Non-integrability and continuation of fixed points of 2n-dimensional perturbed twist maps

Physica D 128 (1999) 70–86

Non-integrability and continuation of fixed points of 2n-dimensional perturbed twist maps Karl Wodnar1 , Simos Ichtiaroglou ∗ , Efi Meletlidou Department of Physics, University of Thessaloniki, Thessaloniki 54006, Greece Received 13 February 1997; received in revised form 5 November 1998; accepted 3 December 1998 Communicated by J.D. Meiss

Abstract In this paper, a simple criterion to prove non-integrability of symplectic, perturbed twist mappings in 2n-dimensions is developed for sufficiently small perturbations. In addition, an upper bound for the number of isolating integrals the system can possess is provided. A criterion for the analytic continuation of isolated periodic orbits in case of a small nonzero perturbation of twist maps is found. The evaluation of their linear stability character by obtaining a simplified expression of the eigenvalues of the Jacobian matrix concludes the theoretical part. The theory is finally applied to a twist map perturbed by the Suris c 1999 Elsevier Science B.V. All rights reserved. potential and also a four-dimensional map with physical interest. Keywords: Symplectic mappings; Hamiltonian systems; Non-integrability; Fixed points; Periodic orbits PACS: 03.20.+i; 05.45.+b; 46.10.+z MSC: 70H05; 58F05; 34C23

1. Introduction In many applications of non-linear dynamical systems one deals with Hamiltonian systems or symplectic maps. Such systems and maps describe, for example, magnetic plasma confinement, the interaction of molecules with an electromagnetic field, chemical reactions etc. Symplectic mappings in two dimensions have been studied widely (e.g. [1–3]). The Poincar´e–Birkhoff theorem proves the existence of at least two periodic orbits for any rational rotation number for a two-dimensional twist map [4,5]. Existence of periodic orbits has also been proved by minimization of the action function [6]. The limits of these periodic orbits for the resonant denominator going to infinity also minimize the action function and are quasiperiodic orbits which form invariant circles or cantori [6–10]. The introduction of the action difference has also been used for the study of the transport of orbits and diffusion [11–14]. Greene [15] ∗ 1

Corresponding author. E-mail: [email protected] Present address: Institute of Astronomy, University of Vienna, T¨urkenschanzstraße 17, A-1180 Vienna, Austria

c 0167-2789/99/$ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 9 8 ) 0 0 3 1 4 - 5

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has studied numerically the persistence of invariant circles in connection with the above mentioned periodic orbits. For a partial justification of Greene’s criterion see [16,17]. The case of 2n-dimensional maps has not been investigated as widely as the two-dimensional case and there are many open problems in that direction. Such maps have been studied through perturbation theory, i.e. through symplectic maps that are small perturbations of an integrable twist map. The KAM theorem for maps [18] guarantees the persistence of invariant tori, the frequency of which satisfies a certain diophantine condition under the perturbation. A significant result concerning the existence of periodic orbits is the Birkhoff–Lewis theorem [19]. A generalization of the Poincar´e–Birkhoff theorem has been proved by Arnol’d ([20], app. 9). Bernstein and Katok [21] prove the continuation of periodic orbits for perturbations of an integrable twist map using variational techniques. Symmetric periodic orbits have been found by Kook and Meiss [22], while the existence of quasiperiodic orbits in a generalization of the four-dimensional semistandard map has also been investigated by Bollt and Meiss [23]. MacKay and Meiss [24] studied the linear stability of minimizing and minimax periodic orbits in the two-dimensional case and Howard and MacKay [25] described the general properties of 2n-dimensional linear symplectic maps. Another important aspect in the dynamics of symplectic maps is the behaviour of homoclinic orbits that are perturbations of an integrable homoclinic loop [26,27]. The non-degenerate critical points of the action function, created through first order perturbation of the generating function and defined on an n-dimensional homoclinic loop of the unperturbed system, give rise to transverse homoclinic orbits in the perturbed system [28–31]. The derivative of the action function of a perturbed system is a generalized homoclinic Mel’nikov vector. The existence of heteroclinic orbits and transport in perturbations of the integrable Suris map is studied in [32]. The homoclinic Mel’nikov vector is also found through other approaches for perturbed homoclinic loops [33–35]. If the unperturbed map does not possess a homoclinic loop, the appearance of transverse homoclinic orbits in the perturbed system is an exponentially small phenomenon and the above theory does not apply. Lazutkin [36] developed a technique to prove the existence of such orbits for the case of the standard map (see also [37]). In this paper, a simple criterion to prove an obstruction to integrability of a perturbed twist mapping under certain conditions as well as a closely related theory on the fixed point structure are provided for small enough perturbations in systems of arbitrary even dimension. The integrable part of the map is generic in the sense that every integrable symplectic map can be written in this form in suitable action-angle variables. The essential steps of the procedure are the implicit formulation of the twist map in terms of a generating function of the second type and an explicit representation of the q-fold iterated map found perturbatively. In particular, we consider the resonant tori of the integrable part. Each resonant torus carries periodic orbits of the same period. With the help of the above explicit representation, we construct the action function related to the first order perturbation, evaluated on these unperturbed periodic orbits. Its restriction to one particular resonant torus is a smooth function of the angle variables. On the other hand, this function is not even continuous with respect to the actions i.e. varying within the set of tori. We do not consider it on the tori that carry quasiperiodic motion, since due to the ergodicity of the motion it reduces to a constant function. Integrability of the perturbed map corresponds to the existence and independence of n smooth functions invariant under the map in an open domain of phase space. The non-constancy of the action function leads to their dependence on the respective resonant tori. If this happens on a finite number of tori, this is not crucial since smooth or analytic functions can be dependent on such a set, but still remain independent in open domains of phase space. If on the contrary, this happens on a dense set of such tori then the smooth integrals are dependent in an open domain and cannot contribute to the further restriction of orbits of the perturbed system on invariant surfaces. Moreover, if the supposed integrals are analytic then their dependence on a set that is even less than dense may guarantee their dependence in an open domain. Such a set – as well as a dense one – is called a key set. In addition to the non-integrability criterion, a second criterion is found, which yields an upper bound to the maximum number of analytic integrals that may exist for the system under consideration. The above results are related to previous work on Hamiltonian flows [38–40]. The first criterion, especially for the two degrees of freedom case, is closely related to the Poincar´e theorem of non-integrability for Hamiltonian flows (see [40]), which shows that a Hamiltonian system is generically non-integrable. The criteria developed in [38–40] are more easily applicable and relate non-integrability to the continuation of periodic orbits, i.e. Poincar´e–Mel’nikov–Arnol’d subharmonic theory.

