- Email: [email protected]
A geometric proof of separatrix crossing results
Nonlinear Analysis 56 (2004) 1047 – 1070
www.elsevier.com/locate/na
A geometric proof of separatrix crossing results Shui-Nee Chowa;∗ , Todd Youngb a School
of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA of Mathematics, Ohio University, Athens, OH 45701-2979, USA
b Department
Received 1 August 2003; accepted 1 November 2003
Abstract We investigate the adiabatic invariance of the action variable of one degree of freedom time-dependent Hamiltonian systems when trajectories cross a separatrix and give relatively simple, geometrically motivated proofs. ? 2003 Elsevier Ltd. All rights reserved.
1. Introduction Consider a family of one degree of freedom Hamiltonian systems, with Hamiltonian function H (q; p; ), where ∈ [0; 1] is a parameter. By the drifting or perturbed system we mean the nonautonomous 6ow on R2 de7ned by q˙ = Hp (q; p; ); p˙ = −Hq (q; p; ); ˙ = ;
(1)
where ¿ 0 is a small parameter. The case = 0 we call the static or unperturbed system. Denote by X (·; t; ) the time-dependent vector 7eld de7ned by (1) and use
(·; t; ) to denote the 6ow generated by (1). We will consider H such that each the static system possesses a hyperbolic equilibrium and associated separatrix cycle which depends on . We study orbits of the perturbed system that cross the separatrix cycle. For = 0, it is well known that for ‘typical’ H , most trajectories of (1) are periodic. It is a classical theorem of Liouville that in regions where all the solutions are periodic one may introduce canonical ‘action-angle’ variables. In these coordinates, the action ∗
Corresponding author. E-mail addresses: [email protected] (S.-N. Chow), [email protected] (T. Young).
0362-546X/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2003.11.002
1048
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
variables are constant and the angle variables proceed linearly. The action variable in R2 is simply I (q; p; ) =
1 A(q; p; ); 2
(2)
where A(q; p; ) is the area in the plane enclosed by the periodic orbit. At integral levels where solutions are not periodic, the action-angle variables are not de7ned. This typically occurs at the energy level set of a hyperbolic equilibrium. Such a level set separates regions in which the action-angle variables are well de7ned and so is called a separatrix. In general, the action variable experiences a jump discontinuity across a separatrix curve. In the drifting system, solutions are not periodic and so the action-angle variables must be de7ned in terms of the static system. In the drifting system, the action variable is an adiabatic invariant, which means that it is constant to order O() for time periods of the order O(1=). However, solutions of the perturbed system may happen to cross a separatrix and the action variable experiences a jump due to the discontinuity in the de7nition of the action. Further, the techniques used to show that the action is an adiabatic invariant are only valid for solutions which are bounded away from a separatrix. Systems with separatrix crossings were studied extensively by a number of researchers, notably, [1,4–6,8–12]. These studies all rely on extending the averaging method to a neighborhood of the separatrix. They are not presented as rigorous results, but as careful estimates. The most complete derivation seems to appear in [8]. They found that for all but a set of initial condition of size not more than O(e−1= ) the action at the separatrix crossing experiences not only a jump caused by the discontinuity, but also a small shift which depends on the ‘phase’ at which the crossing occurs. For small, this phase is extremely sensitive to initial conditions, so for practical purposes one may treat the phase as a random quantity, uniformly distributed at each point in the space of initial conditions. There is however disagreement about the size of the small jump; some estimate that is of order O(), while others assert that it is O( ln −1 ). The excluded set of order O(e−1= ) in this theory corresponds to those initial conditions which pass too closely to the hyperbolic equilibrium. The mechanism for crossing a separatrix is the following. Suppose that the system has separatrices in a symmetric ‘Fig. 8’ con7guration. Let Y () be 1=2 times the area of one of the lobes enclosed by a separatrix. We will suppose that Y () is an increasing function, so that the area enclosed by the lobes increases in time. Since the system is Hamiltonian, area in phase space is preserved and it follows that the orbits of a positive measure set of initial conditions must enter the lobes. Example 1. Polynomial double-well potential: H (q; p; ) =
p2 q2 q4 − + : 2 2 4
(3)
As ¿ 0 changes, the size of the wells changes and some orbits will be captured or escape from the wells. The separatrix is the energy level of the unstable equilibrium at (0; 0).
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1049
Example 2. Pendulum with slowly varying length: H (q; p; ‘) =
p2 − ‘g sin2 (q): 2‘2
(4)
Here the length of the pendulum ‘ may serve as a parameter. If ‘ is decreased, energy is put into the system. If ‘ becomes suJciently small then the motion will change from back and forth oscillations to spinning completely over. In particular, suppose that an initial condition (q0 ; p0 ) is outside the separatrices. Let I0 = I (q0 ; p0 ; 0). Suppose further that the total area enclosed by the separatrices at some time becomes equal to I0 . Since I is approximately preserved, it follows that the solution at that time will be in some vicinity of the separatrices. The parameter value ∗ at which 2Y (∗ ) = I0 is called the pseudo-crossing parameter. Unless the solution comes very close to the equilibrium, it will enter one of the lobes and the action variable will jump to approximately I0 =2. The separatrix will then continue to expand, but the action variable will again be nearly constant once the solution is suJciently far from the separatrix. The 7nal value of the action variable will be I0 =2 plus a small correction LI which we call the shift. According to Cary et al. [5] up to order 2 , the correction term depends on a phase variable s and has the remarkably simple form LI ≈ −
dY ln |2 sin( s)|; ! d
0 ¡ s ¡ 1;
(5)
provided the Hamiltonian has the form !(p2 − q2 )=2 + H3 (q; p; ): Cary et al. [5] also showed that any slowly varying Hamiltonian system with a hyperbolic equilibrium can be put into this form up to any order of . In (5), the derivative of Y with respect to is assumed to be constant. The results of other studies diMer from (5). Most notably, some estimates possess terms of order O( ln −1 ). Our analysis explains this discrepancy in the order of LI . In this paper, we formulate and prove various rigorous results about this phenomenon based on geometric arguments rather than the averaging method. We de7ne a phase variable 0 ¡ ¡ 1 that controls whether or not the orbit is captured, and the size of the shift. Using a relatively simple, geometric argument we show the following results. Theorem 1. The measure of the set of initial conditions that are not captured is of order −1= e O : 2 Theorem 2. Given any 0 ¡ ¡ 1, LI 1 dY ∗ → (Sc () + S()) d as 0, where Sc () = −2 − ln − ln(1 − ) and S() is a bounded function.
1050
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
We 7nd numerically that |S()| is less than 1. These results have 7xed and → 0. If however, one 7xes, small and lets vary, there is a small range of values close to either 0 or 1 where the shift is of order O( ln −1 ), thus clarifying the ambiguity in the theory (Section 4.5). 2. Local linearization and consequences 2.1. Linearization We will assume that H depends C k -smoothly on all variables, where k ¿ 3, k = ∞, or, k = ! (analytic). The key ingredient in our treatment of separatrix crossing is the following linearization lemma. Lemma 1. Suppose that H ∈ C k (R2 × [0; 1]; R) where k = 3; : : : ; ∞; !. Suppose that for each , the unperturbed system has a hyperbolic equilibrium at (q∗ ; p∗ ) and Df((q∗ ; p∗ ); ) has eigenvalues ±(), where () is uniformly bounded away from zero. Then there exist neighborhoods U of (q∗ ; p∗ ) such that on U there exists an, area preserving, time dependent, C 1 -smooth, change of coordinates (·; ; ) : U → R2 , along with a reparameterization of time, that puts system (1) into the form x˙ = x; y˙ = −y:
(6)
Here may be taken to be any positive number. Further, the image of the parameterdependent neighborhood U under may be taken to be a uniform neighborhood U of (0; 0). We will postpone the proof of this lemma until Section 5. Note that past works, in order to use averaging, have employed analytic changes of variable, that were not necessarily canonical, thus ignoring the main geometric feature of the problem. Let Ws and Wu denote the cross-sections of the local stable and unstable manifolds of the equilibrium. We will de7ne these manifolds precisely in Section 5. In the linearized coordinates Ws and Wu coincide with the coordinate axes. A by-product of the proof of Lemma 1 is the following. Lemma 2. Ws and Wu converge to the local separatrices as → 0. Further, the convergence is uniform in the C k topology. By the C k topology we mean the topology of the functions whose graphs are the local manifolds, including dependence on . 2.2. Return maps Let U be as in Lemma 1. Fix d ¿ 0 so that the square D bounded by the lines {x = d}, {x = −d}, {y = d}, and {y = −d} is contained in the interior of U . We will
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1051
Lobe 2 U Σ4 D Σ1
Σ3
Σ2
Lobe 1
Fig. 1. Schematic drawing of the separatrix and a neighborhood of the equilibrium for = 0. Inside U the 6ow is linearized. We consider 7rst-hit maps de7ned by the 6ow between the boundaries of a square D ⊂ U , i.e. %i , i = 1; : : : ; 4. In the linearized coordinates all the 7rst-hit maps are isomorphisms, even for the drifting system ( ¿ 0). This is a consequence of the linearization and the symplectic character of the 6ow.
