- Email: [email protected]
The threshold for global diffusion in the kicked Harper map
17 June 2002
Physics Letters A 298 (2002) 330–334 www.elsevier.com/locate/pla
The threshold for global diffusion in the kicked Harper map Susumu Shinohara 1 Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan Received 8 November 2001; received in revised form 22 April 2002; accepted 2 May 2002 Communicated by A.R. Bishop
Abstract A systematic study on the threshold for global diffusion in the kicked Harper map is presented. Using the method we previously developed for the quadratic non-twist map, the phase diagram is numerically determined, which reveals the fractal structure of the boundary between the diffusive and non-diffusive phases. The connection between the phase diagram of the kicked Harper map and that of the quadratic non-twist map is discussed. 2002 Elsevier Science B.V. All rights reserved. PACS: 05.45.Ac; 05.45.Pq Keywords: Kicked Harper map; Non-twist maps; Hamiltonian systems; Breakup of invariant tori
In two-dimensional area-preserving maps of a cylinder, globally diffusive motion is possible in the absence of invariant tori. Detection of the threshold for global diffusion, or breakup of the last invariant torus is one of the important problems in the studies of Hamiltonian systems [1]. Several methods, such as Greene’s method [2] and the renormalization-group method [3], were proposed to accurately determine such thresholds. For a class of non-twist maps, i.e., maps that do not satisfy the twist condition, we proposed a numerical method by which the threshold can be efficiently determined [4].
E-mail address: [email protected] (S. Shinohara). 1 Present address: Department of Physical Sciences, Ritsumeikan University, Kusatsu 525-8577, Japan.
In this Letter, we determine the threshold for global diffusion in the so-called kicked Harper (KH) map, which has been a prototype model in studies of both classical and quantum chaos (e.g., [5]). Our previous method enables us to determine the phase diagram, which reveals the fractal structure of the boundary between the diffusive and non-diffusive phases. The KH map is defined by TL,K :
p = p + K sin x, mod 2π, x = x − L sin p , mod 2π,
(1)
where x, p ∈ [−π, π), and L and K are real parameters. Since TL,K is conjugate to the maps TL,−K , −1 (see Appendix A), it is sufficient to T−L,K and TK,L consider the fundamental parameter region defined by {(L, K) | 0 L ∞, 0 K L}. We restrict attention to this region in what follows. For these parameter
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 5 7 9 - 0
S. Shinohara / Physics Letters A 298 (2002) 330–334
values, invariant tori encircle the phase space in the x direction and prevent the diffusion in the p direction. An important property of the KH map is the violation of the twist condition: ∂x ∂p
has constant sign.
(2)
This condition is assumed in the proof of Moser’s twist theorem [6], and the violation of this condition generates peculiar phase space phenomena such as separatrix reconnection and meandering invariant tori [4,7–10]. In the integrable limit (K = 0), the KH map has two invariant tori along which the twist condition is violated, i.e., p ≡ π/2 and p ≡ −π/2. These invariant tori are called shearless tori. For finite perturbation strength (K = 0), the shearless torus is defined as the invariant torus which gives the extremum of the rotation number [9]. One of the notable properties of non-twist maps is the robustness of invariant tori around the shearless torus. For sufficiently large perturbation strength, these robust invariant tori form a torus layer separating strongly chaotic regions. As an example, Fig. 1 describes the phase space of the KH map for L = 4.0 and
331
K = 0.6, where one can see two well-localized torus layers located around the shearless tori at p ≈ ±π/2. In order to study the phase space of the KH map TL,K , it is convenient to note the following two types of symmetry: (i) TL,K is reversible [11], i.e., TL,K can be factorized as TL,K = IL IK with IK2 = IL2 = 1, where IK and IL are defined by p = p + K sin x, IK : (3) x = −x, p = p, IL : (4) x = −x − L sin p. (ii) TL,K commutes with the map N with N 2 = 1 defined by p = −p − π, N: (5) x = x − π, i.e., TL,K N = NTL,K . The phase space region with x > 0 and that with x < 0 are related through the map N . Moreover, either of the two shearless tori is an invariant set of N . As it was shown in Ref. [4], for a non-twist map with the symmetries (i) and (ii), one can derive the exact location of the shearless tori. Namely, the shearless tori go through the points defined by the solutions of the following equations: IK x = Nx,
(6)
IL x = Nx,
(7)
= (x, p) ∈ [−π, π)2 . The solutions of Eqs. (6)
where x and (7) are, respectively, given by π K π xK = + , Π + − , 2 2 2 π K π , − ,Π + + 2 2 2 π K π + ,Π − − , 2 2 2 π K π , − ,Π − + 2 2 2 Fig. 1. The phase space of the KH map for L = 4 and K = 0.6. The points x K and x L defined by Eqs. (8) and (9) are plotted by circles (•) and squares (), respectively.