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Although we know already from the work of Poincar´e that a Hamiltonian system is generically non-integrable, the criteria proved in this work can be applied directly to a specific map and prove its non-integrability. Apart from the methods that prove the existence of homoclinic intersections mentioned above, there are no other non-integrability criteria applicable to specific symplectic maps known up to now. As will become apparent below, the existence of an infinite number of non-identically vanishing terms in the Fourier expansion of the perturbing function is convenient, although not always necessary for the application of the theory. This is however a generic property of a periodic function and therefore, the present results also show the genericity of non-integrability for perturbed symplectic maps. Going further, we prove the analytic continuation of periodic orbits for small perturbations by means of the implicit function theorem. In addition, in order to determine the linear stability of these orbits, we reduce the calculation of the eigenvalues of the Jacobian matrix to a linear space of half the dimension of the original system. The theory is well suited to prove the existence of a finite, but arbitrarily large number of isolated periodic orbits for generic systems, compare e.g. to ([41], p. 216) and [42]. A similar theorem on the continuation of periodic orbits for Hamiltonian flows was proved by Poincar´e [43] ([44], sections 42,79). By exploiting the symplectic structure of the Jacobian matrix of the mapping through generalizing the symmetry of the related characteristic polynomial to the matrix itself, we succeed in reducing the length of the proof considerably, compared to Poincar´e’s work. We refer the reader to ([45], p. 12, 140) for statements related to the general form of the implicit function theorem involving analytic functions as it will be heavily used in this paper and to [46] for a modern treatment of this fundamental theorem, however, involving just C ∞ functions. In both theorems proved in this work, the gradient of the action function with respect to the angles plays a fundamental rˆole. If it does not vanish identically on a dense set of resonant invariant tori of the unperturbed map, then the perturbed map is non-integrable, while its simple zeroes correspond to the periodic points continued under the perturbation. If this vector function does not vanish identically, it possesses zeroes that are generally expected to be simple. If on the other hand it vanishes identically, first order perturbation cannot show either non-integrability or the continuation of periodic orbits for the perturbed map. If, for example, the perturbed map remains integrable, utmost a non-key set of new isolated periodic points will be created while most resonant invariant tori will simply deform and continue to carry non-isolated periodic orbits. The theory is demonstrated along with three examples. In the integrable Suris map [47], the non-integrability criterion obviously fails as it should, and it is implicitly shown that no first order perturbation theory can prove the non-integrability of the standard map. Indeed, since these two maps are identical up to order one in the perturbation parameter, if it was possible to prove non-integrability of the standard map by first order perturbation theory, this would imply non-integrability of the Suris map which is obviously wrong. In the second example, we prove the non-integrability of a twist map perturbed by the Suris potential and study the bifurcation of periodic points. Finally, we prove the non-existence of a complete set of analytic integrals in a four-dimensional map with physical interest [48].

2. Perturbed twist maps We consider a one parameter family of discrete dynamical systems of dimension 2n represented through the family of canonical, i.e. symplectic mappings  0 w , J ) mod 2π (component-wise) w = w¯ ε (w (1) Gε : w, J ) J 0 = J¯ ε (w w , J ) 7→ (w w 0 , J 0 ), DE = T n × JG , JG ⊆ Rn , JG open, analytic in its parameter G ε : DE → T n × Rn , (w w , J ). Moreover, assume G 0 to be of twist type, i.e. in canonical ε ∈] − ε1 , ε1 [, ε1 > 0 and on DE with respect to (w w, J ) action-angle variables (w  0 α (JJ ) w = w + 2πα G0 : J0 =J,

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where α (JJ ) is the rotation vector. The above form is general for integrable mappings, since all such mappings can be reduced to it by transforming to action-angle variables in the open domain of bounded motion, e.g. [49]. For sufficiently small ε, G ε can be implicitly defined through a second type generating function w , J 0 ) + O(ε2 ) w , J 0 ) = w · J 0 + ST (JJ 0 ) +εS1 (w S = S(w {z } |

(2)

 ∂S ∂ST ∂S1   =w+ + ε 0 + O(ε2 ),  w0 = J0 J0 ∂J ∂J ∂JJ Gε :  ∂S ∂S 1  J = =J0 +ε + O(ε2 ). w w ∂w ∂w

(3)

S0 :=

as

w , J 0 ) and in ε ∈]−ε2 , ε2 [, 0 < ε2 ≤ ε1 . Subsequently, G ε is analytic in an open domain D = T n ×J , J ⊆ JG in (w 0 0 α (JJ ) := (∂ST /∂JJ )(JJ 0 ). If the non-degeneracy condition we shall use the standard notation 2πα   2   α ∂ ST ∂α 6= 0 (4) Det 2π 0 = Det 2 ∂JJ ∂JJ 0 holds in D we speak of a – perturbed, if ε 6= 0 – (implicit) twist mapping. G ε being canonical, its Jacobian matrix   ∂ w¯ ∂ w¯  ∂w w ∂JJ   (4) DGε =   ∂ J¯ ∂ J¯  w ∂w

