the transformed Hamiltonian). From its corresponding Hamiltonian equations, it is seen that this system generically presents a bifurcating two parametric family of quasiperiodic solutions. We specify this result in the following. Proposition 2 If the coecient a in the expansion of is dierent from zero, there exists a real analytic function, (J J 2 ), dened in a neighborhood
4 of (0 0), such that the parameterization T J J 2 : 8 >< >: i = i (J J 2 )t 0 i i = 2 r = (J J 2 ) I = J I 2 =2J 2 (J J 2 ) p r =0 (7) with (J J 2 2 j T J J 2 (8) 2 (J J 2 )=! 2 J 3 j T J J 2 (9) denes a (twoparameter) family of quasiperiodic solutions of the Hamiltonian (6). Moreover, if ; d a J > 0 and jj 2 jjj j 2=3 (0) for J small enough, then the solutions (7) correspond to a 2parameter family of real 2Dinvariant tori which are nondegenerated, provided b ; d2 a 6=0 () Assuming the reality conditions (0), we can investigate the stabilityofthe real invariant tori through their normal eigenvalues (characteristic exponents). They result to be J J 2 = p 2dJ O 2 (J J 2 ): (2) The formulas (2), (3) and (4), yield to the next proposition. Proposition 3 Under the conditions of proposition 2, the type of the bifurcation is determined by the sign of the coecient a of in (2). More precisely, Case. If a < 0: inverse bifurcation (hyperbolic invariant tori unfold around stable orbits). Case 2. If a>0: direct bifurcation (elliptic invariant tori unfold around complex unstable orbits). The proofs of propositions 2 and 3 can be found at [4]. This bifurcation resembles the classical AndronovHopf's one in the sense that stable (elliptic)
5 objects unfold around lower dimensional unstable ones, and conversely, unstable (hyperbolic) higher dimensional manifolds may appear around stable objects. Preservation of the invariant tori: main ideas The unfolding shown by proposition 2, and also the normal character of the bifurcating tori have been obtained and analyzed from the normal form H 0 only. The next step is to investigate the persistence of these invariant tori when the remainder H (supposed to be an small perturbation), is added and the full Hamiltonian is considered. To do so, we apply basically the Kolmogorov method adapted to lower dimensional tori (see [2]). Here, we shall briey outline the method followed. First, we select a \good" (from the point of view of Diophantine conditions on the frequencies 2 and on the normal eigenvalues ) invariant torus of the family, say T J J 2 with J J 2 xed, and expand the initial Hamiltonian, H (0) = H around it so we introduce new actions I 0T = (I 0 I0 2 ), and a new normal coordinates z T =(x y) through, r = x ; y p r = 2 x 2 y I = J I 0 I 2 =2J 2 I 0 2 where, = (J J 2 )and is the positive determination of the characteristic exponents (2). Note that the selected initial torus corresponds then to I 0 = I 0 2 = x = y = 0). From now on,we drop the primes from the new actions and, with this coordinates, the above mentioned expansion can be expressed as, H (0) ( x I y) =a T I 2 IT C (0) I 2 zt B (0) z I T E z H (0) with T = ( 2 ) and H (0) (x I I 2 y). If we dene, H 0 (x I I 2 y)=h 0 ~a()b T () z ~c T () I 2 IT e C()I 2 zt e B()z 2 IT e E()z (3) holding the terms of degree greater than 2 in x ; y J I 2J 2 I 2 2 x 2 y and in the same way H ( 2 x I I 2 y), from the perturbative term H in (5), then the dierent coecients which appear in the expansion (3) can be identied as a = H 0 ( ), ~a() =H ( ), and so on. Actually, these coecients depend also on the parameters (J J 2 ), but as they are held xed, this dependence is not explicitly written.
6 Now, we include the term 2 IT C()I ~ in H 0 and dene, ^H (0) =~a()b T ()z ~c T () I 2 zt B()z ~ 2 IT E()z ~ and therefore, H (0) may be written shortly as, H (0) =a T I 2 IT C (0) I 2 zt B (0) z I T E z H (0) ^H (0) (4) (0) Remark 4 We note that, in (4), the term ^H gathers all the terms which avoid the existence of the initial invariant torus in the following sense: if it were not present, x = I = I 2 = y = 0 would still be a solution of the complete Hamiltonian system, winding a two dimensional {and reducible in the normal directions{, invariant torus. In other words, the selected torus of the unperturbed system would then be preserved. Thus, the idea is to eliminate, by a sequence of canonical changes, the successive terms ^H (i),which appear at every step of the reduction process. The generating function S, of these changes must be taken of the form, S = T d()e T () z f T () I 2 zt G()z I T F ()z: (5) The averages of f and g with respect are d() = 0, f() = 0. G() is a symmetric matrix verifying G 2 () =0and T =( 2 ) is a parameter to be adjusted conveniently. From this expression for the generating function, it is possible to set and solve the corresponding homological equations to nd the unknowns d e f G F and check out that, at each step the norm (i) of the corresponding ^H decays quadratically, so the perturbative scheme is quadratically convergent and thus, the invariant torus is preserved \but slightly deformed". To prove this convergence, we follow the ideas about preservation oflower dimensional tori given in [2]. Acknowledgments. The rst author acknowledges the Centre de Recerca Matematica (CRM) where this research was carried out. This work has been partially supported by the Catalan grant CIRIT number 2000SGR00027, by the Spanish grant number BFM and by the INTAS grant References [] T. J. Bridges, R. H. Cushman, and R. S. Mackay, Fields Ins. Comm., 995, 4, 6{79. [2] A. Jorba and J. Villanueva, Nonlinearity, 997, 0, 783{882. [3]M. Olle, J. R. Pacha, and J. Villanueva, preprint (submitted to proceedings of IV International symposium /workshop HAMSYS 200). [4] M. Olle, J. R. Pacha, and J. Villanueva, preprint (to be submitted to Celestial Mech. and Dynamical Astron.).
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