assume that the subsquare in D given by x ¿ 0, y ¿ 0 is contained in lobe 1 and the subsquare given by x ¡ 0, y ¡ 0 is contained in lobe 2. Now consider the maps between the boundaries of D induced by the 6ow. We denote the boundaries of D by {%i } where %1 = {x = d} ∩ D, %2 = {y = −d} ∩ D, the numbering continues in the clockwise direction, see Fig. 1. In these coordinates, the local maps are trivial. In particular, consider a point on %4 given by (x; d) with x ¿ 0. Under the 6ow, the orbit of this point 7rst intersects %1 exactly at (d; x). A point on %4 with x ¡ 0 is mapped to (−d; −|x|). The local map starting at %4 is similar. We may summarize the local maps as follows: L2 : %2 → %1 ∪ %3 : (x; −d) → (sign(x)d; −|x|); L4 : %4 → %1 ∪ %3 : (x; d) → (sign(x)d; |x|):
(7)
Along with the local map, one also deduces from (6) that the local transit time for a point starting at (x; ± d) is exactly T‘ (x) =
d 1 ln : |x|
(8)
1052
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
The 6ow outside D induces smooth 7rst-hit maps G1 : %1 → %2 and G3 : %3 → %4 . For the unperturbed system conservation of energy implies that, in the local coordinates, G1 and G2 are isomorphisms. In particular G1 : (d; y) → (−y; −d); G3 : (−d; y) → (−y; d)
(9)
for = 0. On the other hand, for ¿ 0, the global maps might perhaps depend on y, and/or . The next proposition follows easily from the symplectic nature of the 6ow and the linearization. Proposition 1. The maps G1 and G2 have the form G1 : (d; y) → (−y + *; −d); G3 : (−d; y) → (−y − *; d);
(10)
where * = *(; ). Proof. Note from the linear form of the Hamiltonian in U that the instantaneous 6ux across each of the boundaries of D is a constant, namely d. By instantaneous 6ux, we mean X · n, P where nP is a normal vector to the boundary. Now consider a small line segment , in %1 connecting arbitrary points (d; y1 ) and (d; y2 ). For a small time interval -, let R- ={ t (,) : −- 6 t 6 0}. For each point xP ∈ Rlet y( ˜ x) P denote the y-coordinate of the point on , in the local orbit of x. P Next let Tg (y) be the 7rst value of time for which Tg (y) (d; y) ∈ %2 , i.e. the time of transition for the global map. Now let P (x) P : xP ∈ R- }: R- = { Tg (y(˜ x))
The integral invariant of PoincarRe–Cartan [2], i.e. the form p dq − H dt, applied to Rgiven us that the areas A and A , of R and R , respectively, are related by A = A − H dt; /
where / is a curve in R2 × R corresponding to the boundary of R and its transition time coordinate. We partition / into four parts, namely the image of , which we denote by , , the backward image of , under the 6ow by −-, we denote this by ,- and two ‘end’ segments. The line integrals of H d- across , and ,−exactly cancel since H is constant along 6ow lines in these local coordinates and dt on , will be exactly the negative of dt on corresponding points on ,−. Further, - is constant on the ends, so dt ≡ 0 there. Thus, we have A = A . Note that the ‘thicknesses’ of both R and R are exactly the same namely d exp(-). Taking the limit as - 0 shows that the length of the image , is exactly the same as the length of ,. The time of global transition, Tg , from %1 to %2 and from %3 to %4 might depend on y, , and . Standard results on the smooth dependence on parameters of 7nite time
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1053
solutions of initial value problems imply that Tg (y; ; ) → Tg (y; ; 0) uniformly in the C k topology as → 0. 3. Capture of orbits 3.1. Size of initial conditions It is known that the classical adiabatic invariance of the actions can extended to an -dependent neighborhood of the separatrix. In particular, the adiabatic invariance holds as long as ln h−1 h is small [8]. In light of this, we can assume that our initial conditions satisfy h ≈ O(1 ); where 0 ¡ 11. The initial conditions we study will begin on %4 with x-coordinate approximately x∗ =1 . We will see that this implies that the relevant parameter intervals that we must consider becomes arbitrarily small as 0. This will allow us to prove two key simplifying facts: the values of dY=d and Tg (x; ) are arbitrarily close to constants. 3.2. Expansion of the separatrices Next we determine *. Note that in the previous section, we have not proved that * is constant in time only that * is a constant for points starting at %1 at the same time. We denote this constant by *(; ). We now show that * is directly proportional to the rate of expansion of the separatrices. Recall that Y () denotes 1=2 times the area of each lobe. Proposition 2. The following limit holds in C k ([0; 1]): *(; ) 2 dY → () d d as → 0. Proof. Let Ws denote the stable manifold of the equilibrium at parameter . Consider the branch of Ws that 7rst intersects %4 starting from the equilibrium and the segment of that branch from the equilibrium and to the 7rst intersection with %1 . Call this segment W . Consider the region bounded by W , the line segment from (0; 0) to (d; 0), and the segment, ,1 along %1 from (d; 0) to W , see Fig. 2. The area enclosed by this region is approximately the area enclosed by the separatrix lobe. The region is not invariant in t, but orbits may cross the boundary only at ,1 . Note that the length of ,1 is precisely *(; ). Note that the 6ux per unit length across the boundary of ,1
1054
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
Fig. 2. The region enclosed by W , ,1 , and the line segment from the origin to (d; 0) has approximately the same area as lobe 1 for the unperturbed system. Solutions may enter this region only by passing through the segment ,1 .
is a constant, namely d. Thus, the total 6ux across the segment ,1 is d*. Let Y˜ () denote 1=2 times the area of the region, then we have that 2 d Y˜ *(; ) ≡ : d d Now we note that as Y˜ → Y as → 0. In fact, from Lemma 2, we have that W converges uniformly in the C k topology to the corresponding segment of the separatrix as → 0. Thus Y˜ () → Y () in C k ([0; 1]). We will suppose that an orbit begins on %4 with x-coordinate x−n . We will label each successive intersection with %2 or %4 by xi , where xi is the absolute value of the x-coordinate on %2 or %4 . Along with {xi } we have an associated sequence of times {ti } and parameter values {i }. Following each intersection xi , we have an intersection with %1 ∪%3 which we denote by yi , where yi is the absolute value of the y-coordinate of the intersection point. Denote the corresponding intersection times by {tPi } and parameter values by {Pi }. We will denote by x0 the last intersection of the orbit with %2 or %4 before the crossing. We will continue following the orbit up to step x n+1 . Below we will let n be the minimal integer such that x−n ¿ x∗ ≡ 1 . With this convention, we have that yi = xi for all i. Also we have that xi+1 = xi − *(tPi )
for i ¡ 0
(11)
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1055
and xi+1 = xi + *(tPi )
for 1 6 i:
Now we de7ne our phase parameter, , by y0 x0 = = : *(tP0 ) *(tP0 )
(12) (13)
Note that G1 at time tP0 maps (d; *(tP0 )) to (0; −d). Proposition 1 further implies that x1 = *(tP0 ) − x0 , so x1 1−= : (14) *(tP0 ) We consider the orbit in three parts, the approach, the crossing and the departure. By the approach we mean the transition from x−n ≈ 1 to x0 . By crossing, we mean the segment of the orbit from x0 to y1 . We call the transition from y1 to yn+1 the departure. Note that we have the same number of steps in the approach and departure. 3.3. Approach time of
First, we give bounds on n. Let a and b be strict lower and upper bounds on values 2 dY d d
in the entire parameter interval under consideration. Then from Proposition 2, we have that *(; y) a¡ ¡b (15) for small. From (11) we have n *(x−i ; P−i ) = x−n − x0 ≈ 1 : (16) i=1
Combining (15) and (16) we obtain na ¡ 1 ¡ nb: We thus have the following. Proposition 3. For all 0 6 6 1, the number of steps n needed for the approach satis
n−1 n−1 1 d ln + Tg (xi ; i ): xi i=0
i=0
1056
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
Recall that 0 is the parameter value at the last crossing of @D before the crossing begins. Proposition 4. If we suppose that n is the minimal integer such that x−n ¿ x∗ ≡ 1 , then 0 − −n → 0 as → 0. This limit is uniform in . Proof. Let TP g be the maximum value of Tg (x; ). Let * be the minimum value of *(y; ). Then the approach time t0 − t−n satis7es t0 − tn 6
n 1 d ln + nTP g x0 + i* i=1
6
n d 1 ln + nTP g *(i + x0 =*) i=1
n n d 1 x0 + nTP g 6 ln − ln i + * * i=1
6
n d ln + nTP g : *
(17)
Therefore 0 − −n 6
n d ln + nTP g : *
Propositions 2 and 3 imply that the right-hand side of the above equation goes to zero as → 0. Since the domains shrink to zero as → 0, the functions *(y; ) and Tg (y; ) are approximately constant during the approach. Thus, we may replace them by constants in the rest of this section. We have then the following simpli7cation of the approach time: t0 − t−n =
n 1 d + nTg ln x0 + i* i=1
=
n 1 d ln + nTg *( + i) i=1
=
n 1 d 1 ln − ln(i + ) + nTg : * i=1
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1057
We may bound the sum in the last expression by an integral and obtain the estimate t0 − t−n ¡
1 d n+ n+ 1 ln − ln(n + ) + − + nTg : *
(18)
3.4. Value of the action away from the separatrices Let h denote the value of the Hamiltonian. It is well known that T dI = ; dh 2 where T is the period of the periodic orbit for the unperturbed system. Inside the 7rst lobe consider orbits of the unperturbed system that intersect %4 at a point (x; −d). Since h = xy inside U , we have d dI T (x; ): =− 2 dx The time spent by such an orbit inside D is given exactly by (8) and the time spent outside D is the function Tg (x; ). Thus, dI d d d = − ln − Tg (x; ): dx 2 |x| 2 It follows that the action variable of the periodic orbit starting at (x; ±d) on %2 or %4 inside one of the lobes is given by d d x xd 1 + ln − Tg (x; ) d x: (19) I1; 2 = Y () − 2 |x| 2 0 Similarly, the action for an orbit outside the separatrices starting at (x; d) on %4 is given by d x I3 1 d + Tg (x; ) d x: (20) = Y () + xd 1 + ln 2 2 x 2 0 For the perturbed system, we will only measure the instantaneous action when orbits cross %2 or %4 and use the corresponding orbit of the unperturbed system. Because of the variable change, the point (x; d) for the drifting system is not exactly (x; d) for the static system. Since the assignment of the action variable to the perturbed system is not unique and we are only interested in measuring the change in the action this ambiguity is not important. For discussion of higher-order corrections to the assignment of the action variable to the perturbed system, see [8].