and
L π π xL = Π + − ,+ , 2 2 2
(8)
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S. Shinohara / Physics Letters A 298 (2002) 330–334
L π π ,− , Π + + 2 2 2 L π π ,+ , Π − − 2 2 2 π π L ,− , Π − + 2 2 2
(9)
where Π is the projection from R to [−π, π) defined by Π : x + 2πk → x, x ∈ [−π, π), k ∈ Z. In addition to the symmetries described above, the KH map has one more symmetry: (iii) TL,K commutes with the map M with M 2 = 1 defined by p = p − π, M: (10) x = −x − π, i.e., TL,K M = MTL,K . The phase space region with p > 0 and that with p < 0 are related through the map M. In particular, M maps the upper shearless torus to the lower one, and vice versa. Because robust torus layers are formed around the shearless tori, the points x K and x L are always located at the inside of the torus layers as far as they persist. If the trajectories of x K and x L are bounded (non-diffusive) in the p direction, there should exist an invariant torus, while if these trajectories are unbounded (diffusive), there should be none. Namely, by investigating the behavior of these trajectories, one can determine the existence and non-existence of a torus in the phase space. Using this criterion, we numerically determine the existence and nonexistence of a torus. The phase diagram is presented in Fig. 2, where the points marked in gray and those in white, respectively, indicate the existence and non-existence of a torus in the corresponding phase space. Thus, the boundary between the gray and white regions corresponds to the threshold for global diffusion. In the numerical determinations, a trajectory is regarded to be bounded if its whole deviation in the p direction without taking the modulus does not exceed 2π up to 105 iterations. In Fig. 2 it can be observed that the boundary is formed of innumerable smooth curves. Each of these corresponds to the parameter values at which x K or x L becomes periodic or eventually periodic.
Fig. 2. The phase diagram of the KH map. Note that because of the symmetry of the diagram, we omit the region above the line K = L (see text). The curves Cn/2 are define by Eq. (11). For each region Dn , the maximal point of the boundary, (Ln , Kn ), is indicated by an arrow.
Such correspondence has been studied in detail for another non-twist map [4]. In the present Letter, this correspondence is illustrated for only a few curves. In particular, Cn/2 denotes the set of parameter values at which x K becomes periodic with rotation number −n/2 (n = 1, 3, 5, . . .), i.e., 2 Cn/2 = (L, K) | R −n TˆL,K xK = xK nπ = (L, K) | K = 2 arccos (11) , L where R(x, p) = (x − 2π, p), and TˆL,K is the KH map (Eq. (1)) without taking the modulus with respect to the variable x. Fig. 2 indicates the close agreement between the curves Cn/2 and boundary segments. Let us denote by Dn the region between the two curves Cn−1/2 and Cn+1/2 . Partitioning the L–K plane with Dn ’s, one finds in each Dn a similar boundary structure, whose height decays as n is increased. In order to characterize this decay, the maximal point of the boundary in each Dn can be used as a point of focus. (Ln , Kn ) denotes the point on the boundary which has the largest K-value in Dn . The relation between Ln and Kn is presented in Fig. 3, where the plots can be well fitted as Kn = aL−b n ,
a ≈ 2.37, b ≈ 0.504.
(12)
Note that the L range in Fig. 3 is much larger than that in Fig. 2.
S. Shinohara / Physics Letters A 298 (2002) 330–334
Fig. 3. A log–log plot of Kn vs. Ln .