∂JJ

of the related explicit representation clearly satisfies the local symplectic or canonical condition Gε ) =  Gε )T (DG (DG

(5)

at any point of phase space, where   On I n = −II n O n is the symplectic 2n × 2n unit matrix. We call a 2n-dimensional symplectic mapping G ε , represented in any state variables in an open domain D, (ε) completely (analytically) integrable (e.g. [49]) if there exist n functions 8j defined on D and on some open interval of ε around zero, which are analytic in all of their variables and which are integrals, i.e. invariants of the mapping (ε)

(ε)

8j ◦ Gε = 8j ,

j = 1, . . . , n,

if they are independent, (ε) ∇ 8(ε) rank(∇ 1 · · · ∇ 8n ) = n

in D, and if they are in involution within D, i.e. T ∇ 8(ε) ∇ 8(ε) (∇ j )  (∇ k ) = 0,

j, k = 1, . . . , n.

Otherwise such a mapping shall be termed non-integrable. The sufficiency of only n integrals for complete integrability rests on a reasoning analogous to the proof of the Liouville–Arnol’d theorem (e.g. [20], p. 271) which is stated for canonical flows. In subsequent deductions, no explicit use of the involution property will be made.

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As mentioned above, if a mapping is integrable and has bounded motions, its generating function in action-angle variables attains the form of S0 as defined in Eq. (2) in open domains. Thus, according to the second defining equation of G 0 the constant value of J 0 defines an n-dimensional invariant surface on which the motion is confined. The components of J 0 may therefore be regarded as n integrals of the map. By varying J 0 one moves from one such surface to another. These invariant surfaces are n-dimensional tori for open domains of phase space where the motion is bounded. The mapping on each one of them is defined by the angle coordinates and, as can be seen from the first defining equation of G 0 , it depends on the form of the rotation vector. If α (JJ 0 ) is a completely rational vector, then the corresponding torus carries periodic orbits with the same period and we call these tori of G 0 resonant. If some component of α (JJ 0 ) is irrational, then the motion is quasiperiodic. Since the value of α (JJ 0 ) changes continuously from torus to torus due to the non-degeneracy condition, it attains a complete set of rational component values on a dense set of invariant tori. In the next section, we are going to work on this dense set of resonant invariant tori.

3. Non-integrability criteria (ε)

w , J ) = 8j (w w , J ), 2π -periodic in each component of w , analytic in ε Assume the existence of n integrals 8j (w w , J ) in the domain D of G ε and independent in these domains with within ] − ε0 , ε0 [, 0 < ε0 ≤ ε2 , analytic in (w the possible exception of ε = 0. Thus for j = 1, . . . , n, w , J ) = 8j 0 (w w , J ) + ε8j 1 (w w , J ) + O(ε2 ) = 8j (w w 0 , J 0 ). 8j (w Now by acting the mapping Eq. (3) we have   ∂8j 0 ∂S1 0 0 0 0 w , J ) + 8j 1 (w w , J ) = 8j 0 (w w, J ) + ε w, J ) · w , J ) + O(ε2 ), (w (w 8j (w w ∂w ∂JJ 0 w 0 , J 0 ) = 8j 0 (w w + 2πα(JJ 0 ), J 0 ) 8j (w   ∂8j 0 ∂S1 0 0 0 w w α J w + 2πα α (JJ 0 ), J 0 ) · (w , J ) + 8 (w + 2πα (J ), J ) + O(ε2 ), (w +ε j1 w ∂w ∂JJ 0 respectively, and by collecting orders ε 0 and ε1 we obtain the relations w , J 0 ) = 8j 0 (w w + 2πα α (JJ 0 ), J 0 ), 8j 0 (w ∂8j 0 ∂8j 0 ∂S1 ∂S1 w, J 0) − w + 2πα α (JJ 0 ), J 0 ) · w, J 0) · w, J 0) (w (w (w (w w w ∂w ∂w ∂JJ 0 ∂JJ 0 w + 2πα α (JJ 0 ), J 0 ) − 8j 1 (w w , J 0 ). = 8j 1 (w

(6)

(7)

From Eq. (6) 8j 0 are integrals of the mapping generated by S0 , which is obvious from the expansion of 8j with respect to ε. Expanding 8j 0 into a Fourier series X w, J 0) = 8j 0 (w Ak j 0 (JJ 0 ) exp(ikk · w ) k ∈Z n

and applying relation (6) yields X kα Ak j 0 (JJ 0 )[exp(2π ikα kα(JJ 0 )) − 1] exp(ikk · w ) ≡ 0 k ∈Z n

so that either k · α (JJ 0 ) ≡ m ∈ Z

or

Ak j 0 (JJ 0 ) ≡ 0

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in the open domain of definition and for all k 6= O . But if the first alternative is true for at least one k 6= O in any open α (JJ 0 )/∂JJ 0 ) ≡ 0 and this is a contradiction to the non-degeneracy condition (4), therefore, subset of Rn , then Det(∂α ∂8j 0 w , J 0 ) ≡ 0 ⇒ 8j 0 = 8j 0 (JJ 0 ). (w (8) w ∂w We point out that by following Poincar´e ([44], section 81), one can show that the 8j can always be selected such that the 8j 0 are independent, see ([50], p. 43) for a detailed proof. Using Eq. (8) we simplify relation (7) to ∂8j 0 0 ∂S1 w , J 0 ) = 8j 1 (w w + 2πα α (JJ 0 ), J 0 ) − 8j 1 (w w , J 0 ). (w (JJ ) · w ∂w ∂JJ 0 Applying Eq. (9) q ∈ N times iteratively, we obtain w + 2πqα α (JJ 0 ), J 0 ) − 8j 1 (w w, J 0) = 8j 1 (w

q−1  X ∂8j 0 k=0

∂JJ 0

·

∂S1 w ∂w



w + 2π kα α (JJ 0 ), J 0 ). (w

Now let α be of the form pj , pj ∈ Z, q ∈ N , j = 1, . . . , n, αj (JJ 0 ) = q

(9)