1058
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
3.5. Psuedo-crossing parameter ∗ The action of the initial condition x−n ¿ 0 satis7es I3 d d x−n x−n d 1 + ln (x−n ) = Y (−n ) + + Tg (x; 0) d x: 2 x−n 2 0 2 x The integral, 0 −n Tg (x; 0) d x, we denote by TP g . The pseudo-crossing parameter ∗ associated with x−n is de7ned by 2Y (∗ )=I3 (x−n ). It follows that ∗ satis7es ∗ x−n d d d P dY d = 1 + ln T g: + d 2 x 2 −n −n Let m denote the smallest value of dY=d on [0; 1]. Then we have d d P x−n d 1 + ln T g: + m(∗ − −n ) 6 2 x−n 2 Let *P = (x−n − x0 )=n, note that a 6 *P 6 b where a and b are, respectively, the minimum and maximum values of *. Thus, d P d d P + 1 + ln ∗ − −n 6 (x0 + *n) Tg P 2 m 2 x−n + *n d d P d P *( + n) 1 + ln − ln + n + T g: = P 2 m 2 * P → 0 as → 0. Thus, we have the following. Using Propositions 2 and 3, we see that *n Proposition 5. Suppose that ∗ is the pseudo-crossing parameter for x−n , then ∗ − −n → 0 as → 0. This limit is uniform in . In light of this result, we may treat Tg and dY=d (as well as *) as constants on (−n ; ∗ ). 3.6. Crossing time We say that the crossing is the transition from x0 to y1 . The time required for the crossing is thus tP1 − t0 =
1 1 d d ln + ln + Tg x0 x1
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
=
1 d 1 d ln + ln + Tg * (1 − )*
=
2 d 1 ln − ln (1 − ) + Tg : *
1059
(21)
Here * is exactly *(P0 ) and Tg is Tg (y0 ; P0 ). Recall that P1 = tP1 , then the crossing occurs between 0 and P1 . Eq. (21) implies: Proposition 6. As → 0 P1 () − 0 () → 0 for each . Note that this limit is not uniform in . Thus, we see that is eMectively constant during the crossing itself for most . Thus 0 , which we can call the actual crossing parameter, coincides with ∗ . Note that we have not proved that if we de7ne ∗ as the pseudo-crossing parameter for some 7xed x∗ and let 0 be parameter value at the last hit before crossing, then 0 → ∗ as → 0. That is, for a 7xed x∗ we have not shown that the actual crossing parameter 0 converges to the pseudo-crossing parameter. However, an area preservation argument shows that this is true. 3.7. Departure time By the departure we mean the transition from y1 to yn+1 . The next result follows from arguments identical to those for Proposition 4. Proposition 7. Pn − P1 → 0 as → 0. This limit is uniform in . 3.8. Capture and the excluded set We can say that an orbit is captured into one of the separatrices if it reaches yn+1 within a parameter of some 7xed length, say L. From the previous sections, the parameter intervals required for both the approach and departure are negligible. Thus, only the crossing itself contributes to the overall parameter interval. In other words, capture occurs if P1 − 0 ¡ L:
1060
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
Eq. (21) implies that capture occurs if L 2 d 1 ln − ln (1 − ) + Tg ¡ : * Solving, this is equivalent to (1 − ) ¿ eTg
d2 −L= e : *2
The value of * is of order , so that orbits are captured provided (1 − ) ¿ K
e−L= : 2
(22)
Thus, only an exponentially small set of initial conditions, corresponding to either very small, or 1 − very small, fails to cross and we have proved Theorem 1.
4. Estimating the change of the action We now consider the change of the action as the orbit moves from x−n to yn+1 . We only consider points which make this transition in a relatively short parameter interval as in the previous section. Thus, we may treat dY=d and Tg as constants throughout this section. We also treat * as a constant. Outside the separatrices, we consider I3 =2 and we consider I1 , I2 inside lobes 1 and 2. In the symmetric case, this combination of functions is continuous across the separatrix. 4.1. Approach to the crossing We ignore the change in the action from x−n to y−n . The change in the action variable from y−i to y−i+1 , 1 6 i ¡ n is given by 7−i = 12 I3 (y−i+1 ; tP−i+1 ) − 12 I3 (y−i ; tP−i ); This is equal to
1 6 i 6 n:
dY d d + y−i+1 1 + Tg + ln d 2 y−i+1 d d − y−i 1 + Tg + ln 2 y−i dY d 1 d d = Tg + ln + (y−i − *) 1 + Tg + ln d y−i+1 2 y−i+1 d d − y−i 1 + Tg + ln y−i 2
7−i = (tP−i+1 − tP−i )
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1061
d* d 1 d 1 + Tg + ln − ln y−i+1 y−i+1 2 d d d + y−i+1 ln : − ln 2 y−i+1 y−i
=
dY d
Tg +
Using Proposition 2, we obtain 7−i = −
d dY y−i y−i+1 ln + : 2 y−i+1 d
Noting that y−i = y0 + i* and = y0 =* we obtain for 1 6 i 6 n i+ dY −1 + (i + ) ln : 7−i = d i−1+
(23)
It can be shown that the terms of this sequence are of order O(1=i) for i large. 4.2. The crossing Next, we calculate the change in the action from y0 to y1 : 1 70 ≡ I1; 2 (y1 ; P1 ) − I3 (y0 ; P0 ) 2 d d dY d d P P − y1 1 + Tg + ln y0 1 + Tg + ln − : = (t 1 − t 0 ) d 2 y1 2 y0 Noting that y1 = x1 = * − y1 = * − x1 and = y0 =*, we obtain that 1− dY −1 − ln : 70 = d
(24)
4.3. Departure from the crossing The change in the action variable from yi to yi+1 , i ¿ 1 is given by d d dY d d P P − yi+1 1 + Tg + ln yi 1 + Tg + ln 7i = (t i+1 − t i ) + d 2 yi+1 2 yi 1 dY d d d = + Tg ln − (yi + *) 1 + Tg + ln yi+1 d 2 yi+1 d d + yi 1 + Tg + ln : 2 yi Thus, 7i = −
dY yi + * d : + yi ln d 2 yi
1062
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
Noting that yn+1+i = y1 + i and 1 − = x1 =* we obtain for 1 6 i 6 n i+1− dY : −1 + (i − ) ln 7i = i− d
(25)
The terms of this sequence are of order O(1=i) for i large. 4.4. Total shift in the action Combining (24) with the leading terms (i = 1) of (23) and (25) we obtain dY (−3 − ln − ln(1 − ) + (1 + ) ln(1 + ) + (1 − ) ln(2 − )): d The most important term of this is −ln − ln(1 − ). Noting that 1 −ln − ln(1 − ) d = 2; 0
de7ne Sc () = −2 − ln − ln(1 − ): Further, de7ne Sa (; n) = −1 + (1 + ) ln(1 + )
n
−1 + (i + ) ln
i=2
and Sd (; n) = (1 − ) ln(2 − ) +
n
i+ i−1+
−1 + (i − ) ln
i+1− i−
1
1
i=2
Numerically, we 7nd that both of the integrals total shift in the action is given by dY LI = (Sa + Sc + Sd ): d
0
Sa d and
0
:
Sd d are zero. The (26)
Using integral estimates it can be shown that both Sa and Sd are of order O(ln n). However, Sa is positive and Sd is negative. Asymptotically, the terms of the sums are negatives of each other, thus we expect signi7cant cancellation to occur. In fact, a simple calculation shows that the ith term of Sa (; n) + Sd (; n) is of order O(i−2 ), uniformly in . Thus we have Proposition 8. There exists a continuous function S() such that the following limit lim Sa (; n) + Sd (; n) = S()
n→∞
exists uniformly in . We have now proved Theorem 2.
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1063
12 Sa Sd Sa + Sd Sa + Sc + Sd
10 8
shift
6 4 2 0 −2 −4 −6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
φ
Fig. 3. Numerically computed shift functions Sa (; n), Sd (; n), Sa + Sd , and the total shift function Sa + Sc + Sd . In these plots n was taken to be 50,000. As n increases, Sa and Sd are of order O(ln n). However, Sd (; n) + Sa (; n) is relatively small and in fact Sd (; n) + Sa (; n) approaches a limiting function S() as n → ∞.
See Fig. 3. Using high precision arithmetic, we 7nd that |Sa () + Sd ()| ¡ 0:2 for n from 10 up to 109 . Since Sa (; n) and Sd (; n) are of order ln n, we have that for any 8, 0 ¡ 8 ¡ 1, |Sa (; n) − Sa (; n8 )| ≈ 1 − 8: |Sa (; n)| Thus, most of the weight of Sa and Sd belongs to the 7rst n8 terms, i.e. the terms corresponding to the portion of the orbit closest to the crossing. Thus we see that for small, the total shift is well approximated by LI =
dY ∗ (Sc () + S()): d
(27)
If (1 − ) is very small, then 0 and P1 may diMer signi7cantly. In that case our assumptions of this section do not hold. We treat this case in the next section. 4.5. The case (1 − ) small Suppose that the diMerence between 0 and P1 is signi7cant if it is greater than some number c ¿ 0. Repeating the calculation of Section 3.8, we 7nd that P0 − 0 ¿ c if (1 − ) ¡ K
e−c= ; 2
1064
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
where the constant K is the same as in Eq. (22). Thus, the analysis of the previous section is not valid for K
e−c= e−L= ; ¡ (1 − ) ¡ K 2 2
(28)
but initial conditions with in this range are captured. We will produce an estimate of the shift in the action for such . First, we observe that only one of or 1 − may be small. We will only treat the case where is small; the other case is similar. If is small, then x0 is small and the passage from x0 to y0 requires a long time. The previous analysis is valid for the transition from y−n to y−1 and from y0 to yn+1 . We need only to re-examine the transition from y−1 to y0 . The shift in this transition is given exactly by 1 1 I3 (y0 ; P0 ) − I3 (y−1 ; P−1 ) 2 2 d d d d + = Y (P0 ) − Y (P−1 ) + y0 1 + ln y−1 1 + ln 2 y0 2 y−1 y0 y−1 d + Tg (x; P0 ) d x − Tg (x; P−1 ) d x 2 0 0 d dy0 y0 + * d* P P 1 + ln = Y (0 ) − Y (−1 ) + − ln 2 y0 2 y0 + * y0 y−1 d + Tg (x; P0 ) d x − Tg (x; P−1 ) d x 2 0 0 dY P d = Y (P0 ) − Y (P−1 ) + (−1 ) − ln + (1 + ) ln(1 + ) − 1 + ln d * y0 y−1 d + Tg (x; P0 ) d x − Tg (x; P−1 ) d x : 2 0 0
7−1 =
In this calculation, *=*(P−1 ). Note that we may replace P−1 by 0 . The entire approach plus the 7rst half of the crossing, i.e. from y−n to y0 is summarized by dY d 7a = (0 ) − ln + ln + Sa (; n) d * y0 y−1 d + Tg (x; P0 ) d x − Tg (x; 0 ) d x : (29) 2 0 0 For the transition from y0 to yn+1 , the parameter value is approximately P1 . The shift due to this transition may calculated to be 7d =
dY P (1 )(−2 + ln − ln(1 − ) + Sd (; n)): d
(30)
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1065
Combining (29) and (30), we get the total shift in the case is small. Since the coeJcients of Sa and Sd are diMerent, the cancellation does not occur. Since both Sa and Sd are of order O( ln −1 ), the correction term has this order, rather than order O().