The periodic structure of the phase diagram and Eq. (12) can be explained from the following simple argument. In the KH map, robust tori appear around the non-twist regions (p ≈ ±π/2), where the KH map can be approximated as p = p + K sin x, 2 x = x ∓ L 1 − π8 ± π2 p − 12 p 2 , p ≈ ±π/2, (13) provided the K-value is small. Putting π L P= (14) p∓ , X = ∓x, 2 2 L , ω = L, = ∓K (15) 2 and taking the modulus 2π with respect to the variable X, we obtain the standard form of the quadratic nontwist (QN) map P = P + sin X, (16) X = X + ω − P 2 , mod 2π, where X ∈ [−π, π), P ∈ R. Being different from the KH map, the QN map is periodic only in the X direction. The QN map is a representative example of nontwist maps and has been intensively studied in previous works [4,7,9,10]. In particular, we determined the phase diagram for the QN map [4]. Since the QN map is invariant under the transformations (P , ) → (−P , − ) and ω → ω + 2π , the phase diagram of the QN map is invariant under the two transformations → − and ω → ω + 2π . Fig. 4 reveals a fundamental region of the phase diagram for the QN map, where
333
Fig. 4. The phase diagram of the QN map. The maximal point of the boundary, (ω, ˜ ), ˜ is indicated by an arrow.
each point on the boundary between the gray and white regions represents the threshold for global diffusion. The boundary points for the QN map and those for the KH map are related through Eq. (15). Namely, by mapping the former to the L–K plane by Eq. (15), the latter should be approximately reproduced. This provides an explanation for the periodic pattern of the boundary presented in Fig. 2. Moreover, Eq. (12) can be derived as follows. Let us denote by (ω, ˜ ˜ ) the maximal point of the boundary depicted in Fig. 4. Because of the periodicity of the boundary, the maximal points are written as (ω˜ + 2πn, ˜ ) (n = 0, 1, 2, . . .) for the full range of ω. By Eq. (15), these points are mapped to the L–K plane as (L˜ n , K˜ n ) = (ω˜ +
2πn, ˜ 2/(ω˜ + 2πn)). Hence, we obtain K˜ n = a L˜ −b n ,
a=
√
2 ˜ ≈ 2.33, b = 0.5,
(17)
where we put ˜ ≈ 1.65, which is determined from Fig. 4. The values of a and b are in close agreement with those obtained in Eq. (12). In summary, the threshold for global diffusion in the KH map was systematically determined in the two-dimensional parameter plane. This was achieved by deriving the exact location of the shearless tori in the phase space. The phase diagram clearly reveals the fractal structure of the boundary between the diffusive and non-diffusive phases. Some properties of the boundary were explained by approximating the KH map by the non-twist map with quadratic nonlinearity.
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S. Shinohara / Physics Letters A 298 (2002) 330–334
Acknowledgements This work was supported by Waseda University Grant for Special Research Projects No.2000A-530. Appendix A We define the maps U , V and W as follows: U (x, p) = (x + π, p), W (x, p) = (−p, −x).
V (x, p) = (x, p + π), (A.1)
It is easy to verify −1 W −1 . TL,K = U TL,−K U −1 = V T−L,K V −1 = W TK,L
(A.2) Therefore, TL,K is conjugate to TL,−K , T−L,K and TK,L .
References [1] R.S. MacKay, J.D. Meiss (Eds.), Hamiltonian Dynamical Systems: A Reprint Selection, Adam Hilger, London, 1987.
[2] J.M. Greene, J. Math. Phys. 20 (1979) 1183. [3] C. Chandre, M. Govin, H.R. Jauslin, H. Koch, Phys. Rev. E 57 (1998) 6612. [4] S. Shinohara, Y. Aizawa, Prog. Theor. Phys. 97 (1997) 379; S. Shinohara, Y. Aizawa, Prog. Theor. Phys. 100 (1998) 219. [5] G.M. Zaslavsky, M.Yu. Zakharov, R.Z. Sagdeev, D.A. Usikov, A.A. Chernikov, Sov. Phys. JETP 64 (1986) 294; R. Artuso, G. Casati, F. Borgonovi, L. Rebuzzini, I. Guarneri, Int. J. Mod. Phys. B 8 (1994) 207; P. Lebœuf, Physica D 116 (1998) 8. [6] J.K. Moser, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II (1962) 1. [7] J.E. Howard, S.M. Hohs, Phys. Rev. A 29 (1984) 418. [8] J.E. Howard, J. Humpherys, Physica D 80 (1995) 256. [9] D. del Castillo-Negrete, J.M. Greene, P.J. Morrison, Physica D 91 (1996) 1; D. del Castillo-Negrete, J.M. Greene, P.J. Morrison, Physica D 100 (1997) 311. [10] G. Corso, A.J. Lichtenberg, Physica D 131 (1999) 1. [11] R. DeVogelaere, On the structure of symmetric periodic solutions of conservative systems, in: S. Lefschetz (Ed.), Contributions to the Theory of Nonlinear Oscillations, Princeton University Press, Princeton, NJ, 1958.
