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On the scenario of reconnection in non-twist cubic maps
Chaos, Solitons and Fractals 30 (2006) 1260–1264 www.elsevier.com/locate/chaos
On the scenario of reconnection in non-twist cubic maps Gheorghe Tigan Department of Mathematics, ‘‘Politehnica’’ University of Timisoara, Pta Victoriei, No. 2, 300006 Timisoara, Timis, Romania Accepted 2 September 2005
Communicated by Prof. M.S. El Naschie
Abstract In this paper, we study the reconnection process in the dynamics of cubic non-twist maps, introduced in [Howard JE, Humpherys J. Nonmonotonic twist maps. Physica D 1995; 256–76]. In order to describe the route to reconnection of the involved Poincare´–Birkhoff chains we investigate an approximate interpolating Hamiltonian of the map under study. Our study reveals that the scenario of reconnection of cubic non-twist maps is different from that occurring in the dynamics of quadratic non-twist maps. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Area preserving non-twist maps are models for Poincare´ maps associated to sections in an energy manifold of an isoenergetically degenerate two degree of freedom Hamiltonian system. Such Hamiltonian systems describe a large variety of physical systems (see [1,4,8]). During the last decade numerical and theoretical studies of quadratic non-twist maps [3,1,5,6] revealed a global bifurcation of the invariant manifolds of two distinct regular hyperbolic periodic orbits having the same rotation number. This bifurcation is called reconnection after the name of a similar phenomenon in which can be involved the magnetic field lines in tokamaks or in magnetosphere. Such a bifurcation was also identified in a three parameter family of cubic non-twist maps [3]. In [4] is studied an integrable two degree of freedom Hamiltonian with inflection degeneracy. For a rigorous investigation of local and global bifurcations occurring in a family of area preserving maps defined on an annulus T ½a; b (T denotes the circle identified with [0, 2p)) one derives an approximate interpolating Hamiltonian of the map under study (see for example [7] where such a method has been used to study bifurcations in quadratic nontwist maps). In this paper, we study reconnection in the cubic non-twist area preserving diffeomorphism of the annulus T R; f : ðx; yÞ 7! ðx0 ; y 0 Þ: x0 ¼ x þ Xða; b; y 0 Þðmod 2pÞ; y 0 ¼ y þ k sin x;
E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.09.026
ð1Þ
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where the rotation number function X is a cubic map depending on two parameters a, b > 0, i.e. X(a, b; y) = y ay2 + by3, and k > 0 is a perturbation parameter. We recall that an area preserving diffeomorphism g : T R ! T R; g : ðx; yÞ 7! ðx0 ; y 0 Þ is a twist map if oyx 0 5 0 (oy denotes the partial derivative with respect to y). Twist property is a basic assumption of KAM theorem, as well as for the Aubry–Mather theory [2]. The map (Eq. (1)) is a non-twist map because it violates the twist condition. While for quadratic non-twist maps a KAM-type theorem was proved [7], for cubic non-twist maps, as far as we know, no theoretical result was derived. Our aim is to study its dynamics as well as the route to reconnection.
2. Properties of cubic non-twist map The motion in the unperturbed map (Eq. (1)), i.e. the map corresponding to k = 0: x0 ¼ x þ y ay 2 þ by 3 ðmod 2pÞ; y 0 ¼ y;
ð2Þ
occurs along the circle y = cst. The rotation number of an orbit starting at (x, y) is q ¼ lim
n!1
Xn X ¼ Xða; b; yÞ=ð2pÞ; 2np
ð3Þ
where (Xn, Yn) is the orbit of the point (X, Y) = (x, y) under the lift of the map (the map defined on R2 having the same expression, without modulo 2p for the first component). The map (Eq. (2)) violates the twist condition for the parameter values (a, b) such that a2 3b P 0, along the circles: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ a2 3b a a2 3b c1 : y ¼ ; c2 : y ¼ . ð4Þ 3b 3b At the same time along the circle c1 the rotation number has a global minimum, while along c2, a global maximum. These circles are called twistless or shearless circles. Let us denote by xm, xM: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ a2 3b a a2 3b ; xM :¼ ; ð5Þ xm :¼ 3b 3b the points of minimum, respectively maximum, for the rotation number function X. For y 2 (1, xM) the unperturbed map has positive, twist (the rotation number function is increasing), for y 2 (xM, xm) has a negative twist (the rotation number function is decreasing), while for y 2 (xm, +1) has again a positive twist (Fig. 1). The orbits lying on the circles y = y0 with X(a, b; y0)/2p a rational number in lowest terms, p/q, are periodic orbits. If such a periodic orbit lies in a region of monotone twist property of the map, after a slight perturbation, it gives rise generically to at least two periodic orbits of the same rotation number, one elliptic and the second regular hyperbolic. Elliptic, points are surrounded by invariant circles, and hyperbolic points are connected by heteroclinic connections. Such a pair of periodic orbits and the associated invariant sets form a Poincare´–Birkhoff chain. For y 2 (xM, xm) can exist three circles y = cst of the unperturbed map, on which lie periodic orbits of the same rotation number p/q. Our aim is to study the bifurcations of the periodic orbits or the invariant manifolds belonging to three distinct
Fig. 1. The graph of the rotation number function X(a, b; y) for a = 2.5 and b = 1.26.