5. Proof of local linearization We prove Lemma 1 via several propositions; 7rst we 7x the unstable equilibrium at the origin. Proposition 9. Suppose that the unperturbed system is as in Lemma 1. Then there exists a neighborhood of U of (q∗ ; p∗ ) and an area preserving, time dependent, C k -smooth, change of coordinates (·; ; ) : U → U , where U is a
pP = p − p∗∗ (; ):
(31)
For the unperturbed system, simply taking q∗∗ (; 0) = q∗ and p∗∗ (; 0) = p∗ 7xes the equilibrium at (0; 0) for each . However, since the change of variables depends on , it is time dependent for system (1). It is easy to check that the change of variables (31) is generated by the generating function S(q; p) P = (pP + p∗∗ (; ))(q − q∗∗ (; )): For a time-dependent change of variable, the new Hamiltonian is given by H1 (q; P p) P +
@ S; @t
where H1 (q; P p) P = H (q; p) and thus ∇H1 (0; 0) = 0, i.e. (0; 0) is an equilibrium point for H1 . However, the second term, (@=@t)S depends on the derivatives of (q∗∗ ; p∗∗ ). For (31) applied to (1) this results in the system of equations dq∗∗ @H1 qP˙ = − ; @pP d ∗∗
dp @H1 − : pP˙ = − @qP d Thus we seek C k smooth functions (q∗∗ (; ); p∗∗ (; )) so that
dq∗∗ @H ∗∗ ∗∗ = (q ; p ; ); d @pP
(32)
1066
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
dp∗∗ @H ∗∗ ∗∗ = (q ; p ; ); @qP d
(33)
where we require q∗∗ (; 0) = q∗
and
p∗∗ (; 0) = p∗ :
(34)
In the analytic category, there exist a unique solution to (33) and (34) [8]. For k = 3; : : : ; ∞, and 7xed, there is a solution of (33) for any initial data (q∗∗ (0; ); p∗∗ (0; )). To 7nd a smooth family of such solutions which also satisfy (34) is more delicate. Consider the rescaling = t. This leads to the original time-dependent equations in the following form: dq∗∗ @H ∗∗ ∗∗ (q ; p ; t); = @pP dt dp∗∗ @H ∗∗ ∗∗ (q ; p ; t): = dt @qP
(35)
Note that for = 0 the curve segment given by (q∗ ; p∗ ; ) is normally hyperbolic. It is well known that normally hyperbolic manifolds persist under smooth perturbations. In the present case, we may not apply this principle directly since the perturbation is time dependent. If we try to treat the time-dependent system as autonomous, by making t the third variable, then the original region considered in the problem is not invariant (since this time interval is 7nite). We may circumvent these diJculties by extending the time direction of the phase space to (−∞; ∞). Given k we may C k -smoothly extend H (q; p; ) to ∈ R in such a way that the system de7ned by H has a hyperbolic equilibrium (q∗ ; p∗ ) for each ∈ R. In fact, we can take (q∗ ; p∗ ) to be 7xed beyond a neighborhood of [0; 1] and we can make the system linear on a neighborhood of (q∗ ; p∗ ) for outside a neighborhood of [0; 1]. Now let M = {(q∗ ; p∗ ; ) : ∈ R}. Clearly, M is a uniform normally hyperbolic manifold when =0. In the extended system this manifold persists, is C k -smooth and depends C k -smoothly on the perturbation, i.e. on . Let (q∗∗ (); p∗∗ (); ) be the coordinates of the perturbed manifold. This (q∗∗ (); p∗∗ ()) satis7es both (33) and (34). Next we perform a time change which 7xes the eigenvalues of the stationary point. Proposition 10. Suppose that the unperturbed system is as in Lemma 1 and suppose the system has been transformed using Proposition 9. Then given any ¿ 0, there exists a smooth time reparameterization -(t; ; ) such that for the linearization of the unperturbed system at the equilibrium, (0; 0), has eigenvalues ±. Proof. Here we simply multiply the Hamiltonian function by =(; ). Since H is assumed to be C k -smooth, (; ) depends C k -smoothly on and , see [8]. Proposition 11. Suppose that the unperturbed system is as in Lemma 1 and assume that the system has been transformed using Propositions 9 and 10. Then there exists
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1067
a neighborhood of U of (0; 0) and an, area preserving, time dependent, C ∞ , change of coordinates (·; ; ) : U → U , such that under the transformation the coordinate axes are invariant. Proof. From the construction in the preceding proof, the normally hyperbolic manifold M has stable and unstable manifolds W s and W u which are C k -smooth in all variables. Thus, we may produce a change of variables (q; p; ; ) → (x; y; ; ) so that the stable manifold coincides with the y = 0 plane and the unstable manifold coincides with the x = 0 plane. We need only to show that this transformation can be accomplished canonically. First, for each (; ) let A; ∈ SL(2) be the area preserving linear transformation, with eigenvalues {1; 1} that puts the stable and unstable manifolds tangent to the x = 0 and y = 0 planes along the line (0; 0; ). This is canonical by de7nition. Next consider that W s is given as the graph of a function (x; ) ∈ C k . Consider the transformation xP = x; yP = y − (x; ):
(36)
This transformation is clearly canonical and makes the plane x = 0 invariant. Finally, we repeat the above procedure for W u . Note that up to this point the coordinate changes have been both canonical and as smooth as the original system. Now in the 7nal step, we employ a variant Hartman’s construction [7] to linearize the 6ow on the whole neighborhood of 0. This step may be only C 1 -smooth. An example given by Belitskii [3] shows that it might be no more than C 1 -smooth. Proposition 12. Suppose that the unperturbed system is as in Lemma 1 and assume that the system has been transformed using Propositions 9–11. Then there exists a neighborhood of U of (0; 0) and an area preserving, time dependent, C 1 , change of coordinates (·; ; ) : U → U , such that under the transformation the @ow on U is given by (6). The change of variables is smooth away from the coordinate axes. Proof. Consider the extended system. Let %0 = {(x; y; ) ∈ U × R : y = d} and de7ne %t = t (%0 ) ∩ U;
0 6 t ¡ + ∞:
Let y(t) P = de−t . Then {%t } is a smooth foliation of {0 ¡ y 6 d}. We may represent this foliation by y = yP + (x; y); P where is a smooth function and where the coordinate yP associated with %t we take to be y(t). P Extend the foliation to W u by de7ning (x; 0) = 0. It is at W u that the foliation might not be more than C 1 -smooth.
1068
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
Let (x; P y) P be new coordinates de7ned by the generating function S(x; y; P ; ) = xyP +
(x; y; P ; );
where @ = : @x The resulting transformation x = xP +
@ (x; y); P @yP
y = yP + (x; y) P
(37)
‘straightens’ each of the surfaces %t . This produces coordinates in which the 6ow in linear in the x-direction. Since the transformation is area preserving and the 6ow is Hamiltonian, the resulting 6ow is in fact linear in the y-direction as well. Now we need only show that the transformation is C 1 -smooth at W u . Since we straightened the stable and unstable manifolds, the equations of motion near the equilibrium are x˙ = Hy (x; y; ) = ( + f(x; y; ))x; y˙ = −Hx (x; y; ) = −( + g(x; y; ))y; ˙ = :
(38)
Here f and g are smooth functions that are zero at (0; 0; ). Let the surface %0 (as above) be given by the graph of y0 = <(x; ; 0) and let the graph of yt = <(x; ; t) be %t as above. Then < must satisfy x˙ = ( + f(x; <; ))x; <˙ = −( + g(x; <; ))<; ˙ = :
(39)
Clearly, provided that we restrict to a small enough neighborhood of (0; 0) then <(t) ¡ K0 e−(+*0 ) <(0); where K ¿ 0 is a constant and |*0 | is small in comparison to . Now consider derivatives of <, which must satisfy x˙ = ( + f(x; <; ))x; <˙ = −( + g(x; <; ) + gy (x; <; )<)< + g (x; <; )<; ˙ = :
(40)
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1069
Here the last term approaches zero exponentially. The term gy (x; <; )< also approaches zero exponentially, so < approaches zero. Lastly,
(41)
Once again this implies exponential decrease of |
Further reading S. Aranson, G. Belitskii, E. Zhuzhoma, Introduction to the qualitative theory of dynamical systems on surfaces, in: Translations of Mathematical Monographs, Vol. 153, American Mathematical Society, Providence, RI, 1996. V.I. Arnol’d, Geometrical methods in the theory of ordinary diMerential equations, Grundlehren der mathematischen Wissenschaften, Vol. 250, 2nd Edition, Springer, New York, 1988.
1070
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
D. Diminnie, R. Haberman, Slow passage through a saddle-center bifurcation, J. Nonlinear Sci. 10 (2000) 197–221. N. Lebovitz, Bifurcation and unfolding in systems with two timescales, Ann. New York Acad. Sci. 617 (1990) 73–86. N. Lebovitz, A. Pesci, Dynamic bifurcation in Hamiltonian systems with one degree of freedom, SIAM J. Appl. Math. 53 (1995) 1117–1133. A. Neishtadt, On passage through resonance in the problem with two frequencies, Soviet Math. Dokl. 22 1975.
www.elsevier.com/locate/na
A geometric proof of separatrix crossing results Shui-Nee Chowa;∗ , Todd Youngb a School
of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA of Mathematics, Ohio University, Athens, OH 45701-2979, USA
b Department
Received 1 August 2003; accepted 1 November 2003
Abstract We investigate the adiabatic invariance of the action variable of one degree of freedom time-dependent Hamiltonian systems when trajectories cross a separatrix and give relatively simple, geometrically motivated proofs. ? 2003 Elsevier Ltd. All rights reserved.