Physics Letters A 298 (2002) 330–334 www.elsevier.com/locate/pla
The threshold for global diffusion in the kicked Harper map Susumu Shinohara 1 Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan Received 8 November 2001; received in revised form 22 April 2002; accepted 2 May 2002 Communicated by A.R. Bishop
Abstract A systematic study on the threshold for global diffusion in the kicked Harper map is presented. Using the method we previously developed for the quadratic non-twist map, the phase diagram is numerically determined, which reveals the fractal structure of the boundary between the diffusive and non-diffusive phases. The connection between the phase diagram of the kicked Harper map and that of the quadratic non-twist map is discussed. 2002 Elsevier Science B.V. All rights reserved. PACS: 05.45.Ac; 05.45.Pq Keywords: Kicked Harper map; Non-twist maps; Hamiltonian systems; Breakup of invariant tori
In two-dimensional area-preserving maps of a cylinder, globally diffusive motion is possible in the absence of invariant tori. Detection of the threshold for global diffusion, or breakup of the last invariant torus is one of the important problems in the studies of Hamiltonian systems [1]. Several methods, such as Greene’s method [2] and the renormalization-group method [3], were proposed to accurately determine such thresholds. For a class of non-twist maps, i.e., maps that do not satisfy the twist condition, we proposed a numerical method by which the threshold can be efficiently determined [4].
E-mail address: [email protected] (S. Shinohara). 1 Present address: Department of Physical Sciences, Ritsumeikan University, Kusatsu 525-8577, Japan.
In this Letter, we determine the threshold for global diffusion in the so-called kicked Harper (KH) map, which has been a prototype model in studies of both classical and quantum chaos (e.g., [5]). Our previous method enables us to determine the phase diagram, which reveals the fractal structure of the boundary between the diffusive and non-diffusive phases. The KH map is defined by TL,K :
p = p + K sin x, mod 2π, x = x − L sin p , mod 2π,
(1)
where x, p ∈ [−π, π), and L and K are real parameters. Since TL,K is conjugate to the maps TL,−K , −1 (see Appendix A), it is sufficient to T−L,K and TK,L consider the fundamental parameter region defined by {(L, K) | 0 L ∞, 0 K L}. We restrict attention to this region in what follows. For these parameter
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 5 7 9 - 0
S. Shinohara / Physics Letters A 298 (2002) 330–334
values, invariant tori encircle the phase space in the x direction and prevent the diffusion in the p direction. An important property of the KH map is the violation of the twist condition: ∂x ∂p
has constant sign.
(2)
This condition is assumed in the proof of Moser’s twist theorem [6], and the violation of this condition generates peculiar phase space phenomena such as separatrix reconnection and meandering invariant tori [4,7–10]. In the integrable limit (K = 0), the KH map has two invariant tori along which the twist condition is violated, i.e., p ≡ π/2 and p ≡ −π/2. These invariant tori are called shearless tori. For finite perturbation strength (K = 0), the shearless torus is defined as the invariant torus which gives the extremum of the rotation number [9]. One of the notable properties of non-twist maps is the robustness of invariant tori around the shearless torus. For sufficiently large perturbation strength, these robust invariant tori form a torus layer separating strongly chaotic regions. As an example, Fig. 1 describes the phase space of the KH map for L = 4.0 and
331
K = 0.6, where one can see two well-localized torus layers located around the shearless tori at p ≈ ±π/2. In order to study the phase space of the KH map TL,K , it is convenient to note the following two types of symmetry: (i) TL,K is reversible [11], i.e., TL,K can be factorized as TL,K = IL IK with IK2 = IL2 = 1, where IK and IL are defined by p = p + K sin x, IK : (3) x = −x, p = p, IL : (4) x = −x − L sin p. (ii) TL,K commutes with the map N with N 2 = 1 defined by p = −p − π, N: (5) x = x − π, i.e., TL,K N = NTL,K . The phase space region with x > 0 and that with x < 0 are related through the map N . Moreover, either of the two shearless tori is an invariant set of N . As it was shown in Ref. [4], for a non-twist map with the symmetries (i) and (ii), one can derive the exact location of the shearless tori. Namely, the shearless tori go through the points defined by the solutions of the following equations: IK x = Nx,
(6)
IL x = Nx,
(7)
= (x, p) ∈ [−π, π)2 . The solutions of Eqs. (6)
where x and (7) are, respectively, given by π K π xK = + , Π + − , 2 2 2 π K π , − ,Π + + 2 2 2 π K π + ,Π − − , 2 2 2 π K π , − ,Π − + 2 2 2 Fig. 1. The phase space of the KH map for L = 4 and K = 0.6. The points x K and x L defined by Eqs. (8) and (9) are plotted by circles (•) and squares (), respectively.
