Critical number in scattering and escaping problems in classical mechanics

Critical number in scattering and escaping problems in classical mechanics

Salvador Addas-Zanata*and Clodoaldo Grotta-Ragazzo^†

Instituto de Matema´tica e Estatı´stica, Universidade de Sa¯o Paulo, R. do Mata˜o 1010, CEP 05508-900, Sa˜o Paulo-SP, Brazil 共Received 17 January 2001; revised manuscript received 16 July 2001; published 24 September 2001兲

Scattering and escaping problems for Hamiltonian systems with two degrees of freedom of the type kinetic plus potential energy arise in many applications. Under some discrete symmetry assumptions, it is shown that important quantities in these problems are determined by a relation between two canonical invariant numbers that can be explicitly computed.

DOI: 10.1103/PhysRevE.64.046216 PACS number共s兲: 05.45.⫺a

I. INTRODUCTION

Invariant tori are very important in the global dynamics of two degree of freedom Hamiltonian systems. They split the phase space in unconnected components. For several physi- cally relevant systems, the existence of such invariant tori, and therefore the understanding of the phase-space structure, can only be achieved through numerical investigation. In this paper, an analytic criterion for the existence of certain fami- lies of invariant tori is presented. These families are impor- tant in some problems of escaping from and scattering off a potential well.

The class of systems such that our results apply can be described as follows. The Hamiltonian function is of the type kinetic plus potential energy,

H⫽¹2共p_x²⫹p_y²兲⫹V共x,y兲, 共1兲 where V has two critical points: a minimum P_mand a saddle- point P_s. The energy of the saddle point, which will be called critical energy and will be denoted E_cr, is V( P_s)

⫽E_cr. For energy values below E_cr the corresponding energy-level sets have two distinct components, one bounded and one unbounded. The bounded component projects to the configuration space (x,y ) inside what will be called the po- tential well共see Fig. 1 for a topological representation of the level curves of V兲. For H⫽E_cr, these two components touch at the equilibrium corresponding to P_s and for H⬎E_cr, the two components merge into a single unbounded component.

Notice that E_crplays an important role in the dynamics. For energy values below E_cr, there is always a large quantity共in a measure sense兲of bounded orbits trapped inside the poten- tial well, while for energy values above E_cr, this may not occur.

An important example that belongs to the above class is the Hamiltonian system for the motion of a charged particle in the field of a magnetic dipole 共see, for instance关1兴兲, also called the Stormer system. Due to the rotation symmetry of the magnetic field, this system can be reduced to two degrees

of freedom. For a given value of angular momentum and in convenient time and length scales, its Hamiltonian function is

H⫽1

2共p_x²⫹p_y²兲⫹V共x,y兲, V共x, y兲⫽1

2

冉

^{1x⫺共x2⫹xy2兲3/2}

冊

^2,

共2兲 where x is the radial coordinate and y is the coordinate along the dipole axis. It is well known that V satisfies the above properties, so the level curves of the Stormer potential V are topologically as in Fig. 1: V has two critical points, a mini- mum P_m and a saddle-point P_s, such that V( P_s)⫽E_cr

⫽1/32 and for energy values below E_cr, the corresponding energy-level sets have two distinct components, one bounded and one unbounded. For H⫽E_cr, these two components touch at the equilibrium corresponding to P_s and for H

⬎E_cr, the two components merge into a single unbounded component.

In this paper, two different problems will be considered for systems similar to the Stormer system, contained in the class defined below expression共1兲. The first is the so-called

‘‘escaping problem’’ or ‘‘escaping from a potential well,’’

that is: for a given distribution of initial conditions inside the potential well with fixed energy E⬎E_cr to determine the amount of solutions that remain inside the potential well af- ter time t⬎0. This type of escaping problem had been re-

*Current address: Department of Mathematics, Princeton Univer- sity, Fine Hall-Washington Road, Princeton, NJ 08544-1000. Email address: szanata@math.princeton.edu

†Email address: ragazzo@ime.usp.br

FIG. 1. Topological representation of the level sets of a typical potential function for the class we are considering. The shaded re- gion corresponds to an energy level set below the energy of the saddle point Ps.

cently considered by Contopoulus et al. 关2兴 and Kandrup et al.关3兴in a series of papers that were the main motivation for the present paper. Escaping problems appear in several branches of physics and chemistry 共see, for instance, 关3,4兴 and references therein兲. The second problem addressed in this paper is the so-called ‘‘scattering problem.’’ For in- stance, consider the Stormer system and a family of initial conditions parametrized by an ‘‘impact-parameter’’ b with constant energy E⬎E_cr, see Fig. 1. Most of these initial conditions will be scattered off the potential well and will become asymptotically free as t tends to infinity. These scat- tered solutions can be characterized by their asymptotic angle ␾(b) as shown in Fig. 1. In this case, the scattering problem consists in determining the angle␾(b) as a function of the impact-parameter b. The scattering problem for other systems 共or even for this one兲 can be defined in terms of other input-output variables instead of b→␾(b). Scattering problems have been extensively studied, in particular in the context of chaotic systems. The reader can find many refer- ences on the subject in the special volume, Chaos 3, issue 4 共1993兲 共in particular, see关5兴for a review兲and also关6兴. The scattering problem for the Stormer system has been consid- ered, for instance, in 关7兴.

From a theoretical point of view, escaping and scattering problems are better understood for systems that either are integrable or have fully hyperbolic recurrent sets. The sys- tems considered here are mostly in between these two ex- treme cases. In the context of the Stormer system, our main result for the escaping problem can be summarized in the following way. For E larger than but close to E_crsolutions initially in the potential well can escape to infinity. This cer- tainly happens for solutions on the x axis with initial velocity parallel to it. Nevertheless, escape may not occur for most solutions, if there exists a torus that projects to the configu- ration space like the one shown in Fig. 4共a兲. This torus blocks the ‘‘exit’’ from the potential well for ‘‘most’’ solu- tions. To be more precise about what we mean by ‘‘most solutions,’’ it is necessary to consider the limit E→E_cr and to introduce the concept of a family of blocking tori. A fam- ily of blocking tori is a set of invariant tori, one torus T_Efor each E苸(E_cr,E¯ ), for some E¯⬎E_crsufficiently close to E_cr, such that each T_E splits the phase space into one bounded component and one unbounded component and such that T_E tends to a curve C as E→E_cr. Each torus T_E is called a blocking torus. By T_E→C as E→E_crwe mean convergence with respect to the Hausdorff metric, namely,

max兵^dist共z,C兲: z苸T_E其^⫹max兵^dist共z,T_E兲: z苸C其^→0 as E→E_cr.