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Poincare´–Birkhoff chains created after a slight perturbation, as one of the shape parameters a, b > 0, a2 3b > 0 defining the rotation number function X varies.
3. Reconnection scenario A useful tool in analyzing the changes in the topology of invariant manifolds of the involved p/q-type hyperbolic periodic orbits is analysis of interpolating Hamiltonian associated to the map F. The approximate interpolating Hamiltonian associated to the map F is H a;b;k ðx; yÞ ¼ y 2 =2 þ ay 3 =3 by 4 =4 k cos x. It defines the vector field oH oH XH ¼ ; ¼ ðy ay 2 þ by 3 ; k sin xÞ; oy ox
ð6Þ
ð7Þ
which is reversible with respect to the involution R(x, y) = (x, y), i.e. R XH = XH R. The fixed point set, Fix(R), consists in the lines x = 0 and x = p, called symmetry lines. The equilibrium points of XH lying on the symmetry lines are called symmetric. In order to describe the scenario of reconnection and the local bifurcations of the equilibrium points, we analyze the position on the symmetry lines of the equilibrium points, their stability type and bifurcations occurring as a varies and b, k are fixed in the parametric space (a, b, k), with a, b, k > 0. The equilibrium points of XH are solutions of equation XH = 0. If a2 4b > 0, the Hamiltonian system has six equilibrium points: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! a þ a2 4b a þ a2 4b a a2 4b a a2 4b P 1 ð0; 0Þ; P 2 ðp; 0Þ; P 3 0; ; P 4 p; ; P 5 0; ; P 6 p; ; 2b 2b 2b 2b If a2 4b < 0 and a2 3b > 0 (Fig. 2) the vector field XH has only two equilibrium points: P1(0, 0), P2(p, 0), while for a = 4b it has four equilibrium points: P 1 ð0; 0Þ; P 2 ðp; 0Þ; A 0; 2a ; B p; 2a . Using a computer algebra system to calculate the eigenvalues of the Jacobian matrix D(x,py)X evaluated at each ffiffiffi H p ffiffiffi equilibrium point (x, y) we get that at the points P1(0, 0) and P2(p, 0) the eigenvalues are k ; i k , i.e. P1(0, 0) is hyperbolic (saddle equilibrium) and P2(p, 0) is elliptic (center). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2 2 2 2 At P3 and P4 the eigenvalues are ða 4bþa2b a 4bÞk, respectively i ða 4bþa2b a 4bÞk. Hence, the point P3 is hyperbolic and the point P4 is elliptic. For the last two points P5 and P6, the eigenvalues are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða a2 4b ða2 4bÞÞk ða a2 4b ða2 4bÞÞk and ; i 2b 2b 2
that is, P5 is elliptic and P6 is hyperbolic. In the case a2 4b = 0 the two eigenvalues are zero and a bifurcation of equilibrium points occurs.
Fig. 2. The existence domains of the equilibrium points.
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Fig. 3. The reconnection scenario of the upper two Poincare´–Birkhoff chains. The a parameter is (a) a = 2.163, (b) a = 2.245, (c) a = 2.2832, (d) a = 2.343 and (e) a = 2.4068.