1. Introduction Consider a family of one degree of freedom Hamiltonian systems, with Hamiltonian function H (q; p; ), where ∈ [0; 1] is a parameter. By the drifting or perturbed system we mean the nonautonomous 6ow on R2 de7ned by q˙ = Hp (q; p; ); p˙ = −Hq (q; p; ); ˙ = ;
(1)
where ¿ 0 is a small parameter. The case = 0 we call the static or unperturbed system. Denote by X (·; t; ) the time-dependent vector 7eld de7ned by (1) and use
(·; t; ) to denote the 6ow generated by (1). We will consider H such that each the static system possesses a hyperbolic equilibrium and associated separatrix cycle which depends on . We study orbits of the perturbed system that cross the separatrix cycle. For = 0, it is well known that for ‘typical’ H , most trajectories of (1) are periodic. It is a classical theorem of Liouville that in regions where all the solutions are periodic one may introduce canonical ‘action-angle’ variables. In these coordinates, the action ∗
Corresponding author. E-mail addresses: [email protected] (S.-N. Chow), [email protected] (T. Young).
0362-546X/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2003.11.002
1048
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
variables are constant and the angle variables proceed linearly. The action variable in R2 is simply I (q; p; ) =
1 A(q; p; ); 2
(2)
where A(q; p; ) is the area in the plane enclosed by the periodic orbit. At integral levels where solutions are not periodic, the action-angle variables are not de7ned. This typically occurs at the energy level set of a hyperbolic equilibrium. Such a level set separates regions in which the action-angle variables are well de7ned and so is called a separatrix. In general, the action variable experiences a jump discontinuity across a separatrix curve. In the drifting system, solutions are not periodic and so the action-angle variables must be de7ned in terms of the static system. In the drifting system, the action variable is an adiabatic invariant, which means that it is constant to order O() for time periods of the order O(1=). However, solutions of the perturbed system may happen to cross a separatrix and the action variable experiences a jump due to the discontinuity in the de7nition of the action. Further, the techniques used to show that the action is an adiabatic invariant are only valid for solutions which are bounded away from a separatrix. Systems with separatrix crossings were studied extensively by a number of researchers, notably, [1,4–6,8–12]. These studies all rely on extending the averaging method to a neighborhood of the separatrix. They are not presented as rigorous results, but as careful estimates. The most complete derivation seems to appear in [8]. They found that for all but a set of initial condition of size not more than O(e−1= ) the action at the separatrix crossing experiences not only a jump caused by the discontinuity, but also a small shift which depends on the ‘phase’ at which the crossing occurs. For small, this phase is extremely sensitive to initial conditions, so for practical purposes one may treat the phase as a random quantity, uniformly distributed at each point in the space of initial conditions. There is however disagreement about the size of the small jump; some estimate that is of order O(), while others assert that it is O( ln −1 ). The excluded set of order O(e−1= ) in this theory corresponds to those initial conditions which pass too closely to the hyperbolic equilibrium. The mechanism for crossing a separatrix is the following. Suppose that the system has separatrices in a symmetric ‘Fig. 8’ con7guration. Let Y () be 1=2 times the area of one of the lobes enclosed by a separatrix. We will suppose that Y () is an increasing function, so that the area enclosed by the lobes increases in time. Since the system is Hamiltonian, area in phase space is preserved and it follows that the orbits of a positive measure set of initial conditions must enter the lobes. Example 1. Polynomial double-well potential: H (q; p; ) =
p2 q2 q4 − + : 2 2 4
(3)
As ¿ 0 changes, the size of the wells changes and some orbits will be captured or escape from the wells. The separatrix is the energy level of the unstable equilibrium at (0; 0).
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1049
Example 2. Pendulum with slowly varying length: H (q; p; ‘) =
p2 − ‘g sin2 (q): 2‘2
(4)
Here the length of the pendulum ‘ may serve as a parameter. If ‘ is decreased, energy is put into the system. If ‘ becomes suJciently small then the motion will change from back and forth oscillations to spinning completely over. In particular, suppose that an initial condition (q0 ; p0 ) is outside the separatrices. Let I0 = I (q0 ; p0 ; 0). Suppose further that the total area enclosed by the separatrices at some time becomes equal to I0 . Since I is approximately preserved, it follows that the solution at that time will be in some vicinity of the separatrices. The parameter value ∗ at which 2Y (∗ ) = I0 is called the pseudo-crossing parameter. Unless the solution comes very close to the equilibrium, it will enter one of the lobes and the action variable will jump to approximately I0 =2. The separatrix will then continue to expand, but the action variable will again be nearly constant once the solution is suJciently far from the separatrix. The 7nal value of the action variable will be I0 =2 plus a small correction LI which we call the shift. According to Cary et al. [5] up to order 2 , the correction term depends on a phase variable s and has the remarkably simple form LI ≈ −
dY ln |2 sin( s)|; ! d
0 ¡ s ¡ 1;
(5)
provided the Hamiltonian has the form !(p2 − q2 )=2 + H3 (q; p; ): Cary et al. [5] also showed that any slowly varying Hamiltonian system with a hyperbolic equilibrium can be put into this form up to any order of . In (5), the derivative of Y with respect to is assumed to be constant. The results of other studies diMer from (5). Most notably, some estimates possess terms of order O( ln −1 ). Our analysis explains this discrepancy in the order of LI . In this paper, we formulate and prove various rigorous results about this phenomenon based on geometric arguments rather than the averaging method. We de7ne a phase variable 0 ¡ ¡ 1 that controls whether or not the orbit is captured, and the size of the shift. Using a relatively simple, geometric argument we show the following results. Theorem 1. The measure of the set of initial conditions that are not captured is of order −1= e O : 2 Theorem 2. Given any 0 ¡ ¡ 1, LI 1 dY ∗ → (Sc () + S()) d as 0, where Sc () = −2 − ln − ln(1 − ) and S() is a bounded function.
1050
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
We 7nd numerically that |S()| is less than 1. These results have 7xed and → 0. If however, one 7xes, small and lets vary, there is a small range of values close to either 0 or 1 where the shift is of order O( ln −1 ), thus clarifying the ambiguity in the theory (Section 4.5). 2. Local linearization and consequences 2.1. Linearization We will assume that H depends C k -smoothly on all variables, where k ¿ 3, k = ∞, or, k = ! (analytic). The key ingredient in our treatment of separatrix crossing is the following linearization lemma. Lemma 1. Suppose that H ∈ C k (R2 × [0; 1]; R) where k = 3; : : : ; ∞; !. Suppose that for each , the unperturbed system has a hyperbolic equilibrium at (q∗ ; p∗ ) and Df((q∗ ; p∗ ); ) has eigenvalues ±(), where () is uniformly bounded away from zero. Then there exist neighborhoods U of (q∗ ; p∗ ) such that on U there exists an, area preserving, time dependent, C 1 -smooth, change of coordinates (·; ; ) : U → R2 , along with a reparameterization of time, that puts system (1) into the form x˙ = x; y˙ = −y:
(6)
Here may be taken to be any positive number. Further, the image of the parameterdependent neighborhood U under may be taken to be a uniform neighborhood U of (0; 0). We will postpone the proof of this lemma until Section 5. Note that past works, in order to use averaging, have employed analytic changes of variable, that were not necessarily canonical, thus ignoring the main geometric feature of the problem. Let Ws and Wu denote the cross-sections of the local stable and unstable manifolds of the equilibrium. We will de7ne these manifolds precisely in Section 5. In the linearized coordinates Ws and Wu coincide with the coordinate axes. A by-product of the proof of Lemma 1 is the following. Lemma 2. Ws and Wu converge to the local separatrices as → 0. Further, the convergence is uniform in the C k topology. By the C k topology we mean the topology of the functions whose graphs are the local manifolds, including dependence on . 2.2. Return maps Let U be as in Lemma 1. Fix d ¿ 0 so that the square D bounded by the lines {x = d}, {x = −d}, {y = d}, and {y = −d} is contained in the interior of U . We will
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1051
Lobe 2 U Σ4 D Σ1
Σ3
Σ2
Lobe 1
Fig. 1. Schematic drawing of the separatrix and a neighborhood of the equilibrium for = 0. Inside U the 6ow is linearized. We consider 7rst-hit maps de7ned by the 6ow between the boundaries of a square D ⊂ U , i.e. %i , i = 1; : : : ; 4. In the linearized coordinates all the 7rst-hit maps are isomorphisms, even for the drifting system ( ¿ 0). This is a consequence of the linearization and the symplectic character of the 6ow.