The main result in this paper is to give a computable crite- rion for the existence of blocking tori in a class of systems that include the Stormer system. The implications of the ex- istence of blocking tori for scattering problems is the follow- ing. A blocking torus restrict the access of all incoming so- lutions to a small phase-space region inside the potential well. Since most of the recurrent dynamics happens inside the potential well and recurrence is the main source of sin-

gularities in scattering functions, one expects that the regu- larity of these functions may strongly depend on the exis- tence or not of a family of blocking tori.

Our criterion for the existence of blocking tori is a con- sequence of some previous mathematical and numerical in- vestigations on systems with ‘‘saddle-center loops’’关8–11兴. Part of this work is presented in Sec. II. In Sec. III the results presented in Sec. II are applied to the Stormer systems and to three other systems that were studied in 关3,2,12兴. Section IV is a conclusion where we point out the main ideas in this paper.

II. THE CRITICAL NUMBER

In order to simplify the presentation, this section is di- vided in several subsections.

A. Hypotheses and the saddle-center loop map Although the results described here are valid for a wider class of systems 共see references below兲 in this section, we only consider Hamiltonian systems with two degrees of free- dom of the form kinetic plus potential energy:

H⫽¹2关p₁²⫹p₂²兴⫹U共q₁,q₂兲, 共3兲 where,

共a兲 U is analytic, 共b兲 U共q₁,q₂兲⫽¹2关⫺␯^2q1

2⫹␻^2q2

2兴⫹O„共q₁²⫹q₂²兲^3/2…, with ␻⬎0,␯⬎0,

共c兲⳵q2U共q₁,0兲⫽0, for any value of q₁,

共d兲 equation U共q₁,0兲⫽0 has a nontrivial nondegenerated solution q_1c and no solutions in 共0,q_1c兲.

The potential of the Stormer system 共2兲 satisfies these hypotheses with q₁⫽x, q₂⫽y , and with (q₁,q₂)⫽(0,0) re- placed by P_s. The above systems have two important prop- erties

共1兲 the origin is an equilibrium point of saddle-center type, namely, it is associated to a pair of real ⫾␯^{and a pair} of imaginary ⫾i␻ eigenvalues,

共2兲the phase-space plane兵^q1, p₁,q₂⫽p₂⫽0其is invariant and it contains an orbit ⌫ homoclinic to the origin p₁⫽p₂

⫽q₁⫽q₂⫽0.

The invariant set given by the saddle-center equilibrium and the homoclinic orbit⌫will be called saddle-center loop.

From a topological point of view, a saddle-center loop is similar to a periodic orbit 共of ‘‘infinite period’’兲 in phase space. Moreover, for system 共3兲, the saddle-center loop is placed exactly where one can expect a possible family of blocking tori to accumulate 共think about the Stormer sys- tem兲. These facts suggest two things: first, one can use Poin- care´ sections to study the saddle-center loop and second, families of blocking tori may be related to invariant circles

on the induced Poincare´ map. However, some difficulties appear when one tries to define a Poincare´ map for a saddle- center loop due to the presence of the unstable equilibrium in the loop. In order to overcome this difficulty, a Poincare´ map for the saddle-center loop will be defined by the composition of two distinct maps: one related to the passage of solutions near the saddle-center equilibrium denoted by L and another related to the traveling of solutions near ⌫denoted by G. In order to define these mappings, it is first necessary to study the dynamics near the saddle-center equilibrium. The qua- dratic part of H is given by

H_L⫽¹2 关p₁²⫹p₂²兴⫹¹2 关⫺␯^2q1 2⫹␻^2q2

2兴. 共4兲

This implies that the origin has a one-dimensional stable manifold W^s, a one-dimensional unstable manifold W^u, and a two-dimensional center manifold W^c. Within the linear approximation, these manifolds are given by W^s⫽兵^p1⫽

⫺␯^q1, p₂⫽q₂⫽0其^{, Wu}⫽兵^p1⫽␯^q1, p₂⫽q₂⫽0其^{, Wc}⫽兵^p1

⫽q₁⫽0,p₂,q₂其. The part of W^swith q₁⬎0 (q₁⬍0) will be denoted as W_⫹^s (W_⫺^s ) and the part of W^u with q₁⬎0 (q₁

⬍0) will be denoted as W_⫹^u (W_⫺^u). Notice that⌫coincides either with W_⫹^s and W_⫹^u or with W_⫺^s and W_⫺^u 共this follows from a simple analysis in the invariant 兵^{q1, p1}其 ^{plane兲. To} simplify the presentation, it is assumed that⌫coincides with W_⫹^s and W_⫹^u . Now, let ⌺1, ⌺2, and ⌺3 be three planar Poincare´ sections 共three-dimensional兲 transversal to W_⫹^s , W_⫹^u , and W_⫺^u , respectively, as shown in Fig. 2. Sections

⌺1,⌺2, and⌺3, are locally parametrized by q₂, p₂, and E, where E is again the energy. In order to simplify the compu- tations, it is convenient to define canonical variables Q₁, P₁,Q₂, P₂

q₁⫽共P₁⫹Q₁兲 1

冑

²␯^{, p1⫽共P1⫺Q1兲}

冑

␯

&,

q₂⫽共P₂⫹Q₂兲 1

冑

²␻^{, p2⫽共P2⫺Q2兲}

冑

␻

& , in such a way that H_L becomes

H_L⫽⫺␯^P1Q₁⫹␻^I, where I⫽¹2 关␻^⫺1p2

2⫹␻^q2

2兴⫽¹2 关P₂²⫹Q₂²兴.

Using these coordinates and the approximation H_L to H, one can parametrize⌺1 as

Q₁⫽␦⫽const⬎0, P₁⫽␻^I⫺E

␯␦ ^.