In the following we want to describe the local changes in the topology of the invariant sets of the Hamiltonian system, when b, k > 0 are fixed and a varies. The systems whose phase portraits pffiffiffiffiffi pffiffiffiffiffiare illustrated in different figures correspond to b = 1.26 and k = 0.0147. Fig. 3(a) displays the case a 2 ð 3b; 4bÞ, (we work on the non-twist domain a2 3b > 0) when we have only two equilibrium points P1h(0, 0) and P2e(p, 0) (h stands for hyperbolic, e for elliptic), while p (b)ffiffiffiffiffidisplays the case a2 4b = 0 when other two equilibrium points A 0; 2a ; B p; 2a are born. Increasing a slightly from 4b, that is a2 4b > 0, the other four equilibrium points P3, P4, P5 and P6 are born through a saddle-center bifurcation Fig. 3(c). The new born equilibrium points form two dimerized island chains, that is, the elliptic points are surrounded by homoclinic circles to the corresponding hyperbolic points. Between these chains the trajectories of the Hamiltonian vector field are not graphs of real functions of x, but they are meanders, Fig. 3(c). Observe that on both symmetry lines there exist now three equilibrium points. Each two neighboring points on the same symmetry line have opposite stability type. Increasing further the parameter a, the equilibrium points lying on the same symmetry line (such points lie within two different chains) go away and at a threshold, called reconnection threshold, the hyperbolic points of the two chains get connected by common branches of invariant manifolds (Fig. 3(d)). Such a threshold is called reconnection threshold. In order to get this threshold we impose that the hyperbolic equilibrium points P3h and P6h to belong to the same energy level set, that is Ha,b,k(P3h) = Ha,b,k(P6h). This leads to a :¼ a1rec = 2.343. Increasing a beyond the reconnection threshold the dimerized island chains transform into Poincare´–Birkhoff chains, and the lower two PB chains approach each other (Fig. 3(e)). The reconnection of the lower two Poincare´–Birkhoff chains occurs at a :¼ a2rec = 2.79. In this case, the stable/unstable manifolds of these points have a common branch C3 = Wu(P1h) \ Ws(P6h), C4 = Ws(P1h) \ Ws(P6h). After reconnection, i.e. for a lightly greater than a2rec, the above described scenario occurs in a reverse order (Fig. 4). We note that fixing the shape parameter b the simultaneous reconnection of the three chains can occur. Imposing Ha,b,k(P1h) = Ha,b,k(P3h) = Ha,b,k(P6h), we get through symbolic computation the following reconnection curve: 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < k ¼ 1 a ða2 4bÞ a2 4b 3 24 b ð8Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 6b þ a4 6a2 b þ 48b3 k þ 4ab a2 4b a3 a2 4b ¼ 0.
Fig. 4. The phase portraits of the Hamiltonian system at the second reconnection threshold (a) and beyond it (b)–(d).
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Fig. 5. Scenario of reconnection of the all three Poincare´–Birkhoff chains. The a parameter is (a) a = 2.3, (b) a = 2.3812 and (c) a = 2.6.
Solving numerically these equations for b = 1.26 one get k = 0.0248 and a = 2.3812. This implies that in order to describe the reconnection scenario of the all three chainspfor ffiffiffiffiffi bp= ffiffiffiffiffi1.26 we2should keep the value of k at krec = 0.0248. Letting now the a parameter to vary, we getpthat if a 2 ð 3b ; 4bÞ and a 4b = 0 the scenario is the same as above, ffiffiffiffiffi Figs. 3(a)–(c) and 5(a). Increasing a beyond 4b the all three chains get connected when any two saddle distinct points P1h, P3h, P6h are connected by a common arc of their invariant manifolds, Fig. 5(b). The threshold of reconnection is a3rec = 2.3812. Increasing a further we get again the scenario in reverse order, Figs. 4(b)–(d) and 5(c).
4. Conclusion A three-parameter cubic non-twist map was analyzed in this paper. By using an approximate interpolating Hamiltonian of the map we have described the reconnection process of any two neighboring chains in the case when two parameters are fixed and the other one varies. By numerical computations we got the two thresholds of reconnection. Also we remarked that the three chains can be simultaneously reconnected. In this case only a parameter of the system can be arbitrary fixed. The reconnection thresholds of the other two parameters are found.
Acknowledgments This work was supported through a European Community Marie Curie Fellowship and in the framework of the CTS, contract number HPMT-CT-2001-00278. The author is grateful to Professor Emilia Petrisor for useful discussions on reconnection process.