(10)

(11)

such that q is relative prime to the largest common divisor of pj . Then on the resonant torus of the unperturbed w + 2π kα α (JJ 0 ), J 0 ), k = 0, . . . , q − 1, part G0 which is defined by Eq. (11) and which carries q-periodic orbits, (w corresponds to a representative segment of such a q-periodic orbit. This form of α can be achieved on a dense set of actions J 0 , i.e. on a dense set of resonant tori of the unperturbed map, because of the non-degeneracy condition (4). For such a choice of α , because of periodicity, the left hand side of Eq. (10) vanishes, so that, using again Eq. (8), q−1

∂8j 0 0 X ∂S1 w + 2πkα α (JJ 0 ), J 0 ) ≡ 0, (w (JJ ) · w ∂w ∂JJ 0

j = 1, . . . , n,

k=0

α (JJ ) ∈ Qn }. w , J 0 ) ∈ DQ := T n × JQ , JQ = {JJ ∈ J |α identically on (w In other words, ∂6 (q) w, J 0) ≡ 0 (w w ∂w holds on DQ where F (JJ 0 )

F (JJ 0 ))j,l=1,... ,n := (F w , J 0 ) := 6 (q) (w

(12)

∂8j 0 ∂Jl0

q−1 X w + 2πkα α (JJ 0 ), J 0 ). S1 (w k=0

J0

w , J ) is called the action function along an orbital segment of length q. See e.g. [7] for the definition of the 6 (q) (w action function in context with type one generating functions. On the resonant torus defined by Eq. (11) q is constant and therefore, the action function is analytic with respect to w , since S1 is an analytic function of its arguments. A different resonant torus of G 0 corresponds in general to a different q. Therefore, we have created a function on DQ which is analytic with respect to w but is only defined on a dense set of values of J 0 . In analogy to the case of continuous systems we define the subharmonic Poincar´e–Mel’nikov–Arnol’d vector of a mapping as ∂6 (q) w , J 0 ). (w w ∂w

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w , J 0 ) on a dense set of J 0 values. Then we Let us suppose now that this vector does not vanish identically in (w would have Det F ≡ 0 on this set which is a contradiction to the independence of integrals 8i0 , i = 1, . . . , n. Consequently, the system determined by G ε cannot have a complete set of independent analytic integrals. The set where the above vector is not identically zero and as a consequence, the vanishing of the determinant of F can be established, may not necessarily be dense. Definition. A key set is defined to be a set of points, such that for every analytic function it holds, if it vanishes on the points of this set, then it vanishes identically, see e.g. ([41], p. 213). w is nonzero on such a key set, then Det F vanishes identically and non-integrability is If the vector ∂6 (q) /∂w proved. This establishes the first part of our criterion. Going further, we differentiate Eq. (12) with respect to w yielding F (JJ 0 ) ≡ O , w , J 0 )F L (w identically on DQ , with L(w w , J 0 ))j,l=1,... ,n := (L

∂2 w , J 0 ). 6 (q) (w ∂wj ∂wl

Now suppose the system represented by mapping G ε has at least s independent integrals 8j , j = 1, . . . , s for F ) ≤ dim ker(L L) and dim ker(L L)+rank(L L) = F (JJ 0 )) ≥ s in D. Because rank(F some fixed s, 1 ≤ s ≤ n, then rank(F n, it follows that L(w w , J 0 )). F (JJ 0 )) ≤ n − rank(L s ≤ rank(F Note that F and its rank depend just on the resonant torus selected, while L also depends on the particular periodic orbit considered. Thus, passing to the maximum rank over all such orbits on the torus determined by J 0 L(w w , J 0 ))|w w ∈ T n }, rJ 0 := max{rank(L L(w w J 0 , J 0 )), thus s ≤ we know there exists at least one orbit with initial w = w J 0 , such that rJ 0 = rank(L 0 F (JJ )) ≤ n − rJ 0 . Note that the supremum taken here is really a maximum, since rank is an integer. Now let rank(F r := min{rJ 0 |JJ 0 ∈ J1 }, where J1 ⊆ JQ is a dense or a key set of resonant J 0 , then we have F (JJ 0 )) ≤ n − r s ≤ rank(F on this set. If now s > n − r we have arrived at a contradiction. Thus, there cannot be more than n − r independent integrals with the prescribed properties. Summarizing, we have proved the following theorem. Theorem 1. Let G ε be the implicit twist mapping defined through a generating function S as in Eq. (2), so that the w , J 0 ) in open domains and analytic in ε in an non-degeneracy condition (4) holds. Assume G ε to be analytic in (w (q) open interval around zero. Let furthermore the action function 6 be defined as 6

(q)

q−1 X w, J ) = w + 2πkα α (JJ 0 ), J 0 ) (w S1 (w 0

k=0

J0

such that J corresponds to a resonant torus, i.e. to a q -periodic orbit of G 0 , α (JJ 0 ) =