Similarly, one gets explicit expressions for ⌺2 and⌺3. The integration of the linearized vector field near the origin gives the linear flow

冉

^{QQPP1122共共共共tttt兲兲兲兲}

冊

^⫽

冉

^{e⫺␯000 t e⫹␯000 t ⫺cossin00共共␻␻t兲t兲 cossin共共00␻␻tt兲兲}

冊

⫻

冉

^{QQPP1212共共共共0000兲兲兲兲}

冊

^{. 共5兲}

This flow induces a discontinuous Poincare´ map L: ⌺1

→⌺2艛⌺3 given by the following relations共see Fig. 2兲. 共i兲 If (Q₂, P₂,E)苸⌺1 is such that ␻^I⬎E 共namely, P₁

⬎0兲 then L(Q₂, P₂,E)⫽(Q₂

⬘

^{, P}2

⬘

^,E

⬘

^)苸⌺2 is given by 关Z and Z

⬘

denote the column vectors (Q₂, P₂) and (Q₂

⬘

^{, P}2

⬘

^), respectively,兴

E

⬘

^{⫽E, Z}

⬘

^⫽R共␪兲Z,

where R(␪) is the rotation matrix of angle␪

R共␪兲⫽

冉

^{cossin␪␪ ⫺cossin␪␪}

冊

^共6兲

and␪⫽␪(I,E) is given by

␪共I,E兲⫽⫺␻

␯^ln

冉

^P␦1

冊

^{⫽␻␯ln共␻I⫺E兲⫺␻␯ln共␦2␯兲.}

共ii兲 If (Q₂, P₂,E)苸⌺1 is such that ␻^I⬍E 共namely, E

⬎0 and P₁⬍0兲 then L(Q₂, P₂,E)⫽(Q₂

⬘

^{, P}2

⬘

^,E

⬘

⁾苸⌺3 is given by the same expressions as in item共i兲

E

⬘

^{⫽E, Z}

⬘

^⫽R共␪兲Z,

except that ␪is given by

␪共I,E兲⫽⫺␻

␯ ^ln

冉

^⫺P␦1

冊

^{⫽␻␯ ln共E⫺␻I兲⫺␻␯ln共␦2␯兲.}

共iii兲 If (Q₂, P₂,E)苸⌺1 is such that ␻^I⫽E 共namely, E

⭓0 and P₁⫽0兲 then the solution that leaves ⌺1 is in the stable manifold of either a periodic orbit in W^c(E⬎0) or the saddle-center equilibrium (E⫽0). So the flow does not de- fine the Poincare´ map. In this case, L(Q₂, P₂,E)

⫽(Q₂

⬘

^{, P}2

⬘

^,E

⬘

) is defined to be in⌺2with coordinates given by (Q₂

⬘

^{, P}2

⬘

^,E

⬘

⁾⫽(Q₂, P₂,E).

FIG. 2. Diagram showing the Poincare´ sections⌺1,⌺2, and⌺3

used to define mappings L and G.

A Poincare´ map G: ⌺2→⌺1 is defined in the usual way by the flow near ⌫. If (Q₂, P₂,E)苸⌺2 then G(Q₂, P₂,E)

⫽(Q₂

⬘

^{, P}2

⬘

^,E

⬘

^)苸⌺1 is given by

E⫽E

⬘

^{, Z}

⬘

⫽C共␦兲Z⫹Eu共␦兲⫹O共E²⫹储Z储²兲, 共7兲 where C(␦) is a two-by-two matrix 关the (Q₂, P₂) compo- nents of the derivative of G with respect to (Q₂, P₂) at (Q₂, P₂,E)⫽(0,0,0)兴 and u(␦) is a vector 关the (Q₂, P₂) components of the derivative of G with respect to E at (Q₂, P₂,E)⫽(0,0,0)兴. The invariance of the plane 兵^{q1, p1,q2⫽0,p2⫽0}其 by the flow implies that G(0,0,E)

⫽(0,0,E) and therefore in this case u(␦⁾⫽0. The flow ex- pression 共5兲 implies that if ␦⫽␦0⫹D varies, where ␦0 is some fixed positive number, then C(␦) changes as C(␦0

⫹⌬)⫽R(K

⬘

^)C(␦0)R(K

⬘

^{) where K}

⬘

苸(0,2␲兴 can assume any value depending on⌬. Writing C(␦0)⫽A₀R(K₀) where A₀is symmetric关polar decomposition of C(␦0)兴and choos- ing⌬conveniently, one concludes that C(␦) can be written as C(␦⁾⫽AR(K) where A is a diagonal matrix. Using the simplectic property of the flow, one gets that the determinant of A is one. Therefore, the expression for the (Q₂, P₂) com- ponents of G, denoted as Gˆ , within the linear approximation and with a convenient choice of␦ is given by

Z

⬘

⫽Gˆ共Z兲⫽AR共K兲Z, where A⫽

冉

^{␣0 1/0␣}

冊

^{, ␣⭓1.}

共8兲 Let⌺1Eand⌺2Ebe the restrictions of⌺1and⌺2, respec- tively, to the energy level set H⫽E, for E close to zero.

Since the dynamics preserves energy, it is convenient to re- strict maps L and G to⌺1Eand⌺2E, respectively. The above expressions for L imply that the (Q₂, P₂) components of L, denoted by Lˆ, restricted to⌺1E are given by

Z

⬘

⫽Lˆ共Z,E兲⫽R共␪兲Z,

where ␪共I,E兲⫽␥^ln储I⫺␻^⫺1E储⫹K

⬘

^, 共9兲 where ␥⫽␻^/␯^{, K}

⬘

^苸关0,2␲) is some fixed number that de- pends on␦^{, and Z}

⬘

belongs to either⌺2E or ⌺3E depending on␻^I⫺E⭓0 or␻^I⫺E⬍0, respectively共if␻^I⫹E then Z

⬘

⫽Z兲. Notice that for E⫽0, Lˆ:⌺10→⌺20, Gˆ :⌺20→⌺10, and the composition F⫽GⴰL: ⌺10→⌺10 is well-defined. In this case, Z

⬘

⫽F(Z)⫽AR(2␥^ln储Z储⫺␥^{ln 2}⫹K

⬘

⫹K)Z and with a simple rescaling Z→␤^{Z, Z}

⬘

^→␤^Z

⬘

^{, where 2}␥^ln␤⫽␥^{ln 2}

⫺K⫺K

⬘

, one can write F in the simple form

Z

⬘

⫽F共Z兲⫽AR共2␥^ln储Z储兲Z, 共10兲 that will be called saddle-center loop map. Several approxi- mations were made in the derivation of F. In particular, the linearized flow was used to describe the dynamics near the equilibrium. So,␦⬎0 must be chosen small. It can be shown 关9,10兴that the real Poincare´ map from⌺10→⌺10differs from the one above by a term bounded by K储Z储², where K is some positive constant.