and
L π π xL = Π + − ,+ , 2 2 2
(8)
332
S. Shinohara / Physics Letters A 298 (2002) 330–334
L π π ,− , Π + + 2 2 2 L π π ,+ , Π − − 2 2 2 π π L ,− , Π − + 2 2 2
(9)
where Π is the projection from R to [−π, π) defined by Π : x + 2πk → x, x ∈ [−π, π), k ∈ Z. In addition to the symmetries described above, the KH map has one more symmetry: (iii) TL,K commutes with the map M with M 2 = 1 defined by p = p − π, M: (10) x = −x − π, i.e., TL,K M = MTL,K . The phase space region with p > 0 and that with p < 0 are related through the map M. In particular, M maps the upper shearless torus to the lower one, and vice versa. Because robust torus layers are formed around the shearless tori, the points x K and x L are always located at the inside of the torus layers as far as they persist. If the trajectories of x K and x L are bounded (non-diffusive) in the p direction, there should exist an invariant torus, while if these trajectories are unbounded (diffusive), there should be none. Namely, by investigating the behavior of these trajectories, one can determine the existence and non-existence of a torus in the phase space. Using this criterion, we numerically determine the existence and nonexistence of a torus. The phase diagram is presented in Fig. 2, where the points marked in gray and those in white, respectively, indicate the existence and non-existence of a torus in the corresponding phase space. Thus, the boundary between the gray and white regions corresponds to the threshold for global diffusion. In the numerical determinations, a trajectory is regarded to be bounded if its whole deviation in the p direction without taking the modulus does not exceed 2π up to 105 iterations. In Fig. 2 it can be observed that the boundary is formed of innumerable smooth curves. Each of these corresponds to the parameter values at which x K or x L becomes periodic or eventually periodic.
Fig. 2. The phase diagram of the KH map. Note that because of the symmetry of the diagram, we omit the region above the line K = L (see text). The curves Cn/2 are define by Eq. (11). For each region Dn , the maximal point of the boundary, (Ln , Kn ), is indicated by an arrow.
Such correspondence has been studied in detail for another non-twist map [4]. In the present Letter, this correspondence is illustrated for only a few curves. In particular, Cn/2 denotes the set of parameter values at which x K becomes periodic with rotation number −n/2 (n = 1, 3, 5, . . .), i.e., 2 Cn/2 = (L, K) | R −n TˆL,K xK = xK nπ = (L, K) | K = 2 arccos (11) , L where R(x, p) = (x − 2π, p), and TˆL,K is the KH map (Eq. (1)) without taking the modulus with respect to the variable x. Fig. 2 indicates the close agreement between the curves Cn/2 and boundary segments. Let us denote by Dn the region between the two curves Cn−1/2 and Cn+1/2 . Partitioning the L–K plane with Dn ’s, one finds in each Dn a similar boundary structure, whose height decays as n is increased. In order to characterize this decay, the maximal point of the boundary in each Dn can be used as a point of focus. (Ln , Kn ) denotes the point on the boundary which has the largest K-value in Dn . The relation between Ln and Kn is presented in Fig. 3, where the plots can be well fitted as Kn = aL−b n ,
a ≈ 2.37, b ≈ 0.504.
(12)
Note that the L range in Fig. 3 is much larger than that in Fig. 2.
S. Shinohara / Physics Letters A 298 (2002) 330–334
Fig. 3. A log–log plot of Kn vs. Ln .