1 p, q

p ∈ Z n, q ∈ N ,

with q being relative prime to the largest common divisor of the pj . Then,

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(i) if within D a dense or a key set of resonant tori of G 0 is found such that for at least one orbit on each one of them ∂6 (q) w , J 0 ) 6≡ 0, (w w ∂w then G ε does not possess a complete set of analytic integrals for ε ∈] − ε0 , ε0 [\{0}, 0 < ε0 ≤ ε2 ; and (ii) if, likewise, on some dense or key set of resonant tori there is at least one orbit on each of them such that ! ∂ 2 6 (q) 0 w, J ) ≥ r ∈ N , (w rank w2 ∂w then G ε cannot possess more than n − r independent integrals being analytic in the same set of ε values. From the defining relation of the action function, notice that if S1 , which is a periodic function with respect to the angle coordinates, possesses an infinite number of coefficients in its Fourier expansion, this will in general result in its non-constancy for an infinite number of rational values of the rotation vector and will lead to the non-vanishing of its gradient on a dense set of resonant tori of the unperturbed map. The above property, i.e. the existence of an infinite number of nonzero Fourier coefficients is a generic property of periodic functions and reveals the genericity of the theorem. 4. Continuation of periodic orbits and linear stability w , J 0 ) and ε in open domains D = T n × J , J ⊆ Rn and Again we assume the implicit map G ε to be analytic in (w ] − ε2 , ε2 [, ε2 > 0, respectively. Expanding with respect to ε we obtain the following explicit first order formulation of the q-fold iterated mapping  (q)  ( (q) α ∂α α (JJ ) + ε ∂6 w , J ) − 2π ∂J J ) ∂6 w , J ) + O(ε2 ) w (q) = w + 2πqα (q) J (w J (J w (w ∂J ∂w Gε : (q) w , J ) + O(ε2 ). J (q) = J − ε ∂6 w (w ∂w w , J ) to be q-periodic, which will meet the requirements for a proper application of The condition for a point (w the implicit function theorem in order to show continuation of the periodicity of the specific point from ε = 0 to small |ε| 6= 0, takes the form ! p0 w (q0 ) (ε, w , J ) − w − 2πp (13) = O , p 0 ∈ Z n , q0 ∈ N , f (ε, w , J ) := w ,JJ ) J −JJ (q0 ) (ε,w ε

where p 0 , q0 are still arbitrarily selectable constant parameters characterizing the resonance studied, which are not a priori correlated to a specific fixed value of J up to the moment. In order to emphasize this striking difference of the use of p and q in its relation to 6 (q) as used now in the context of continuation – in contrast to its use with respect to non-integrability – we attach zero subscripts to p and q in the discussion of fixed point continuation. Furthermore, the division by ε involved in the J -terms is consistent, because an orbit being continued for small enough |ε| 6= 0 must fulfill Eq. (13) in its original form without division, thus, since ε 6= 0, division is possible and the condition must be still valid in the limit of ε → 0 because of continuity. The necessity of this procedure will w , J ) + O(ε) with become clear below. We have f (ε, w , J ) = f 0 (w   p0 2πq0α (JJ ) − 2πp w, J ) = . f 0 (w ∂6 (q0 ) w (w , J ) w ∂w The implicit function theorem now ensures that when w0, J 0) = O f 0 (w

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w 0 , J 0 ) and also if holds for certain (w On f0 ∂f w 0 , J 0 ) = 2 (q ) (w Det w, J ) ∂(w ∂ 6 0 (w w0, J 0) w2 ∂w

6= 0, (q ) 2 ∂ 6 0 w (w , J ) 0 0 w ∂JJ ∂w α ∂α J ∂JJ (J 0 )

(14)

then there is an open interval of ε values around zero, such that the perturbed twist map G ε has a q0 -periodic orbit w 1 + O(ε2 ), J 0 + εJJ 1 + O(ε2 )) analytically depending on w , J ) = (w w (ε), J (ε)) = (w w 0 + εw with initial condition (w w 0 , J 0 ). We note that condition (14), because of non-degeneracy reduces to ε which for ε = 0 coincides with (w ∂ 2 6 (q0 ) w 0 , J 0 ) 6= 0 (w w2 ∂w and also that without the above division by ε in the periodicity conditions for J this condition would have been void in any case. Since, p 0 and q0 are held fixed during derivation with respect to J , in contrast to p and q in Theorem 1, and are related to J 0 only a posteriori, 6 (q0 ) is an analytic function of w and J . The linear stability of the fixed points is determined by the eigenvalues of the matrix M of the tangential map. Composing this matrix as   A B (q0 ) Gε = M = DG C D Det

with four n × n matrices n

Aij = δij + ε

X ∂αi ∂2 ∂2 − 2π ∂Ji ∂wj ∂Jl ∂wl ∂wj

! 6 (q0 ) + O(ε2 )

l=1

∂αi +ε Bij = 2π q ∂Jj Cij = −ε

∂2 ∂Ji ∂Jj

− 2π

n X ∂αi l=1

∂2 ∂Jl ∂wl ∂Jj

! 6 (q0 ) + O(ε2 )

∂2 6 (q0 ) + O(ε2 ) ∂wi ∂wj

Dij = δij − ε

∂2 6 (q0 ) + O(ε2 ), ∂wi ∂Jj

for i, j = 1, . . . , n, we have from Eq. (5), since M is a symplectic matrix,  T  BT D −B M T = . M −1 = − C T AT −C Let {λ1 , . . . , λn , 1/λ1 , . . . , 1/λn } be the set of eigenvalues of both M and M −1 . We define the matrix N := M + M −1 with eigenvalues ν. Since M , M −1 are simultaneously reduced to triangular form, the eigenvalues of N are ν = λ + 1/λ and have even multiplicity. The matrix N attains the form     2 + O(ε 2 ) O(ε) 2II n + ε2 A + DT B − B T , = N = 2 T + O(ε2 ) 2In + ε2 O(ε 2 ) C − C T AT + D where α ∂ 2 6 (q0 ) ∂α w , J ). (w (JJ ) w2 ∂JJ ∂w Following the method of Yakubovich and Starzhinskii ([51], p. 315) we draw the following conclusions. For ε = 0, w , J ) = νε (w w , J ) − 2. N − νII 2n ) = 0 defines ν as a function of ε – we set εβε (w clearly ν = 2, therefore – since Det(N w , J ) = −2π 2 (w