The results presented in this section are due to several authors, especially Lerman 关13兴 and Mielke, Holmes, and O’Reilly 关14兴. The normal form theorem used to study the integrable nonlinear dynamics near the saddle-center loop is due to Moser关15兴with a supplement of Ru¨ssmann关16兴 共see also 关17兴, Appendix 8兲. Some early ideas related to the dy- namics in systems with saddle-center equilibria using Moser’s theorem can be found in Conley关18,19兴, Churchill, Pecelli, and Rod关12兴and Churchhill and Rod关20兴 共see, also earlier work by the same authors cited in these references兲, and Llibre, Martinez, and Simo´关21兴. Mappings G, L, and the saddle-center loop map F as presented here are due to Ler- man 关13兴and Mielke, Holmes, and O’Reilly 关14兴, indepen- dently.

B. The invariants␣and␥

The dynamics of the saddle-center loop map depends on the two parameters ␥ ^and␣. The parameter␥⫽␻^/␯ ^{is the} ratio between the modulus of two eigenvalues ␻^and␯ ^and therefore is invariant under any change of coordinates 共it is invariant even under time reparametrization兲. The parameter

␣is harder to compute and is related to the flow linearized at the orbit ⌫. Let q¯₁(t) be the q₁ component of ⌫(t). The (q₂, p₂) components of the vector field linearized at⌫are

q˙₂⫽p₂,

p˙₂⫽⫺关␻²⫹S兵^¯q1共t兲其兴q₂, 共11兲 where

␻²⫹S共q₁兲⫽␻²⫹O共q₁兲⫽⳵q₂,q₂V共q₁,0兲. In the variables (Q₂, P₂) these equations become

Q˙

2⫽␻^P2, P˙

2⫽⫺关␻⫹␻^⫺1S兵^¯q1共t兲其兴Q₂. 共12兲 For a given ␦⬎0, let t_⫺(␦^{) and t}⫹(␦) be two time values such that the (Q₁, P₁) coordinates of⌫关t_⫺(␦⁾兴are共0,␦兲and the (Q₁, P₁) coordinates of⌫关t_⫹(␦⁾兴are共␦^{, 0}兲. Let␺^(t_⫺^,t) be a fundamental system of solutions of Eq. 共12兲 such that

␺^(t_⫺^,t_⫺) is the identity matrix. Then, matrix C(␦^{) appear-} ing in Eq. 共7兲is given by ␺^(t_⫺^,t_⫹^{) and} ␣共␦兲is the square root of the largest eigenvalue of C(␦^)C(␦^{)†, where C(}␦^)† denotes the transpose of C(␦⁾ 关geometrically, ␣共␦兲 is the largest semiaxis of an ellipsis that is the image of a unit circle by C(␦⁾兴. In principle, the value of ␣共␦兲depends not only on ␦ but also on the coordinates used near the saddle- center equilibrium. However, a computation using Moser co- ordinates关or assuming as true the linear flow approximation 共5兲兴shows that␣共␦兲does not depend on␦. Still, this property depends on this special system of coordinates. In order to present a characterization of ␣ that does not depend on a special coordinate system, one defines ␦⫽␦0 and ␦n

⫽␦0e^⫺2␲n/␥. Then, one takes the limit as n→⬁ to get:␦n

→0, t_⫾→⫾⬁, and␺^(t_⫺^,t_⫹⁾→␺⁽⫺⬁,⫹⬁): TW^c→TW^c, where TW^c is the tangent manifold to W^cat the origin共this

limit is explained in detail in关8兴, Theorem 3兲. The trace of

␺共⫺⬁, ⫹⬁兲 does not depend on coordinate systems any- more, but depends on the choice of ␦0. Nevertheless, one can prove共as in关8兴, Theorems 3 and 4兲that the maximum of the trace of ␺共⫺⬁, ⫹⬁兲 for ␦0 varying in the interval 关␦^{e⫺2␲/␥,}␥兴 is equal to ␣⫹␣^⫺1 and it does not depend either on ␦ or on the coordinate system. This construction gives an invariant definition for␣. In order to compute␣^{, it} is necessary to take a special limit␺共⫺⬁,⫹⬁兲of the funda- mental solution of Eq.共12兲. For the particular systems in this paper, this limit gets a very familiar form. Notice that both equations 共11兲and 共12兲 are equivalent to second-order dif- ferential equations for q₂ and Q₂, respectively. Moreover, both equations for q₂and Q₂ are the same and are given by q¨₂⫽⫺关␻²⫹S兵^q¯1共t兲其兴q₂, 共13兲 where S关¯q₁(t)兴 decays exponentially fast to zero as t

→⫾⬁. Equation 共13兲 is a one-dimensional Schro¨dinger equation similar to those appearing in one-dimensional quan- tum scattering problems. It has a complex-valued solution␾ with the following asymptotic behavior:

␾共t兲→Aˆ e^i␻t⫹Be^⫺i␻t as t→⫺⬁,

␾共t兲→e^i␻t as t→⬁, Aˆ ,B complex. 共14兲 Aˆ and B are the usual scattering coefficients of quantum mechanics. In关8兴 共Theorem 4兲it is shown that the maximum with respect to ␦0 of the trace of the limit␺共⫺⬁, ⫹⬁兲 共as discussed above兲, which is equal to␣⫹␣^⫺1, is also equal to 2

冑

兩B兩²⫹1. Therefore, one gets that␣⭓1 is given by

␣⫽兩B兩⫹

冑

兩B兩²⫹1, 共15兲 where 兩B兩 is determined by the one-dimensional scattering problem共13兲–共14兲.