References [1] del-Castillo-Negrete D, Greene JM, Morrison PJ. Area preserving nontwist maps: periodic orbits and transition to chaos. Physica D 1996;91:1–23. [2] Katok A, Hasselblatt B. Introduction to the modern theory of dynamical systems. Cambridge University Press; 1995. [3] Howard JE, Humpherys J. Nonmonotonic twist maps. Physica D 1995;80:256–76. [4] Voyatzis G, Ichtiaroglou S. Degenerate bifurcations of resonant tori in Hamiltonian systems. Int J Bif Chaos 1999;9:849–63. [5] Petrisor E. Nontwist area preserving maps with reversing symmetry group. Int J Bif Chaos 2001;11:497–511. [6] Petrisor E. Reconnection scenarios and the threshold of reconnection in the dynamics of nontwist maps. Chaos, Solitons & Fractals 2002;14:117–27. [7] Simo´ C. Invariant curves of analytic perturbed nontwist area preserving maps. Regul Chaotic Dyn 1998;3:180–95. [8] Apte A, Wurm A, Morrison PJ. Renormalization and destruction of 1/c2 tori in the standard nontwist map. Chaos 2003;13.
On the scenario of reconnection in non-twist cubic maps Gheorghe Tigan Department of Mathematics, ‘‘Politehnica’’ University of Timisoara, Pta Victoriei, No. 2, 300006 Timisoara, Timis, Romania Accepted 2 September 2005
Communicated by Prof. M.S. El Naschie
Abstract In this paper, we study the reconnection process in the dynamics of cubic non-twist maps, introduced in [Howard JE, Humpherys J. Nonmonotonic twist maps. Physica D 1995; 256–76]. In order to describe the route to reconnection of the involved Poincare´–Birkhoff chains we investigate an approximate interpolating Hamiltonian of the map under study. Our study reveals that the scenario of reconnection of cubic non-twist maps is different from that occurring in the dynamics of quadratic non-twist maps. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction Area preserving non-twist maps are models for Poincare´ maps associated to sections in an energy manifold of an isoenergetically degenerate two degree of freedom Hamiltonian system. Such Hamiltonian systems describe a large variety of physical systems (see [1,4,8]). During the last decade numerical and theoretical studies of quadratic non-twist maps [3,1,5,6] revealed a global bifurcation of the invariant manifolds of two distinct regular hyperbolic periodic orbits having the same rotation number. This bifurcation is called reconnection after the name of a similar phenomenon in which can be involved the magnetic field lines in tokamaks or in magnetosphere. Such a bifurcation was also identified in a three parameter family of cubic non-twist maps [3]. In [4] is studied an integrable two degree of freedom Hamiltonian with inflection degeneracy. For a rigorous investigation of local and global bifurcations occurring in a family of area preserving maps defined on an annulus T ½a; b (T denotes the circle identified with [0, 2p)) one derives an approximate interpolating Hamiltonian of the map under study (see for example [7] where such a method has been used to study bifurcations in quadratic nontwist maps). In this paper, we study reconnection in the cubic non-twist area preserving diffeomorphism of the annulus T R; f : ðx; yÞ 7! ðx0 ; y 0 Þ: x0 ¼ x þ Xða; b; y 0 Þðmod 2pÞ; y 0 ¼ y þ k sin x;
E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.09.026
ð1Þ
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where the rotation number function X is a cubic map depending on two parameters a, b > 0, i.e. X(a, b; y) = y ay2 + by3, and k > 0 is a perturbation parameter. We recall that an area preserving diffeomorphism g : T R ! T R; g : ðx; yÞ 7! ðx0 ; y 0 Þ is a twist map if oyx 0 5 0 (oy denotes the partial derivative with respect to y). Twist property is a basic assumption of KAM theorem, as well as for the Aubry–Mather theory [2]. The map (Eq. (1)) is a non-twist map because it violates the twist condition. While for quadratic non-twist maps a KAM-type theorem was proved [7], for cubic non-twist maps, as far as we know, no theoretical result was derived. Our aim is to study its dynamics as well as the route to reconnection.