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Dividing the above determinant by ε2n , which is justified by the necessity of continuity with respect to ε at ε = 0, we obtain 2 − βε I n O(1) = 0. O(ε) 2 T − βε I n + O(ε) w 0 , J 0 ) is an eigenvalue of 2 (w w 0 , J 0 ) we have If β0 (w w , J ) = β0 (w w 0 , J 0 ) + O(ε1/ l ), βε (w

l ∈ N,

which depends analytically on some fractional power of ε. Thus, w , J ) = 2 + εβ0 (w w 0 , J 0 ) + O(ε1+1/ l ), νε (w and consequently, w, J ) = 1 ± λ = λε (w

√ w 0 , J 0 ) + O(εmin(1,1/2+1/ l) ), ερ(w

l∈N

w 0 , J 0 ) = β0 (w w 0 , J 0 ). It follows that the linear stability of the periodic orbit continued is just determined where ρ 2 (w α /∂JJ )(JJ 0 ) and the n × n Hessian (∂ 2 6 (q0 ) /∂w w 2 )(w w 0 , J 0 ). Note, that because by the n × n (Hessian) twist matrix (∂α of Eq. (14), ρ is always different from zero, so the continued periodic orbits are always isolated. The linear stability analysis is thus reduced to consideration of a linear space of half the dimension of the system. The result is the following theorem. Theorem 2. Let G ε be the implicit twist mapping defined through a generating function S as in Theorem 1, so that the non-degeneracy condition holds. Then for all resonant tori of the unperturbed mapping G 0 determined by J 0 such that α (JJ 0 ) =

1 p , q0 0

p 0 ∈ Z n , q0 ∈ N ,

w 0 , J 0 ) is analytically continued to (w w , J ) = (w w 0 , J 0 ) + O(ε) for a q0 -periodic orbit of G 0 containing the point (w w has a simple zero in (w w 0 , J 0 ), i.e. small enough |ε| 6= 0, if the vector ∂6 (q0 ) /∂w ∂6 (q0 ) w0, J 0) = O , (w w ∂w

Det

∂ 2 6 (q0 ) w 0 , J 0 ) 6= 0, (w w2 ∂w

0 ) at such a point are of the G(q 6 (q) being defined as in Theorem 1. Moreover the eigenvalues of the Jacobian DG ε form √ w , J ) = 1 ± ερ(w w 0 , J 0 ) + O(εmin(1,1/2+1/ l) ), l ∈ N , λε (w

where ρ 2 are the eigenvalues of −2π

α ∂ 2 6 (q0 ) ∂α w0, J 0) (JJ 0 ) (w w2 ∂JJ ∂w

and λε are analytic functions of a fractional power of ε. w 0 , J 0 ) for sufficiently In the following, we consider the linear stability of mapping G ε with respect to β0 = β0 (w small |ε| in more detail. If at least one β0 is complex, instability of complex type, results. If all β0 are real and simple with at least one β0 such that εβ0 > 0, we have real instability. If all β0 are real and simple with εβ0 < 0 we encounter the only certain case of linear stability based on the framework of Theorem 2. In the latter two cases, if some β0 is multiple, complex instability may result, depending on the behaviour of the higher order terms in ε. There are certain remarkable properties that link the non-integrability criterion proved in the previous section w (q = q(JJ 0 )) is not and the continuation of periodic points from the integrable to the perturbed map. If ∂6 (q) /∂w

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identically zero on a dense set of resonant tori then, since it is a periodic function of the angles with zero mean with respect to the latter, it will in general possess simple zeroes on every torus of this set. The perturbed system will be non-integrable and these tori will dissolve under the perturbation to isolated periodic orbits as Theorem w is identically zero with the 2 establishes. On the contrary, if the perturbed map is also integrable, then ∂6 (q) /∂w possible exception of a set of resonant tori which is not a key set. The latter tori will in general produce isolated periodic orbits due to Theorem 2 (see Section 5.1 below). Note that, in view of Theorem 1, if the perturbed mapping possesses at least one integral, then for a resonant torus in general position, Det L = 0 and Theorem 2 is not applicable. It may happen, as in the case of the standard map, that the perturbed map is non-integrable and has an infinity of isolated periodic points which cannot be predicted by O(ε) perturbation theory. Note further that in the case of n = 1 we have ρ 2 (w0 , J0 ) = −2π

dα ∂ 2 6 (q0 ) (w0 , J0 ). (J0 ) dJ ∂w 2

Since 6 (q0 ) is 2π -periodic with respect to w, two consecutive simple zeroes of ∂6 (q0 ) /∂w correspond in general to a maximum and a minimum of 6 (q0 ) , i.e. positive and negative ρ 2 respectively, assuming positive twist direction (dα/dJ )(J0 ) > 0. Thus, we obtain the known result of Poincar´e–Birkhoff [4,5] for small enough ε > 0.