The number␣ can be also understood in a different and more geometric way. The center manifold of the saddle- center equilibrium is foliated by periodic orbits ␨E, one for each energy E⬎0. In the linear flow approximation共5兲␨E is a circle of radius

冑

^2E/␻ ^{in the plane} 兵^Q1⫽P1

⫽0,Q₁, P₂其. The stable共unstable兲manifold of␨Eis a cylin- der that intersects⌺1E (⌺2E) at a circle C_1E (C_2E) with the same radius. The global mapping G maps C_2E苸⌺2E to a closed curve C_1E

⬘

苸⌺1E that is approximately an ellipsis if E⬎0 is small. In general, C_1E

⬘

intersects the circle C_1Etrans- versally. Let␤

⬘

denote the area outside C_1E and inside C_1E

⬘

and let␤ be the area inside C_1E. Since G restricted to⌺2E

preserves area, then 0⭐␤

⬘

^/␤⬍1. Let ␤* be the limit of

␤

⬘

^(E)/␤^{(E) as E}→0. Then one gets

cos

冉

^␤*␲2

冊

^{⫽␣⫹2␣⫺1. 共16兲}

Notice that if the unstable and stable manifolds of␨E coin- cide then␣⫽1. This is a necessary condition for integrability obtained in关13兴and关14兴. Notice that in terms of the scatter- ing coefficient B, a necessary condition for integrability is

B⫽0, namely, a resonance in the ‘‘quantum scattering prob- lem’’ 共13兲–共14兲. The number ␣ first appeared in 关13兴 and 关14兴. An invariant characterization of it was given in关8兴that was the source for most of the presentation above.

C. A class of examples where␣can be easily computed In general, it is not easy to solve the scattering equation 共13兲. In most cases this can only be done numerically. Nev- ertheless, there is an important class of potentials where this problem can be solved analytically 关8兴. This class is de- scribed in the following. Suppose that the potential V in Eq.

共3兲satisfies the following additional hypotheses:

V共q₁,0兲⫽⫺¹2 ␯^2q1

2⫹ ␦

n⫹1q₁^n⫹1,

⳵q2,q2V共q₁,0兲⫽␻²⫹␤^q1

n⫺1. 共17兲

This implies that the solution homoclinic to the origin satis- fies q₂⫽p₂⫽0 and

q¨₁⫽⫹␯^2q1⫺␦^q1 n.

It can be checked that the q₁ component of⌫is given by q₁共t兲⫽C sech^{2/共n⫺1兲}

冉

^{t␯共n2⫺1兲}

冊

^,

where C⫽

冉

^{共n⫹21␦兲␯2}

冊

^{1/共n⫺1兲.}

This implies that Eq. 共13兲becomes

q¨₂⫽⫺

冋

^{␻2⫹␤共n⫹2␦1兲␯2sech2}

冉

^{t␯共n2⫺1兲}

冊册

^{q2. 共18兲}

This is a well-known one-dimensional Schro¨dinger equation problem that can be solved using hyper geometric functions.

The modulus of B is given in 关22兴, Sec. 25, Eq. 4, as 兩B兩²⫽cos²关共␲^/2兲

冑

␩兴

sinh²共␲⑀兲 , if ␩⬎0, 兩B兩²⫽cosh²关共␲^/2兲

冑

⫺␩兴

sinh²共␲⑀兲 , if ␩⬍0, 共19兲 where

⑀⫽ 2␻

共n⫺1兲␯^, ␩⫽␤

␦

8共n⫹1兲 共n⫺1兲²⫹1.

Then, ␣ follows from these expressions and from ␣⫽兩B兩

⫹

冑

兩B兩²⫹1 关Eq.共15兲兴.

D. Families of blocking tori and the critical number In the introduction, a family of blocking tori was defined using the topology of the energy-level sets of the Stormer system as a model. That definition can be easily generalized

to systems that satisfy the hypotheses of Sec. II A. In this case, a blocking torus splits the phase space into two con- nected components but none of them has to be necessarily bounded. For these systems, the existence of a family of blocking tori accumulating at ⌫ is implied by the existence of a family of invariant curves accumulating at the trivial fixed point of the saddle-center map F, Eq.共10兲. Before ex- plaining this claim, it is necessary to study the dynamics of F. The following two paragraphs were essentially taken from 关11兴.

Map F has among other symmetries a remarkable one: it is invariant under discrete dilation Z→e^␲/␥Z 关namely, F(e^␲/␥Z)⫽e^␲/␥F(Z)兴. In particular, this symmetry implies that the fixed point Z⫽(0,0)⫽0គ is stable when ␣ ^{is suffi-} ciently close to␣⫽1. In fact, for ␣⫽1, F leaves all circles centered at 0គ invariant共F is just a rotation in each circle兲. Far from the origin, F is analytic and the KAM 共Kolmogorov, Arnold, and Moser兲 theorem can be used to prove that, if 兩␣⫺1兩is sufficiently small, then invariant closed curves sur- rounding 0គ exist. Since there is one such a curve there are infinitely many, due to the dilation symmetry, and these curves accumulate at 0គ 共where F is not differentiable兲.

The dilation symmetry also suggests a beautiful mecha- nism for the instability of 0គ when␣is large. For␣⬎1, if F has a fixed point p distinct from 0គ, then it has an infinite family of fixed points given by e^k␲/␥p, k苸Z. Indeed, an ex- plicit computation shows that F has four families of fixed points. The first family is denoted by z_k, k苸Z, where z_k

⫽e^␲k/␥z₀. The elements of the second family are given by

⫺z_k. The element z₀ has coordinates

P₂⫽exp

冉

^{␥␦⫺4␲␥}

冊

^{cos␦, Q2⫽exp}

冉

^{␥␦⫺4␲␥}

冊

^sin␦,

where ␦⫽arctan(1/␣⁾苸(0,␲/4). The linearization of F around z_k and⫺z_k, for all k, is associated to the same pair of eigenvalues (␭,␭^⫺1), determined by␭⫹␭^⫺1⫽2⫹2␥⁽␣