2. Properties of cubic non-twist map The motion in the unperturbed map (Eq. (1)), i.e. the map corresponding to k = 0: x0 ¼ x þ y ay 2 þ by 3 ðmod 2pÞ; y 0 ¼ y;
ð2Þ
occurs along the circle y = cst. The rotation number of an orbit starting at (x, y) is q ¼ lim
n!1
Xn X ¼ Xða; b; yÞ=ð2pÞ; 2np
ð3Þ
where (Xn, Yn) is the orbit of the point (X, Y) = (x, y) under the lift of the map (the map defined on R2 having the same expression, without modulo 2p for the first component). The map (Eq. (2)) violates the twist condition for the parameter values (a, b) such that a2 3b P 0, along the circles: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ a2 3b a a2 3b c1 : y ¼ ; c2 : y ¼ . ð4Þ 3b 3b At the same time along the circle c1 the rotation number has a global minimum, while along c2, a global maximum. These circles are called twistless or shearless circles. Let us denote by xm, xM: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ a2 3b a a2 3b ; xM :¼ ; ð5Þ xm :¼ 3b 3b the points of minimum, respectively maximum, for the rotation number function X. For y 2 (1, xM) the unperturbed map has positive, twist (the rotation number function is increasing), for y 2 (xM, xm) has a negative twist (the rotation number function is decreasing), while for y 2 (xm, +1) has again a positive twist (Fig. 1). The orbits lying on the circles y = y0 with X(a, b; y0)/2p a rational number in lowest terms, p/q, are periodic orbits. If such a periodic orbit lies in a region of monotone twist property of the map, after a slight perturbation, it gives rise generically to at least two periodic orbits of the same rotation number, one elliptic and the second regular hyperbolic. Elliptic, points are surrounded by invariant circles, and hyperbolic points are connected by heteroclinic connections. Such a pair of periodic orbits and the associated invariant sets form a Poincare´–Birkhoff chain. For y 2 (xM, xm) can exist three circles y = cst of the unperturbed map, on which lie periodic orbits of the same rotation number p/q. Our aim is to study the bifurcations of the periodic orbits or the invariant manifolds belonging to three distinct
Fig. 1. The graph of the rotation number function X(a, b; y) for a = 2.5 and b = 1.26.
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Poincare´–Birkhoff chains created after a slight perturbation, as one of the shape parameters a, b > 0, a2 3b > 0 defining the rotation number function X varies.
3. Reconnection scenario A useful tool in analyzing the changes in the topology of invariant manifolds of the involved p/q-type hyperbolic periodic orbits is analysis of interpolating Hamiltonian associated to the map F. The approximate interpolating Hamiltonian associated to the map F is H a;b;k ðx; yÞ ¼ y 2 =2 þ ay 3 =3 by 4 =4 k cos x. It defines the vector field oH oH XH ¼ ; ¼ ðy ay 2 þ by 3 ; k sin xÞ; oy ox
ð6Þ
ð7Þ
which is reversible with respect to the involution R(x, y) = (x, y), i.e. R XH = XH R. The fixed point set, Fix(R), consists in the lines x = 0 and x = p, called symmetry lines. The equilibrium points of XH lying on the symmetry lines are called symmetric. In order to describe the scenario of reconnection and the local bifurcations of the equilibrium points, we analyze the position on the symmetry lines of the equilibrium points, their stability type and bifurcations occurring as a varies and b, k are fixed in the parametric space (a, b, k), with a, b, k > 0. The equilibrium points of XH are solutions of equation XH = 0. If a2 4b > 0, the Hamiltonian system has six equilibrium points: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! a þ a2 4b a þ a2 4b a a2 4b a a2 4b P 1 ð0; 0Þ; P 2 ðp; 0Þ; P 3 0; ; P 4 p; ; P 5 0; ; P 6 p; ; 2b 2b 2b 2b If a2 4b < 0 and a2 3b > 0 (Fig. 2) the vector field XH has only two equilibrium points: P1(0, 0), P2(p, 0), while for a = 4b it has four equilibrium points: P 1 ð0; 0Þ; P 2 ðp; 0Þ; A 0; 2a ; B p; 2a . Using a computer algebra system to calculate the eigenvalues of the Jacobian matrix D(x,py)X evaluated at each ffiffiffi H p ffiffiffi equilibrium point (x, y) we get that at the points P1(0, 0) and P2(p, 0) the eigenvalues are k ; i k , i.e. P1(0, 0) is hyperbolic (saddle equilibrium) and P2(p, 0) is elliptic (center). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2 2 2 2 At P3 and P4 the eigenvalues are ða 4bþa2b a 4bÞk, respectively i ða 4bþa2b a 4bÞk. Hence, the point P3 is hyperbolic and the point P4 is elliptic. For the last two points P5 and P6, the eigenvalues are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða a2 4b ða2 4bÞÞk ða a2 4b ða2 4bÞÞk and ; i 2b 2b 2
that is, P5 is elliptic and P6 is hyperbolic. In the case a2 4b = 0 the two eigenvalues are zero and a bifurcation of equilibrium points occurs.
Fig. 2. The existence domains of the equilibrium points.
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Fig. 3. The reconnection scenario of the upper two Poincare´–Birkhoff chains. The a parameter is (a) a = 2.163, (b) a = 2.245, (c) a = 2.2832, (d) a = 2.343 and (e) a = 2.4068.