assume that the subsquare in D given by x ¿ 0, y ¿ 0 is contained in lobe 1 and the subsquare given by x ¡ 0, y ¡ 0 is contained in lobe 2. Now consider the maps between the boundaries of D induced by the 6ow. We denote the boundaries of D by {%i } where %1 = {x = d} ∩ D, %2 = {y = −d} ∩ D, the numbering continues in the clockwise direction, see Fig. 1. In these coordinates, the local maps are trivial. In particular, consider a point on %4 given by (x; d) with x ¿ 0. Under the 6ow, the orbit of this point 7rst intersects %1 exactly at (d; x). A point on %4 with x ¡ 0 is mapped to (−d; −|x|). The local map starting at %4 is similar. We may summarize the local maps as follows: L2 : %2 → %1 ∪ %3 : (x; −d) → (sign(x)d; −|x|); L4 : %4 → %1 ∪ %3 : (x; d) → (sign(x)d; |x|):
(7)
Along with the local map, one also deduces from (6) that the local transit time for a point starting at (x; ± d) is exactly T‘ (x) =
d 1 ln : |x|
(8)
1052
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
The 6ow outside D induces smooth 7rst-hit maps G1 : %1 → %2 and G3 : %3 → %4 . For the unperturbed system conservation of energy implies that, in the local coordinates, G1 and G2 are isomorphisms. In particular G1 : (d; y) → (−y; −d); G3 : (−d; y) → (−y; d)
(9)
for = 0. On the other hand, for ¿ 0, the global maps might perhaps depend on y, and/or . The next proposition follows easily from the symplectic nature of the 6ow and the linearization. Proposition 1. The maps G1 and G2 have the form G1 : (d; y) → (−y + *; −d); G3 : (−d; y) → (−y − *; d);
(10)
where * = *(; ). Proof. Note from the linear form of the Hamiltonian in U that the instantaneous 6ux across each of the boundaries of D is a constant, namely d. By instantaneous 6ux, we mean X · n, P where nP is a normal vector to the boundary. Now consider a small line segment , in %1 connecting arbitrary points (d; y1 ) and (d; y2 ). For a small time interval -, let R- ={ t (,) : −- 6 t 6 0}. For each point xP ∈ Rlet y( ˜ x) P denote the y-coordinate of the point on , in the local orbit of x. P Next let Tg (y) be the 7rst value of time for which Tg (y) (d; y) ∈ %2 , i.e. the time of transition for the global map. Now let P (x) P : xP ∈ R- }: R- = { Tg (y(˜ x))
The integral invariant of PoincarRe–Cartan [2], i.e. the form p dq − H dt, applied to Rgiven us that the areas A and A , of R and R , respectively, are related by A = A − H dt; /
where / is a curve in R2 × R corresponding to the boundary of R and its transition time coordinate. We partition / into four parts, namely the image of , which we denote by , , the backward image of , under the 6ow by −-, we denote this by ,- and two ‘end’ segments. The line integrals of H d- across , and ,−exactly cancel since H is constant along 6ow lines in these local coordinates and dt on , will be exactly the negative of dt on corresponding points on ,−. Further, - is constant on the ends, so dt ≡ 0 there. Thus, we have A = A . Note that the ‘thicknesses’ of both R and R are exactly the same namely d exp(-). Taking the limit as - 0 shows that the length of the image , is exactly the same as the length of ,. The time of global transition, Tg , from %1 to %2 and from %3 to %4 might depend on y, , and . Standard results on the smooth dependence on parameters of 7nite time
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1053
solutions of initial value problems imply that Tg (y; ; ) → Tg (y; ; 0) uniformly in the C k topology as → 0. 3. Capture of orbits 3.1. Size of initial conditions It is known that the classical adiabatic invariance of the actions can extended to an -dependent neighborhood of the separatrix. In particular, the adiabatic invariance holds as long as ln h−1 h is small [8]. In light of this, we can assume that our initial conditions satisfy h ≈ O(1 ); where 0 ¡ 11. The initial conditions we study will begin on %4 with x-coordinate approximately x∗ =1 . We will see that this implies that the relevant parameter intervals that we must consider becomes arbitrarily small as 0. This will allow us to prove two key simplifying facts: the values of dY=d and Tg (x; ) are arbitrarily close to constants. 3.2. Expansion of the separatrices Next we determine *. Note that in the previous section, we have not proved that * is constant in time only that * is a constant for points starting at %1 at the same time. We denote this constant by *(; ). We now show that * is directly proportional to the rate of expansion of the separatrices. Recall that Y () denotes 1=2 times the area of each lobe. Proposition 2. The following limit holds in C k ([0; 1]): *(; ) 2 dY → () d d as → 0. Proof. Let Ws denote the stable manifold of the equilibrium at parameter . Consider the branch of Ws that 7rst intersects %4 starting from the equilibrium and the segment of that branch from the equilibrium and to the 7rst intersection with %1 . Call this segment W . Consider the region bounded by W , the line segment from (0; 0) to (d; 0), and the segment, ,1 along %1 from (d; 0) to W , see Fig. 2. The area enclosed by this region is approximately the area enclosed by the separatrix lobe. The region is not invariant in t, but orbits may cross the boundary only at ,1 . Note that the length of ,1 is precisely *(; ). Note that the 6ux per unit length across the boundary of ,1
1054
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
Fig. 2. The region enclosed by W , ,1 , and the line segment from the origin to (d; 0) has approximately the same area as lobe 1 for the unperturbed system. Solutions may enter this region only by passing through the segment ,1 .
is a constant, namely d. Thus, the total 6ux across the segment ,1 is d*. Let Y˜ () denote 1=2 times the area of the region, then we have that 2 d Y˜ *(; ) ≡ : d d Now we note that as Y˜ → Y as → 0. In fact, from Lemma 2, we have that W converges uniformly in the C k topology to the corresponding segment of the separatrix as → 0. Thus Y˜ () → Y () in C k ([0; 1]). We will suppose that an orbit begins on %4 with x-coordinate x−n . We will label each successive intersection with %2 or %4 by xi , where xi is the absolute value of the x-coordinate on %2 or %4 . Along with {xi } we have an associated sequence of times {ti } and parameter values {i }. Following each intersection xi , we have an intersection with %1 ∪%3 which we denote by yi , where yi is the absolute value of the y-coordinate of the intersection point. Denote the corresponding intersection times by {tPi } and parameter values by {Pi }. We will denote by x0 the last intersection of the orbit with %2 or %4 before the crossing. We will continue following the orbit up to step x n+1 . Below we will let n be the minimal integer such that x−n ¿ x∗ ≡ 1 . With this convention, we have that yi = xi for all i. Also we have that xi+1 = xi − *(tPi )
for i ¡ 0
(11)
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1055
and xi+1 = xi + *(tPi )
for 1 6 i:
Now we de7ne our phase parameter, , by y0 x0 = = : *(tP0 ) *(tP0 )
(12) (13)
Note that G1 at time tP0 maps (d; *(tP0 )) to (0; −d). Proposition 1 further implies that x1 = *(tP0 ) − x0 , so x1 1−= : (14) *(tP0 ) We consider the orbit in three parts, the approach, the crossing and the departure. By the approach we mean the transition from x−n ≈ 1 to x0 . By crossing, we mean the segment of the orbit from x0 to y1 . We call the transition from y1 to yn+1 the departure. Note that we have the same number of steps in the approach and departure. 3.3. Approach time of
First, we give bounds on n. Let a and b be strict lower and upper bounds on values 2 dY d d
in the entire parameter interval under consideration. Then from Proposition 2, we have that *(; y) a¡ ¡b (15) for small. From (11) we have n *(x−i ; P−i ) = x−n − x0 ≈ 1 : (16) i=1
Combining (15) and (16) we obtain na ¡ 1 ¡ nb: We thus have the following. Proposition 3. For all 0 6 6 1, the number of steps n needed for the approach satis
n−1 n−1 1 d ln + Tg (xi ; i ): xi i=0
i=0
1056
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
Recall that 0 is the parameter value at the last crossing of @D before the crossing begins. Proposition 4. If we suppose that n is the minimal integer such that x−n ¿ x∗ ≡ 1 , then 0 − −n → 0 as → 0. This limit is uniform in . Proof. Let TP g be the maximum value of Tg (x; ). Let * be the minimum value of *(y; ). Then the approach time t0 − t−n satis7es t0 − tn 6
n 1 d ln + nTP g x0 + i* i=1
6
n d 1 ln + nTP g *(i + x0 =*) i=1
n n d 1 x0 + nTP g 6 ln − ln i + * * i=1
6
n d ln + nTP g : *
(17)
Therefore 0 − −n 6
n d ln + nTP g : *
Propositions 2 and 3 imply that the right-hand side of the above equation goes to zero as → 0. Since the domains shrink to zero as → 0, the functions *(y; ) and Tg (y; ) are approximately constant during the approach. Thus, we may replace them by constants in the rest of this section. We have then the following simpli7cation of the approach time: t0 − t−n =
n 1 d + nTg ln x0 + i* i=1
=
n 1 d ln + nTg *( + i) i=1
=
n 1 d 1 ln − ln(i + ) + nTg : * i=1
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1057
We may bound the sum in the last expression by an integral and obtain the estimate t0 − t−n ¡
1 d n+ n+ 1 ln − ln(n + ) + − + nTg : *
(18)
3.4. Value of the action away from the separatrices Let h denote the value of the Hamiltonian. It is well known that T dI = ; dh 2 where T is the period of the periodic orbit for the unperturbed system. Inside the 7rst lobe consider orbits of the unperturbed system that intersect %4 at a point (x; −d). Since h = xy inside U , we have d dI T (x; ): =− 2 dx The time spent by such an orbit inside D is given exactly by (8) and the time spent outside D is the function Tg (x; ). Thus, dI d d d = − ln − Tg (x; ): dx 2 |x| 2 It follows that the action variable of the periodic orbit starting at (x; ±d) on %2 or %4 inside one of the lobes is given by d d x xd 1 + ln − Tg (x; ) d x: (19) I1; 2 = Y () − 2 |x| 2 0 Similarly, the action for an orbit outside the separatrices starting at (x; d) on %4 is given by d x I3 1 d + Tg (x; ) d x: (20) = Y () + xd 1 + ln 2 2 x 2 0 For the perturbed system, we will only measure the instantaneous action when orbits cross %2 or %4 and use the corresponding orbit of the unperturbed system. Because of the variable change, the point (x; d) for the drifting system is not exactly (x; d) for the static system. Since the assignment of the action variable to the perturbed system is not unique and we are only interested in measuring the change in the action this ambiguity is not important. For discussion of higher-order corrections to the assignment of the action variable to the perturbed system, see [8].