⫺␣^⫺1). This implies that all⫾z_kare hyperbolic. Now, sup- pose the unstable manifold of z₀ intersects transversally the stable manifold of z₁. Then, the dilation symmetry of F implies that the unstable manifold of z_k intersects transver- sally the stable manifold of z_k⫹1 for all k苸Z. The same is valid for the family⫺z_k, k苸Z. So, due to the discrete sym- metry, the existence of a transversal heteroclinic orbit con- necting z₀ to z₁ implies the existence of two infinite hetero- clinic chains, or ‘‘Arnold’s transition chains’’ 关24兴, which implies 共besides a lot of ‘‘chaos’’兲 the instability of 0គ. In order to show that this scenario indeed occurs, it is still nec- essary to show that the unstable manifold of z₀intersects the stable manifold of z₁ transversally. This is related to the nonexistence of homotopically nontrivial invariant curves in the saddle-center loop map. It was proved in 关10兴 that the above transition chain exists if ␥⁽␣⫺␣^⫺1)⬎1. For the real saddle-center loop map, that takes into account the higher- order corrections to F, it was proved 关10兴the existence of a semi-infinite heteroclinic chain connecting points arbitrarily close to the saddle-center loop to points at finite distance of it.

In 关11兴, a numerical study of F was done in order to estimate the region in the parameter space ␥⬎0, ␣⭓1, where invariant circles surrounding 0គ exists. Very curiously the following simple criterion was obtained.

Stability criterion. Let c be the number共stability number兲: c⫽␥共␣⫺␣^⫺1兲. 共20兲 There is a critical value of c, denoted as c_c and called critical number, such that

• If c⬍c_c, then there are invariant circles surrounding 0គ, so all orbits of F are bounded and 0គ is a stable fixed point.

• If c⬎c_cthen there are no invariant circles surrounding 0គ and F has an infinite heteroclinic chain connecting 0គ to in- finity.

An approximated value of c_c, probably overestimated, is c_c⫽1/&.

Remark: The above criterion was obtained numerically.

Both the form of the critical curve and the critical number

c_c⫽1/&are only approximations. The computation near the

breakup of the last invariant curve is delicate. For instance, for some particular values of␥, it was numerically verified a stability-instability transition at c_c⫽0.55关23兴.

Notice that the appearance of the above heteroclinic chain reminds us of共and it is very similar to兲the so-called ‘‘reso- nance overlap criterion’’ of Chirikov关24兴. This idea of reso- nances overlap when applied to the standard map also pro- duces infinite heteroclinic chains. A remarkable property of the saddle-center loop map F is that this infinite heteroclinic chain of F implies a semi-infinite heteroclinic chain for the original Hamiltonian system共3兲 关10兴.

Now, we turn back to the question of the existence of families of blocking tori. All the discussion above supposes the energy E to be zero. Nevertheless, the concept of families of blocking tori requires the existence of invariant tori for E⬎0. In fact, using the KAM theorem 共as, for instance, in 关10兴兲 it can be shown that the existence of sufficiently smooth invariant circles surrounding 0គ for E⫽0 imply the existence of perturbed invariant circles for E⫽0,兩E兩 small.

The statement is as follows. If the above invariant circles for E⫽0 exist, then given an ⑀⬎0 there exists ␦⬎0 such that for兩E兩⬍␦ the mapping GⴰL restricted to⌺1E has an invari- ant circle with a radius less than ⑀. This conclusion clearly implies the existence of a family of blocking tori associated to the saddle-center equilibrium at the origin of system共3兲. So, the stability criterion above can also be used as a crite- rion for the existence of families of blocking tori. More in- formation about invariant curves for E⬎0 can be found in 关10兴 共in particular, in Fig. 4 of关10兴one can find a diagram that motivated the definition of a family of blocking tori兲.

E. The saddle-center scattering map

In Sec. II A we presented maps L and G to describe the dynamics near a saddle-center loop of a system satisfying certain hypotheses. In this section, the dynamics generated by these maps will be studied for a fixed small value of E

⬎0. For simplicity, the index E will be omitted from ⌺1E,

⌺2E, and ⌺3E, and the restriction of the (Q₂, P₂) compo- nents of L and G to the energy-level set E will be denoted also by L and G, respectively. So, the dynamics near the saddle-center loop in this case is determined by the pair of mappings 关see Eqs.共9兲and共8兲兴

L: ⌺1→⌺2艛⌺3,

Z

⬘

⫽L共Z兲⫽R共␥^ln兩储Z储²⫺2␻^⫺1E兩⫹K

⬘

兲Z, and

G: ⌺2→⌺1, Z

⬘

^⫽G共Z兲⫽AR共K兲,

where K and K

⬘

are two constant phases in 共0,2␲兲 and the image of L is in⌺2if储Z储²⫺2␻^⫺1E⭓0 otherwise it is in⌺3

共see Fig. 2兲. For a system satisfying the hypotheses of Sec.

II A, a typical scattering problem, similar to the one for the Stormer system mentioned in the introduction and illustrated in Fig. 1, is the following. The incoming particles are given by a line of initial conditions with fixed energy E⬎0 at q₁

⫽0 with p₁⫽q˙₁⫽v⬎0, v constant satisfying 0⬍v⬍

冑

^2E.

For E small, from the linearized Hamiltonian H_L 共4兲, one gets that this set of initial conditions corresponds to a circle in phase space given by

Q₁⫽⫺ v

冑

²␯^{, P1⫽} v

冑

²␯^{, Q2}

2⫹P₂²⫽2

␻

冉

^E⫺v22

冊

^.

The linearized flow共5兲maps this circle to another circle S in

⌺2 with the same radius. So one can consider S苸⌺2 as the input of the scattering process. It is clear that if the only exit from the potential well is over the saddle-center equilibrium at the origin, then any scattered solution must hit⌺3. There- fore, a suitable output for the scattering problem is the (Q₂, P₂) components共or any function of them兲of the inter- section of an exiting solution with ⌺3.

Under some circumstances, as discussed below, the scat- tering problem presented above is approximately determined by the dynamics of L and G in the following way. The initial condition for the scattering process is given by the circle S 苸⌺2. This circle is mapped by G to an ellipsis G(S) 苸⌺1, see Fig. 3. The points in G(S) have two possible ways. If they are outside the circle 储Z储²⫽2␻^⫺1E 共repre- sented by a dotted circle in Fig. 3兲 then they are mapped back to⌺2 by L. If they are inside this circle, then they are mapped to⌺3 by L. The scattering process for the points in

⌺3 is finished and the corresponding output value are their (Q₂, P₂) components. Moreover, the scattering time in this case may be taken as one, because they were scattered after one cycle of iteration of G and L. Then the points LⴰG(S) 苸⌺2 are mapped again to ⌺2 and so on, see Fig. 3. The result is that from this iteration scheme 共which we call

‘‘saddle-center scattering map’’兲 one can get all scattering functions.