5. Applications 5.1. The integrable Suris map A two-dimensional symplectic map which has been proved to be integrable is the Suris map [47] wn+1 = wn + 4π 2 Jn+1 , Jn = Jn+1 − V 0 (wn ) where V 0 (w) =

dV 1 = − 2 tan−1 dw π



δ sin w 1 + δ cos w



and where w is mod 2π. Its type-two generating function is 2 − V (wn , δ). S = Jn+1 wn + 2π 2 Jn+1

By considering δ as a small parameter and expanding, we get at first order 1 ∂S1 = 2 sin w. ∂w π

(15)

For the invariant circles of period one, corresponding to J = p/2π, p ∈ Z, ∂6 (1) /∂w is equal to Eq. (15), while it is zero for all other resonances. These circles obviously do not constitute a key set so Theorem 1 fails to prove non-integrability, as expected. At this point note that the Suris map in O(δ) terms actually coincide to the standard map. Since the Suris map is integrable, this fact shows that it is not possible to prove non-integrability of the standard map, by using any first order perturbation theory. If it was possible, this would mean non-integrability of the Suris map, which is not the case. Non-integrability of the standard map has been proved by Lazutkin [36], by showing the existence of exponentially small transverse intersections of separatrices. By applying Theorem 2, we √ prove the existence of periodic points δ-close to J0 = p0 /2π and w0 = kπ, k ∈ Z with eigenvalues λ = 1 ± ( δ/π)(−1)(k+1)/2 , retrieving all the isolated fixed points of the Suris map and their stability.

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5.2. The Suris potential As a second application we consider a one-degree of freedom integrable twist map, perturbed by the Suris potential through a small parameter ε, i.e. we consider the generating function 2 − εV (wn , δ) S = Jn+1 wn + 2π 2 Jn+1

where 0 < δ < 1. For ε = 1 we obtain the integrable Suris map, but by Theorem 1 we will show that this map is non-integrable for ε 6= 0 in an open interval around zero. Note that this example is remarkable because integrability is obtained for (at least) two distinct values of the perturbation parameter (ε = 0, 1), enclosing a region of nonintegrability. By using the Fourier expansion of S1 (e.g. [52], p. 15), on the invariant circles with α = 2π J = p/q one obtains   q−1 ∞ X 1 X −1 m ∂6 (q) = m δ (−1)m+1  ei2π kmp/q  eimw + c.c. (16) ∂w 2iπ 2 m=1

k=0

where c.c. denotes the complex conjugate. The coefficients in the above expression are zero, unless m = sq, s integer, and Eq. (16) becomes ∞

1 X −1 sq ∂6 (q) s δ (−1)sq+1 sin(sqw). = 2 ∂w π

(17)

s=1

Eq. (17) shows that, on the dense set of the resonant circles of the unperturbed map, ∂6 (q) /∂w is different from zero and, according to Theorem 1, the perturbed map is non-integrable for ε 6= 0 in an open interval around zero. To apply Theorem 2 for the continuation of periodic points with respect to ε, we need to find the non-degenerate zeroes of Eq. (17), which is written in compact form as   1 δ q0 sin [q0 (π − w)] ∂6 (q0 ) = 2 arctan . (18) ∂w 1 − δ q0 cos [q0 (π − w)] π For each q0 -periodic invariant circle of the unperturbed map, there are 2q0 zeroes of Eq. (18) at   k π, k = −q0 + 1, . . . , 0, 1, . . . , q0 . w = 1− q0 For even k we calculate qδ q0 ∂ 2 6 (q0 ) = − < 0, ∂w2 π 2 (1 − δ q0 ) i.e. the assumptions of Theorem 2 hold and these periodic points are continued under sufficiently small perturbation. Their stability is evaluated by ρ 2 = −4π 2

∂ 2 6 (q0 ) > 0, ∂w 2

so, for ε > 0 the continued points are unstable, while for ε < 0 they are stable. For odd k we have q0 δ q0 ∂ 2 6 (q0 ) >0 = ∂w2 π 2 (1 + δ q0 ) and the stability character of the continued periodic points is the opposite. In Fig. 1 the islands around the continued stable fixed points at some main resonances are shown for ε = −0.1.

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Fig. 1. Some orbits around fixed points of resonances 0/1, 1/4, 1/3, 1/2, 2/3, 3/4 (from below) of the twist map perturbed by the Suris potential, for ε = −0.1.

5.3. The electron cyclotron resonance heating model We consider the following four-dimensional map, which models the two-frequency electron cyclotron resonance heating [48], where µ is a small parameter, µ (sin ψn + ε sin(λψn + χn )), N ε = Jn + µ sin(λψn + χn ), N

Pn+1 = Pn + Jn+1

ψn+1 = ψn −

M σ MJn+1 + , un+1 2u3n+1

χn+1 = χn −

M σ 2 MJn+1 + . un+1 2u3n+1

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P , J are the actions, ψ, χ the corresponding angles, u2 = P − σ J, σ, M, N, ε are constants and λ is an integer. See [48] for the physical interpretation of the various constants. The generating function of the map is S = Pn+1 ψn + Jn+1 χn − 2Mun+1 −

σ MJn+1 + µS1 un+1

where S1 =

1 ε cos ψn + cos(λψn + χn ). N N

The rotation vector is given by   M σ MJ 1 − + , α1 = 2π u 2u3 α2 = −

σ 2 MJ . 4πu3

The sum 6 (q) , evaluated on the resonant tori with α1 =

p1 = s (integer) q

α2 =

p2 (any rational number) q

is different from zero and is given by 6 (q) =

q cos ψ. N

(19)

It is also nonzero on the tori satisfying λp1 + p2 = qs 0 , where s 0 is an integer and hence 6 (q) =

εq cos(λψ + χ). N

For the tori where p1 = sq and λp1 +p2 = qs 0 hold simultaneously, 6 (q) is equal to the sum of the above two terms, while it is identically zero on all other resonant tori of the unperturbed map. Due to the specific form of the rotation vector, the equation α1 = s defines a set of lines in action space, which accumulate at the point (P , J ) = (0, 0). On the other hand, the equation α2 = p2 /q corresponds to dense sets of points on the previously mentioned lines, i.e. the set of points where, due to Eq. (19), it holds that ∂6 (q) 6≡ 0, w ∂w forms a key set around (0,0). Thus, according to Theorem 1, the perturbed map does not possess a complete set of analytic integrals in an open neighbourhood of this point, for µ in an open interval around zero. Continuation of periodic points can be proved only when both conditions p10 = q0 s and λp10 + p20 = q0 s 0 hold simultaneously. It can be easily shown that in this case the rotation vector is an integer vector, i.e. it corresponds to the period-one fixed points of the map. For these points we have 6 (q0 ) =

q0 εq0 cos ψ + cos(λψ + χ ) N N

and therefore the determinant in Eq. (14) is different from zero. These points coincide with the fixed points found in [48], where a detailed stability analysis is also included.