In the following we want to describe the local changes in the topology of the invariant sets of the Hamiltonian system, when b, k > 0 are fixed and a varies. The systems whose phase portraits pffiffiffiffiffi pffiffiffiffiffiare illustrated in different figures correspond to b = 1.26 and k = 0.0147. Fig. 3(a) displays the case a 2 ð 3b; 4bÞ, (we work on the non-twist domain a2 3b > 0) when we have only two equilibrium points P1h(0, 0) and P2e(p, 0) (h stands for hyperbolic, e for elliptic), while p (b)ffiffiffiffiffidisplays the case a2 4b = 0 when other two equilibrium points A 0; 2a ; B p; 2a are born. Increasing a slightly from 4b, that is a2 4b > 0, the other four equilibrium points P3, P4, P5 and P6 are born through a saddle-center bifurcation Fig. 3(c). The new born equilibrium points form two dimerized island chains, that is, the elliptic points are surrounded by homoclinic circles to the corresponding hyperbolic points. Between these chains the trajectories of the Hamiltonian vector field are not graphs of real functions of x, but they are meanders, Fig. 3(c). Observe that on both symmetry lines there exist now three equilibrium points. Each two neighboring points on the same symmetry line have opposite stability type. Increasing further the parameter a, the equilibrium points lying on the same symmetry line (such points lie within two different chains) go away and at a threshold, called reconnection threshold, the hyperbolic points of the two chains get connected by common branches of invariant manifolds (Fig. 3(d)). Such a threshold is called reconnection threshold. In order to get this threshold we impose that the hyperbolic equilibrium points P3h and P6h to belong to the same energy level set, that is Ha,b,k(P3h) = Ha,b,k(P6h). This leads to a :¼ a1rec = 2.343. Increasing a beyond the reconnection threshold the dimerized island chains transform into Poincare´–Birkhoff chains, and the lower two PB chains approach each other (Fig. 3(e)). The reconnection of the lower two Poincare´–Birkhoff chains occurs at a :¼ a2rec = 2.79. In this case, the stable/unstable manifolds of these points have a common branch C3 = Wu(P1h) \ Ws(P6h), C4 = Ws(P1h) \ Ws(P6h). After reconnection, i.e. for a lightly greater than a2rec, the above described scenario occurs in a reverse order (Fig. 4). We note that fixing the shape parameter b the simultaneous reconnection of the three chains can occur. Imposing Ha,b,k(P1h) = Ha,b,k(P3h) = Ha,b,k(P6h), we get through symbolic computation the following reconnection curve: 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < k ¼ 1 a ða2 4bÞ a2 4b 3 24 b ð8Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 6b þ a4 6a2 b þ 48b3 k þ 4ab a2 4b a3 a2 4b ¼ 0.
Fig. 4. The phase portraits of the Hamiltonian system at the second reconnection threshold (a) and beyond it (b)–(d).
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Fig. 5. Scenario of reconnection of the all three Poincare´–Birkhoff chains. The a parameter is (a) a = 2.3, (b) a = 2.3812 and (c) a = 2.6.
Solving numerically these equations for b = 1.26 one get k = 0.0248 and a = 2.3812. This implies that in order to describe the reconnection scenario of the all three chainspfor ffiffiffiffiffi bp= ffiffiffiffiffi1.26 we2should keep the value of k at krec = 0.0248. Letting now the a parameter to vary, we getpthat if a 2 ð 3b ; 4bÞ and a 4b = 0 the scenario is the same as above, ffiffiffiffiffi Figs. 3(a)–(c) and 5(a). Increasing a beyond 4b the all three chains get connected when any two saddle distinct points P1h, P3h, P6h are connected by a common arc of their invariant manifolds, Fig. 5(b). The threshold of reconnection is a3rec = 2.3812. Increasing a further we get again the scenario in reverse order, Figs. 4(b)–(d) and 5(c).
4. Conclusion A three-parameter cubic non-twist map was analyzed in this paper. By using an approximate interpolating Hamiltonian of the map we have described the reconnection process of any two neighboring chains in the case when two parameters are fixed and the other one varies. By numerical computations we got the two thresholds of reconnection. Also we remarked that the three chains can be simultaneously reconnected. In this case only a parameter of the system can be arbitrary fixed. The reconnection thresholds of the other two parameters are found.
Acknowledgments This work was supported through a European Community Marie Curie Fellowship and in the framework of the CTS, contract number HPMT-CT-2001-00278. The author is grateful to Professor Emilia Petrisor for useful discussions on reconnection process.
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