The procedure described in the above paragraph approxi- mates well the real scattering process under two circum-

stances. At first, E⬎0 has to be sufficiently small. At sec- ond, the stability number c of the previous section must be less than the critical one c_c⬇1/&, namely, the system must have a family of blocking tori. If this last condition is not verified, then points initially close to Z⫽(Q₂, P₂)⫽(0,0) can escape to infinity under iterations of LⴰG. Therefore, they will never hit ⌺3. Moreover, mappings G and L ap- proximate the real dynamics only when 储Z储 is not large.

Even when c is large, the scattering results obtained with mappings G and L may be a good approximation to the real scattering in its first few iterates.

We point out that the scattering map defined in 关25兴, which consists of reinjecting the scattered solutions, may be easily obtained in the context of the saddle-center scattering map. This is done through the identification of ⌺3 and⌺2. The result is a mapping from Fˆ : ⌺1→⌺1 given by

Z

⬘

^⫽Fˆ共Z兲⫽AR共␥^ln兩储Z储²⫺2␻^⫺1E兩⫹K

⬘

^⫹K兲Z, where K and K

⬘

are constants in关0,2␲兲and the mapping is the identity if 储Z储²⫺2␻^⫺1E⫽0.

III. APPLICATIONS

The goal in this section is to apply the results presented in the previous one to four examples. The first three are taken from 关3,2,12兴.

A. Example 1

In this example, the Hamiltonian function is H⫽¹2 共p_x²⫹p_y²兲⫹V共x,y兲,

V共x,y兲⫽¹2 共x²⫹y²兲⫺¹3 y³⫹␮^{x2y .}

This is Eq.共3兲of关3兴. For␮⫽1, it is the well-studied He´non- Heiles system 关26,12兴. For any value of␮⭓0, the potential FIG. 3. Diagram showing the sequence of mappings used to define the saddle-center scattering map. Notice that points outside the dotted circle in⌺1are mapped to⌺2and then mapped back to

⌺1by G. Those points inside the dotted circle in ⌺2are mapped outside the potential well to⌺3. For them the scattering process is over.

function has a saddle point at (x, y )⫽(0,1) with an associ- ated critical energy E_cr⫽1/6. With the change of variables y⫽1⫺q₁, x⫽q₂, one gets

V共x, y兲⫽¹2 关⫺q_t²⫹共1⫹2␮兲q₂²兴⫹¹3 q₁³⫺␮^q1q₂². This system satisfies the hypotheses in Sec. II C with: ␯

⫽1, n⫽2, ␦⫽1, ␻⫽

冑

¹⫹2␮^, ␤⫽⫺2␮^{. So,} ⑀

⫽2

冑

¹⫹2␮^, ␩⫽⫺48␮⫹1, and the values of 兩B兩 follow from Eq. 共19兲. For instance, for ␮⫽0, ␮⫽0.43, ␮⫽0.49, and␮⫽1, one obtains兩B兩⫽0,兩B兩⬇0.25, and兩B兩⬇0.89, re- spectively. Using Eq.共15兲, the stability number can be writ- ten in terms of兩B兩 as

c⫽2␥兩B兩,

which implies c⫽0, c⬇0.55, c⬇1/&, and c⬇3.1, for ␮

⫽0,␮⫽0.43,␮⫽0.49, and␮⫽1, respectively. In this case, the criterion for the nonexistence of invariant tori around the saddle-center loop using the critical number c_c⫽1/& gives for ␮ the critical value␮⫽0.49. A more careful numerical analysis of the dynamics of the saddle-center loop map F for the particular parameter values of this problem showed, how- ever, that the breaking up of the last invariant tori of F hap- pens for␮⫽0.43 that corresponds to a c_c⫽0.55关23兴. This is in agreement with numerical simulations of the real flow 关23兴. So, in this example, the critical value of␮is taken as 0.43.

It is interesting to visualize projections of blocking tori in the configuration space. Consider the sequence of Figs. 4 for different parameter values␮and for a fixed value of energy equal to 0.17共above E_cr⫽1/6兲. In Figs. 4共a兲(␮⫽0) and 4共b兲 (␮⫽0.4), we show projections of invariant tori on the con- figuration space that block the escaping channel over the saddle-center equilibrium. In Fig. 4c (␮⫽0.7) we exhibit a single solution starting near the boundary of the potential well that escapes from it. This shows that there is no variant torus near the saddle-center loop blocking the exit of the potential well.

In order to illustrate the dependence of a scattering prob- lem for this system on the parameter␮, consider a scattering function␺ defined in the following way. For a given E⬎0, let兵x⫽b, y⫽1.1, p_x⫽0, p_y⬍0其be a one parameter family of input initial conditions where p_y is determined by H⫽E

⫽const and the impact-parameter b is defined in a maximal interval where V(b,1.1)⭐E. Consider the solution generated by the initial condition associated to b and suppose that for some time T(b) this solution hits the phase-space hypersur- face 兵^y⫽1其 ^{with p}y⬎0 and x⫽␺(b). The functions b

→T(b) and b→␺(b) will be called as time delay and scat- tering return map, respectively. It seems from a sequence of graphs of the scattering return map, obtained through the variation of␮^{for E}⫽0.17 fixed, that near ␮⫽0.43 the rate 共with respect to ␮兲 of loss of regularity of b→␺^{(b) is en-} hanced. Nevertheless, the transition that occurs near ␮

⫽0.43 can be much easily seen through a plot of an averaged time delay function T¯ (␮) as shown in Fig. 5. This function T¯ is obtained taking the average of T(b) over many equally spaced values of b for a fixed value of␮^.