1058
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
3.5. Psuedo-crossing parameter ∗ The action of the initial condition x−n ¿ 0 satis7es I3 d d x−n x−n d 1 + ln (x−n ) = Y (−n ) + + Tg (x; 0) d x: 2 x−n 2 0 2 x The integral, 0 −n Tg (x; 0) d x, we denote by TP g . The pseudo-crossing parameter ∗ associated with x−n is de7ned by 2Y (∗ )=I3 (x−n ). It follows that ∗ satis7es ∗ x−n d d d P dY d = 1 + ln T g: + d 2 x 2 −n −n Let m denote the smallest value of dY=d on [0; 1]. Then we have d d P x−n d 1 + ln T g: + m(∗ − −n ) 6 2 x−n 2 Let *P = (x−n − x0 )=n, note that a 6 *P 6 b where a and b are, respectively, the minimum and maximum values of *. Thus, d P d d P + 1 + ln ∗ − −n 6 (x0 + *n) Tg P 2 m 2 x−n + *n d d P d P *( + n) 1 + ln − ln + n + T g: = P 2 m 2 * P → 0 as → 0. Thus, we have the following. Using Propositions 2 and 3, we see that *n Proposition 5. Suppose that ∗ is the pseudo-crossing parameter for x−n , then ∗ − −n → 0 as → 0. This limit is uniform in . In light of this result, we may treat Tg and dY=d (as well as *) as constants on (−n ; ∗ ). 3.6. Crossing time We say that the crossing is the transition from x0 to y1 . The time required for the crossing is thus tP1 − t0 =
1 1 d d ln + ln + Tg x0 x1
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
=
1 d 1 d ln + ln + Tg * (1 − )*
=
2 d 1 ln − ln (1 − ) + Tg : *
1059
(21)
Here * is exactly *(P0 ) and Tg is Tg (y0 ; P0 ). Recall that P1 = tP1 , then the crossing occurs between 0 and P1 . Eq. (21) implies: Proposition 6. As → 0 P1 () − 0 () → 0 for each . Note that this limit is not uniform in . Thus, we see that is eMectively constant during the crossing itself for most . Thus 0 , which we can call the actual crossing parameter, coincides with ∗ . Note that we have not proved that if we de7ne ∗ as the pseudo-crossing parameter for some 7xed x∗ and let 0 be parameter value at the last hit before crossing, then 0 → ∗ as → 0. That is, for a 7xed x∗ we have not shown that the actual crossing parameter 0 converges to the pseudo-crossing parameter. However, an area preservation argument shows that this is true. 3.7. Departure time By the departure we mean the transition from y1 to yn+1 . The next result follows from arguments identical to those for Proposition 4. Proposition 7. Pn − P1 → 0 as → 0. This limit is uniform in . 3.8. Capture and the excluded set We can say that an orbit is captured into one of the separatrices if it reaches yn+1 within a parameter of some 7xed length, say L. From the previous sections, the parameter intervals required for both the approach and departure are negligible. Thus, only the crossing itself contributes to the overall parameter interval. In other words, capture occurs if P1 − 0 ¡ L:
1060
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
Eq. (21) implies that capture occurs if L 2 d 1 ln − ln (1 − ) + Tg ¡ : * Solving, this is equivalent to (1 − ) ¿ eTg
d2 −L= e : *2
The value of * is of order , so that orbits are captured provided (1 − ) ¿ K
e−L= : 2
(22)
Thus, only an exponentially small set of initial conditions, corresponding to either very small, or 1 − very small, fails to cross and we have proved Theorem 1.
4. Estimating the change of the action We now consider the change of the action as the orbit moves from x−n to yn+1 . We only consider points which make this transition in a relatively short parameter interval as in the previous section. Thus, we may treat dY=d and Tg as constants throughout this section. We also treat * as a constant. Outside the separatrices, we consider I3 =2 and we consider I1 , I2 inside lobes 1 and 2. In the symmetric case, this combination of functions is continuous across the separatrix. 4.1. Approach to the crossing We ignore the change in the action from x−n to y−n . The change in the action variable from y−i to y−i+1 , 1 6 i ¡ n is given by 7−i = 12 I3 (y−i+1 ; tP−i+1 ) − 12 I3 (y−i ; tP−i ); This is equal to
1 6 i 6 n:
dY d d + y−i+1 1 + Tg + ln d 2 y−i+1 d d − y−i 1 + Tg + ln 2 y−i dY d 1 d d = Tg + ln + (y−i − *) 1 + Tg + ln d y−i+1 2 y−i+1 d d − y−i 1 + Tg + ln y−i 2
7−i = (tP−i+1 − tP−i )
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1061
d* d 1 d 1 + Tg + ln − ln y−i+1 y−i+1 2 d d d + y−i+1 ln : − ln 2 y−i+1 y−i
=
dY d
Tg +
Using Proposition 2, we obtain 7−i = −
d dY y−i y−i+1 ln + : 2 y−i+1 d
Noting that y−i = y0 + i* and = y0 =* we obtain for 1 6 i 6 n i+ dY −1 + (i + ) ln : 7−i = d i−1+
(23)
It can be shown that the terms of this sequence are of order O(1=i) for i large. 4.2. The crossing Next, we calculate the change in the action from y0 to y1 : 1 70 ≡ I1; 2 (y1 ; P1 ) − I3 (y0 ; P0 ) 2 d d dY d d P P − y1 1 + Tg + ln y0 1 + Tg + ln − : = (t 1 − t 0 ) d 2 y1 2 y0 Noting that y1 = x1 = * − y1 = * − x1 and = y0 =*, we obtain that 1− dY −1 − ln : 70 = d
(24)
4.3. Departure from the crossing The change in the action variable from yi to yi+1 , i ¿ 1 is given by d d dY d d P P − yi+1 1 + Tg + ln yi 1 + Tg + ln 7i = (t i+1 − t i ) + d 2 yi+1 2 yi 1 dY d d d = + Tg ln − (yi + *) 1 + Tg + ln yi+1 d 2 yi+1 d d + yi 1 + Tg + ln : 2 yi Thus, 7i = −
dY yi + * d : + yi ln d 2 yi
1062
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
Noting that yn+1+i = y1 + i and 1 − = x1 =* we obtain for 1 6 i 6 n i+1− dY : −1 + (i − ) ln 7i = i− d
(25)
The terms of this sequence are of order O(1=i) for i large. 4.4. Total shift in the action Combining (24) with the leading terms (i = 1) of (23) and (25) we obtain dY (−3 − ln − ln(1 − ) + (1 + ) ln(1 + ) + (1 − ) ln(2 − )): d The most important term of this is −ln − ln(1 − ). Noting that 1 −ln − ln(1 − ) d = 2; 0
de7ne Sc () = −2 − ln − ln(1 − ): Further, de7ne Sa (; n) = −1 + (1 + ) ln(1 + )
n
−1 + (i + ) ln
i=2
and Sd (; n) = (1 − ) ln(2 − ) +
n
i+ i−1+
−1 + (i − ) ln
i+1− i−
1
1
i=2
Numerically, we 7nd that both of the integrals total shift in the action is given by dY LI = (Sa + Sc + Sd ): d
0
Sa d and
0
:
Sd d are zero. The (26)
Using integral estimates it can be shown that both Sa and Sd are of order O(ln n). However, Sa is positive and Sd is negative. Asymptotically, the terms of the sums are negatives of each other, thus we expect signi7cant cancellation to occur. In fact, a simple calculation shows that the ith term of Sa (; n) + Sd (; n) is of order O(i−2 ), uniformly in . Thus we have Proposition 8. There exists a continuous function S() such that the following limit lim Sa (; n) + Sd (; n) = S()
n→∞
exists uniformly in . We have now proved Theorem 2.
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1063
12 Sa Sd Sa + Sd Sa + Sc + Sd
10 8
shift
6 4 2 0 −2 −4 −6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
φ
Fig. 3. Numerically computed shift functions Sa (; n), Sd (; n), Sa + Sd , and the total shift function Sa + Sc + Sd . In these plots n was taken to be 50,000. As n increases, Sa and Sd are of order O(ln n). However, Sd (; n) + Sa (; n) is relatively small and in fact Sd (; n) + Sa (; n) approaches a limiting function S() as n → ∞.
See Fig. 3. Using high precision arithmetic, we 7nd that |Sa () + Sd ()| ¡ 0:2 for n from 10 up to 109 . Since Sa (; n) and Sd (; n) are of order ln n, we have that for any 8, 0 ¡ 8 ¡ 1, |Sa (; n) − Sa (; n8 )| ≈ 1 − 8: |Sa (; n)| Thus, most of the weight of Sa and Sd belongs to the 7rst n8 terms, i.e. the terms corresponding to the portion of the orbit closest to the crossing. Thus we see that for small, the total shift is well approximated by LI =
dY ∗ (Sc () + S()): d
(27)
If (1 − ) is very small, then 0 and P1 may diMer signi7cantly. In that case our assumptions of this section do not hold. We treat this case in the next section. 4.5. The case (1 − ) small Suppose that the diMerence between 0 and P1 is signi7cant if it is greater than some number c ¿ 0. Repeating the calculation of Section 3.8, we 7nd that P0 − 0 ¿ c if (1 − ) ¡ K
e−c= ; 2
1064
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
where the constant K is the same as in Eq. (22). Thus, the analysis of the previous section is not valid for K
e−c= e−L= ; ¡ (1 − ) ¡ K 2 2
(28)
but initial conditions with in this range are captured. We will produce an estimate of the shift in the action for such . First, we observe that only one of or 1 − may be small. We will only treat the case where is small; the other case is similar. If is small, then x0 is small and the passage from x0 to y0 requires a long time. The previous analysis is valid for the transition from y−n to y−1 and from y0 to yn+1 . We need only to re-examine the transition from y−1 to y0 . The shift in this transition is given exactly by 1 1 I3 (y0 ; P0 ) − I3 (y−1 ; P−1 ) 2 2 d d d d + = Y (P0 ) − Y (P−1 ) + y0 1 + ln y−1 1 + ln 2 y0 2 y−1 y0 y−1 d + Tg (x; P0 ) d x − Tg (x; P−1 ) d x 2 0 0 d dy0 y0 + * d* P P 1 + ln = Y (0 ) − Y (−1 ) + − ln 2 y0 2 y0 + * y0 y−1 d + Tg (x; P0 ) d x − Tg (x; P−1 ) d x 2 0 0 dY P d = Y (P0 ) − Y (P−1 ) + (−1 ) − ln + (1 + ) ln(1 + ) − 1 + ln d * y0 y−1 d + Tg (x; P0 ) d x − Tg (x; P−1 ) d x : 2 0 0
7−1 =
In this calculation, *=*(P−1 ). Note that we may replace P−1 by 0 . The entire approach plus the 7rst half of the crossing, i.e. from y−n to y0 is summarized by dY d 7a = (0 ) − ln + ln + Sa (; n) d * y0 y−1 d + Tg (x; P0 ) d x − Tg (x; 0 ) d x : (29) 2 0 0 For the transition from y0 to yn+1 , the parameter value is approximately P1 . The shift due to this transition may calculated to be 7d =
dY P (1 )(−2 + ln − ln(1 − ) + Sd (; n)): d
(30)
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1065
Combining (29) and (30), we get the total shift in the case is small. Since the coeJcients of Sa and Sd are diMerent, the cancellation does not occur. Since both Sa and Sd are of order O( ln −1 ), the correction term has this order, rather than order O().