The periodic structure of the phase diagram and Eq. (12) can be explained from the following simple argument. In the KH map, robust tori appear around the non-twist regions (p ≈ ±π/2), where the KH map can be approximated as p = p + K sin x, 2 x = x ∓ L 1 − π8 ± π2 p − 12 p 2 , p ≈ ±π/2, (13) provided the K-value is small. Putting π L P= (14) p∓ , X = ∓x, 2 2 L , ω = L, = ∓K (15) 2 and taking the modulus 2π with respect to the variable X, we obtain the standard form of the quadratic nontwist (QN) map P = P + sin X, (16) X = X + ω − P 2 , mod 2π, where X ∈ [−π, π), P ∈ R. Being different from the KH map, the QN map is periodic only in the X direction. The QN map is a representative example of nontwist maps and has been intensively studied in previous works [4,7,9,10]. In particular, we determined the phase diagram for the QN map [4]. Since the QN map is invariant under the transformations (P , ) → (−P , − ) and ω → ω + 2π , the phase diagram of the QN map is invariant under the two transformations → − and ω → ω + 2π . Fig. 4 reveals a fundamental region of the phase diagram for the QN map, where
333
Fig. 4. The phase diagram of the QN map. The maximal point of the boundary, (ω, ˜ ), ˜ is indicated by an arrow.
each point on the boundary between the gray and white regions represents the threshold for global diffusion. The boundary points for the QN map and those for the KH map are related through Eq. (15). Namely, by mapping the former to the L–K plane by Eq. (15), the latter should be approximately reproduced. This provides an explanation for the periodic pattern of the boundary presented in Fig. 2. Moreover, Eq. (12) can be derived as follows. Let us denote by (ω, ˜ ˜ ) the maximal point of the boundary depicted in Fig. 4. Because of the periodicity of the boundary, the maximal points are written as (ω˜ + 2πn, ˜ ) (n = 0, 1, 2, . . .) for the full range of ω. By Eq. (15), these points are mapped to the L–K plane as (L˜ n , K˜ n ) = (ω˜ +
2πn, ˜ 2/(ω˜ + 2πn)). Hence, we obtain K˜ n = a L˜ −b n ,
a=
√
2 ˜ ≈ 2.33, b = 0.5,
(17)
where we put ˜ ≈ 1.65, which is determined from Fig. 4. The values of a and b are in close agreement with those obtained in Eq. (12). In summary, the threshold for global diffusion in the KH map was systematically determined in the two-dimensional parameter plane. This was achieved by deriving the exact location of the shearless tori in the phase space. The phase diagram clearly reveals the fractal structure of the boundary between the diffusive and non-diffusive phases. Some properties of the boundary were explained by approximating the KH map by the non-twist map with quadratic nonlinearity.
334
S. Shinohara / Physics Letters A 298 (2002) 330–334
Acknowledgements This work was supported by Waseda University Grant for Special Research Projects No.2000A-530. Appendix A We define the maps U , V and W as follows: U (x, p) = (x + π, p), W (x, p) = (−p, −x).
V (x, p) = (x, p + π), (A.1)
It is easy to verify −1 W −1 . TL,K = U TL,−K U −1 = V T−L,K V −1 = W TK,L
(A.2) Therefore, TL,K is conjugate to TL,−K , T−L,K and TK,L .
References [1] R.S. MacKay, J.D. Meiss (Eds.), Hamiltonian Dynamical Systems: A Reprint Selection, Adam Hilger, London, 1987.
[2] J.M. Greene, J. Math. Phys. 20 (1979) 1183. [3] C. Chandre, M. Govin, H.R. Jauslin, H. Koch, Phys. Rev. E 57 (1998) 6612. [4] S. Shinohara, Y. Aizawa, Prog. Theor. Phys. 97 (1997) 379; S. Shinohara, Y. Aizawa, Prog. Theor. Phys. 100 (1998) 219. [5] G.M. Zaslavsky, M.Yu. Zakharov, R.Z. Sagdeev, D.A. Usikov, A.A. Chernikov, Sov. Phys. JETP 64 (1986) 294; R. Artuso, G. Casati, F. Borgonovi, L. Rebuzzini, I. Guarneri, Int. J. Mod. Phys. B 8 (1994) 207; P. Lebœuf, Physica D 116 (1998) 8. [6] J.K. Moser, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II (1962) 1. [7] J.E. Howard, S.M. Hohs, Phys. Rev. A 29 (1984) 418. [8] J.E. Howard, J. Humpherys, Physica D 80 (1995) 256. [9] D. del Castillo-Negrete, J.M. Greene, P.J. Morrison, Physica D 91 (1996) 1; D. del Castillo-Negrete, J.M. Greene, P.J. Morrison, Physica D 100 (1997) 311. [10] G. Corso, A.J. Lichtenberg, Physica D 131 (1999) 1. [11] R. DeVogelaere, On the structure of symmetric periodic solutions of conservative systems, in: S. Lefschetz (Ed.), Contributions to the Theory of Nonlinear Oscillations, Princeton University Press, Princeton, NJ, 1958.