The potential V above has two more saddle points x

⫽⫾(1/2␮

冑

␮⁾

冑

¹⫹2␮⫽⫾¯ , yx ⫽⫺1/(2␮⁾⫽¯ , at energyy E⫽(1⫹3␮^)/(24␮³). There is a homoclinic solution to each saddle-center equilibrium corresponding to each of these FIG. 4. Projections on the configuration space of orbits of the He´non-Heiles family with energy equal to 0.17 and initial condi- tions y (0)⫽0, p_x(0)⫽0, p_y(0)⬎0, and: 共a兲 for ␮⫽0, x(0)

⫽0.12; 共b兲for␮⫽0.4, x(0)⫽0.12; 共c兲for␮⫽0.7, x(0)⫽0.58.

saddle points. These homoclinic solutions are given by x(t)

⫽⫾¯ u(t), y (t)x ⫽¯ u(t), where u satisfies u¨y ⫽⫺u⫹u², u(t)→1 as t→⫾⬁. As above, the coefficients␥^and兩B兩can be computed explicitly. For␮⫽1, the potential is symmetric with respect to rotations of 2␲/3 radians, which implies that

␥^and兩B兩are the same for the three homoclinic saddle-center loops. These saddle-center loops are related to an escaping problem that will be discussed after the next two examples.

B. Example 2

In this example, the Hamiltonian function is

H⫽¹2 共p_x²⫹p_y²兲⫹V共x,y兲, V共x,y兲⫽¹2 共x²⫹y²兲⫺x y². This is Eq.共8.1兲of关12兴and after some rescaling of coordi- nates and time, this is Eq.共2兲of 关3兴. The potential V has a pair of saddle points (x, y )⫽(1/2)(1⫾&) at energy 1/8. The saddle-center equilibria related to these saddle points have eigenvalues⫾1 and⫾i&. There is one homoclinic solution to each saddle-center equilibrium. Introducing coordinates

x⫽ 1

)共qˆ₁⫺&q₂兲, y⫽ 1

)共&qˆ₁⫹q₂兲, qˆ₁⫽) 2⫺q₁, one gets a system satisfying the hypotheses in Sec. II C. The same reasoning as in the previous example gives: ␯⫽1, ␻

⫽&, ␥⫽&, 兩B兩⬇0.259, c⫽2␥兩B兩⬇0.807⬎&. This im- plies that there are no families of blocking tori accumulating at the saddle-center loops.

C. Example 3

In this example, the Hamiltonian function is

H⫽¹2 共p_x²⫹p_y²兲⫹V共x,y兲, V共x,y兲⫽¹2 共x²⫹y²兲⫺¹2 x²y². This is Eq.共8.2兲of关12兴and after some rescaling of coordi- nates and time, this is Eq.共1兲of关3兴. The potential V has four saddle-points (x,y )⫽(⫾1,⫾1) at energy 1/2. The saddle- center equilibria related to these saddle points have eigenval-

ues⫾&and⫾i&(␥⫽1). There is one pair of heteroclinic orbits connecting the equilibria at (x, y )⫽(1,1) and (x,y )

⫽(⫺1,⫺1), and one pair of heteroclinic orbits connecting the equilibria at (x, y )⫽(1,⫺1) and (x,y )⫽(⫺1,1). In prin- ciple, the results in Sec. II A, for homoclinic orbits to saddle- center equilibria do not apply. Nevertheless, there is a simple adaptation关9,10兴of the above-presented results to the case of heteroclinic loops. Again, the scattering problem for the de- termination of ␣ is explicitly solvable and gives: 兩B兩

⬇0.119, c⫽2␥兩B兩⬇0.238⬍1/&. This implies that there are invariant tori accumulating at the saddle-center heteroclinic loop, and therefore, a family of blocking tori for E⬎E_cr, E→E_cr.

D. The escaping problem and the dynamics for examples 1, 2, and 3

The escaping problem for the systems in Examples 1, 2, and 3 was studied by Kandrup et al.关3兴. They have a param- eter⑀in their systems that coincides with the parameter␮ⁱⁿ Example 1 and that in Examples 2 and 3, after some rescal- ing, can be identified with the energy共in fact, what they do is to fix an energy and vary this parameter; after the rescaling, this is equivalent to fix the parameter and vary the energy兲. So, below, the reader can think of⑀as if it were the energy.

In particular, their ⑀0 is equivalent to the energy E_crof the saddle-center equilibrium. In their own words, one of their main conclusions was: 共the following paragraph was taken from the abstract of 关3兴兲:

‘‘For⑀below a critical value⑀0, escapes are impossible energetically. For somewhat higher values, escape is allowed energetically but still many orbits never escape. The escape probability P computed for an arbitrary orbit ensemble de- cays toward zero exponentially. At or near a critical value

⑀1⬎⑀0 there is a rather abrupt qualitative change in the be- havior. Above ⑀1,...,. The transitional behavior observed near⑀1is attributed to the breakdown of some specially sig- nificant KAM tori or cantori... .’’

In principle, it is not easy to relate the results in 关3兴 to those in here. They studied the escaping problem for a range of energies that include values far from the energy E_crof the saddle-center equilibrium, while here, all results concern en- ergies near E_cr. However, there is an interesting fact in their results that makes it possible to establish a connection be- tween their critical⑀1 and the concept of a family of block- ing tori introduced here. For Examples 2 and 3 above, the value of E_cris 0.125 and 0.5, respectively, and the values of

⑀1 they found are equivalent to energies E⫽0.165_⫺^⫹_0.014^0.006and E₁⫽1.176⫾0.0024, respectively. For Example 1, they fixed the energy of the initial conditions E⫽0.167, which implies that escape is energetically possible for ␮⬎␮0⫽1, and found a critical value of␮1 共or⑀1兲of 1.1⫾0.5. Under these circumstances, the saddle-center loops that are relevant for the escaping problem are those presented at the end of Sec.

III A. Their energy is E_cr⫽0.135_⫺^⫹_0.013^0.015 depending on ␮1

⫽1.1⫾0.5. Since ␮1⬇1, these saddle-center loops have numbers␥^and兩B兩‘‘close to’’ those computed for the saddle- center loop on the x axis for ␮⫽1. Therefore they are un- stable. Notice that in Examples 1 and 2, there is no family of blocking tori accumulating on those saddle-center loops that FIG. 5. Graph of the averaged time delay function T¯ (␮).