5. Proof of local linearization We prove Lemma 1 via several propositions; 7rst we 7x the unstable equilibrium at the origin. Proposition 9. Suppose that the unperturbed system is as in Lemma 1. Then there exists a neighborhood of U of (q∗ ; p∗ ) and an area preserving, time dependent, C k -smooth, change of coordinates (·; ; ) : U → U , where U is a
pP = p − p∗∗ (; ):
(31)
For the unperturbed system, simply taking q∗∗ (; 0) = q∗ and p∗∗ (; 0) = p∗ 7xes the equilibrium at (0; 0) for each . However, since the change of variables depends on , it is time dependent for system (1). It is easy to check that the change of variables (31) is generated by the generating function S(q; p) P = (pP + p∗∗ (; ))(q − q∗∗ (; )): For a time-dependent change of variable, the new Hamiltonian is given by H1 (q; P p) P +
@ S; @t
where H1 (q; P p) P = H (q; p) and thus ∇H1 (0; 0) = 0, i.e. (0; 0) is an equilibrium point for H1 . However, the second term, (@=@t)S depends on the derivatives of (q∗∗ ; p∗∗ ). For (31) applied to (1) this results in the system of equations dq∗∗ @H1 qP˙ = − ; @pP d ∗∗
dp @H1 − : pP˙ = − @qP d Thus we seek C k smooth functions (q∗∗ (; ); p∗∗ (; )) so that
dq∗∗ @H ∗∗ ∗∗ = (q ; p ; ); d @pP
(32)
1066
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
dp∗∗ @H ∗∗ ∗∗ = (q ; p ; ); @qP d
(33)
where we require q∗∗ (; 0) = q∗
and
p∗∗ (; 0) = p∗ :
(34)
In the analytic category, there exist a unique solution to (33) and (34) [8]. For k = 3; : : : ; ∞, and 7xed, there is a solution of (33) for any initial data (q∗∗ (0; ); p∗∗ (0; )). To 7nd a smooth family of such solutions which also satisfy (34) is more delicate. Consider the rescaling = t. This leads to the original time-dependent equations in the following form: dq∗∗ @H ∗∗ ∗∗ (q ; p ; t); = @pP dt dp∗∗ @H ∗∗ ∗∗ (q ; p ; t): = dt @qP
(35)
Note that for = 0 the curve segment given by (q∗ ; p∗ ; ) is normally hyperbolic. It is well known that normally hyperbolic manifolds persist under smooth perturbations. In the present case, we may not apply this principle directly since the perturbation is time dependent. If we try to treat the time-dependent system as autonomous, by making t the third variable, then the original region considered in the problem is not invariant (since this time interval is 7nite). We may circumvent these diJculties by extending the time direction of the phase space to (−∞; ∞). Given k we may C k -smoothly extend H (q; p; ) to ∈ R in such a way that the system de7ned by H has a hyperbolic equilibrium (q∗ ; p∗ ) for each ∈ R. In fact, we can take (q∗ ; p∗ ) to be 7xed beyond a neighborhood of [0; 1] and we can make the system linear on a neighborhood of (q∗ ; p∗ ) for outside a neighborhood of [0; 1]. Now let M = {(q∗ ; p∗ ; ) : ∈ R}. Clearly, M is a uniform normally hyperbolic manifold when =0. In the extended system this manifold persists, is C k -smooth and depends C k -smoothly on the perturbation, i.e. on . Let (q∗∗ (); p∗∗ (); ) be the coordinates of the perturbed manifold. This (q∗∗ (); p∗∗ ()) satis7es both (33) and (34). Next we perform a time change which 7xes the eigenvalues of the stationary point. Proposition 10. Suppose that the unperturbed system is as in Lemma 1 and suppose the system has been transformed using Proposition 9. Then given any ¿ 0, there exists a smooth time reparameterization -(t; ; ) such that for the linearization of the unperturbed system at the equilibrium, (0; 0), has eigenvalues ±. Proof. Here we simply multiply the Hamiltonian function by =(; ). Since H is assumed to be C k -smooth, (; ) depends C k -smoothly on and , see [8]. Proposition 11. Suppose that the unperturbed system is as in Lemma 1 and assume that the system has been transformed using Propositions 9 and 10. Then there exists
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1067
a neighborhood of U of (0; 0) and an, area preserving, time dependent, C ∞ , change of coordinates (·; ; ) : U → U , such that under the transformation the coordinate axes are invariant. Proof. From the construction in the preceding proof, the normally hyperbolic manifold M has stable and unstable manifolds W s and W u which are C k -smooth in all variables. Thus, we may produce a change of variables (q; p; ; ) → (x; y; ; ) so that the stable manifold coincides with the y = 0 plane and the unstable manifold coincides with the x = 0 plane. We need only to show that this transformation can be accomplished canonically. First, for each (; ) let A; ∈ SL(2) be the area preserving linear transformation, with eigenvalues {1; 1} that puts the stable and unstable manifolds tangent to the x = 0 and y = 0 planes along the line (0; 0; ). This is canonical by de7nition. Next consider that W s is given as the graph of a function (x; ) ∈ C k . Consider the transformation xP = x; yP = y − (x; ):
(36)
This transformation is clearly canonical and makes the plane x = 0 invariant. Finally, we repeat the above procedure for W u . Note that up to this point the coordinate changes have been both canonical and as smooth as the original system. Now in the 7nal step, we employ a variant Hartman’s construction [7] to linearize the 6ow on the whole neighborhood of 0. This step may be only C 1 -smooth. An example given by Belitskii [3] shows that it might be no more than C 1 -smooth. Proposition 12. Suppose that the unperturbed system is as in Lemma 1 and assume that the system has been transformed using Propositions 9–11. Then there exists a neighborhood of U of (0; 0) and an area preserving, time dependent, C 1 , change of coordinates (·; ; ) : U → U , such that under the transformation the @ow on U is given by (6). The change of variables is smooth away from the coordinate axes. Proof. Consider the extended system. Let %0 = {(x; y; ) ∈ U × R : y = d} and de7ne %t = t (%0 ) ∩ U;
0 6 t ¡ + ∞:
Let y(t) P = de−t . Then {%t } is a smooth foliation of {0 ¡ y 6 d}. We may represent this foliation by y = yP + (x; y); P where is a smooth function and where the coordinate yP associated with %t we take to be y(t). P Extend the foliation to W u by de7ning (x; 0) = 0. It is at W u that the foliation might not be more than C 1 -smooth.
1068
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
Let (x; P y) P be new coordinates de7ned by the generating function S(x; y; P ; ) = xyP +
(x; y; P ; );
where @ = : @x The resulting transformation x = xP +
@ (x; y); P @yP
y = yP + (x; y) P
(37)
‘straightens’ each of the surfaces %t . This produces coordinates in which the 6ow in linear in the x-direction. Since the transformation is area preserving and the 6ow is Hamiltonian, the resulting 6ow is in fact linear in the y-direction as well. Now we need only show that the transformation is C 1 -smooth at W u . Since we straightened the stable and unstable manifolds, the equations of motion near the equilibrium are x˙ = Hy (x; y; ) = ( + f(x; y; ))x; y˙ = −Hx (x; y; ) = −( + g(x; y; ))y; ˙ = :
(38)
Here f and g are smooth functions that are zero at (0; 0; ). Let the surface %0 (as above) be given by the graph of y0 = <(x; ; 0) and let the graph of yt = <(x; ; t) be %t as above. Then < must satisfy x˙ = ( + f(x; <; ))x; <˙ = −( + g(x; <; ))<; ˙ = :
(39)
Clearly, provided that we restrict to a small enough neighborhood of (0; 0) then <(t) ¡ K0 e−(+*0 ) <(0); where K ¿ 0 is a constant and |*0 | is small in comparison to . Now consider derivatives of <, which must satisfy x˙ = ( + f(x; <; ))x; <˙ = −( + g(x; <; ) + gy (x; <; )<)< + g (x; <; )<; ˙ = :
(40)
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
1069
Here the last term approaches zero exponentially. The term gy (x; <; )< also approaches zero exponentially, so < approaches zero. Lastly,
(41)
Once again this implies exponential decrease of |
Further reading S. Aranson, G. Belitskii, E. Zhuzhoma, Introduction to the qualitative theory of dynamical systems on surfaces, in: Translations of Mathematical Monographs, Vol. 153, American Mathematical Society, Providence, RI, 1996. V.I. Arnol’d, Geometrical methods in the theory of ordinary diMerential equations, Grundlehren der mathematischen Wissenschaften, Vol. 250, 2nd Edition, Springer, New York, 1988.
1070
S.-N. Chow, T. Young / Nonlinear Analysis 56 (2004) 1047 – 1070
D. Diminnie, R. Haberman, Slow passage through a saddle-center bifurcation, J. Nonlinear Sci. 10 (2000) 197–221. N. Lebovitz, Bifurcation and unfolding in systems with two timescales, Ann. New York Acad. Sci. 617 (1990) 73–86. N. Lebovitz, A. Pesci, Dynamic bifurcation in Hamiltonian systems with one degree of freedom, SIAM J. Appl. Math. 53 (1995) 1117–1133. A. Neishtadt, On passage through resonance in the problem with two frequencies, Soviet Math. Dokl. 22 1975.