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6. Final remarks In this paper we defined the subharmonic Poincar´e–Mel’nikov–Arnol’d vector for the case of a symplectic nearintegrable twist map and used it for showing both non-integrability and continuation of fixed points by Theorems 1 and 2. Moreover, our work indicates the close connection between the non-existence of integrals and the generation of isolated periodic points in the perturbed map. Related work on non-integrability and the homoclinic Poincar´e– Mel’nikov–Arnol’d vector has been done in [27–35]. For canonical flows, the definition and treatment of these vectors go back to Poincar´e [43], Mel’nikov [26] and Arnol’d [53]. See e.g. Wiggins ([54], p. 117) for a theorem on the rˆole of the subharmonic Mel’nikov vector for the existence of periodic orbits and ([54], p. 483) for a treatment of the homoclinic Mel’nikov vector, both in the context of flows. In order to apply the homoclinic theory one needs the existence of a homoclinic loop in the integrable part. On the other hand, Theorem 1 proves non-integrability, irrespectively of whether such a loop exists or not. The existence of homoclinic loops for ε = 0 does not prevent the application of Theorem 1, since it can be applied in the open domains inside or outside the separatrices, where action-angle variables can be defined. For a specific system of two degrees of freedom Veerman and Holmes [42] proved a theorem which establishes the existence of an arbitrarily large but finite number of isolated fixed points in the neighbourhood of an elliptic fixed point for sufficiently small perturbation away from integrability. From our present work, using further estimates involved in the application of the implicit function theorem not published here, we conclude that under appropriate conditions such a result may be proved for a large class of symplectic maps. However, since the domain of validity in ε of this approach without further assumptions shrinks to zero as the period q of an orbit under consideration tends to infinity, the number of isolated fixed points that can be shown to exist with this method in some open set remains always bounded. From the formulation of Theorem 1, it is obvious that the existence of an infinite number of different resonant terms in the Fourier expansion of S1 with respect to the angles can potentially create a dense set of periodic points of w is not identically zero. The existence, however, of an infinite number of nonzero varying periods where ∂6 (q) /∂w Fourier coefficients in the expansion of a periodic function is a generic property and therefore, we expect that Theorem 1 is applicable in general. On the other hand, if the transformation ω : J 7→ α (JJ ) is such that it creates an infinite set of periodic points of the same or different periods in the neighbourhood of a certain point or of a surface w 6≡ 0, then one can prove non-integrability by considering only this of codimension one or less, where ∂6 (q) /∂w key set, as is done in Section 5.3. In this case S1 has only two non-zero Fourier terms, hence the existence of an infinite set of resonant terms in the perturbing function is not necessary. As far as the standard map is concerned, the form of S1 is very specific with only one nonzero Fourier coefficient and, as is seen in Section 5.1, it cannot be treated by our method. Comparing the results of this paper with the analogous ones of [38,39], for perturbed Hamiltonians of the form H = H0 + εH1 ,

(20)

there is a striking resemblence to the theory developed here visible if the expression hH1 i (the average value of H1 taken over one period) in the quoted papers is replaced by 6 (q) . Indeed, the theory of symplectic integrators (see e.g. [55–57]) shows that the first order symplectic integrator, implicitly defined by a generating function of the second type, establishes a close connection between those two expressions. Particularly, if we deal with a Hamiltonian system of the form ofEq. (20) the first order symplectic integrator takes the form  ∂H0 0 ∂H1   J ) + τ ε 0 (w w, J 0)  w0 = w + τ 0 (J J ∂J ∂JJ ˜ (21) Gε :  ∂H   J = J 0 + τ ε 1 (w w , J 0 ), w ∂w where τ is the integration time step assumed to be a small quantity. Taking now heuristically τ = 2π in Eq. (21) and comparing the result with Eq. (3) we recognize that in Eq. (21) 2π H1 takes the rˆole of S1 in Eq. (3). In this respect, see also [31].

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In literature, extensive use has been made of the generating function approach and related methods in order to describe the dynamics of mappings, see, e.g. [6–14] for the use of the first type generators mostly relying on stationarity of the related action functions with respect to certain orbital variations, as far as fixed points and their linear stability or invariant sets in general are concerned. In our present work we used the more convenient way of stationarity against variations of the initial conditions and preferred to take advantage of the natural occurence of type two generators in the context of perturbed twist maps as long as the perturbation is small enough. Generating functions of the second type and their critical points when expressed in non-mixed form play a decisive rˆole in generalisations of the Poincar´e–Birkhoff fixed point theorem to higher dimensions, see ([20], App. 9) for supplementary remarks. The main difference of our approach lies in the fact that in Arnol’d’s work the generating function has to be modified to yield a non-mixed representation by means of a function that is known to exist, but unknown in explicit terms which results in just an existence statement. In contrast, our result on fixed points is of constructive nature, well suited for practical calculations, however, it is a perturbative one, valid for small ε only.

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