Abstract
A Lagrangian system on π2 that has been studied earlier under a geometrical condition and found to possess a pair of solutions, HΒ±, homoclinic to periodic solutions, vΒ±, of a given homotopy type, is considered further. It is shown with the aid of HΒ± and variational arguments that, in fact, there is a much richer structure of homoclinics and heteroclinics to vΒ±. Indeed, the system admits chaotic solutions.
This paper studies the Lagrangian system on β2:
(LS)
where the Lagrangian L is given by
Assume
and aij also satisfies (V1).
Because of the periodicity of (LS) in q1, q2, it can be viewed as a system in β2 or on β2/β€2 = π2. For V β‘ 0, (LS) was considered by Morse (1) and Hedlund (2). They established the existence of a pair of geodesics (for the Riemannean metric associated with L) lying between adjacent periodic geodesics in a given homotopy class on π2 and heteroclinic to these periodic geodesics. When the potential V is present, the situation becomes more complicated due to the equilibrium solutions at β€2 given via (V2). Under further geometrical conditions, there has been some work on the existence of zero energy periodic, heteroclinic, and homoclinic solutions of (LS) in refs. 3β6. In particular in ref. 6, it was shown that a geometrical condition led to a pair of periodic solutions v+, vβ of (LS), and to homoclinics to v+, vβ lying in the region between v+ and vβ. The goal of this paper is to show that, in the setting of ref. 6, there is a much richer set of homoclinic and heteroclinic solutions of (LS). Indeed there, is a full symbolic dynamics of these and other solutions. Thus, (LS) admits chaotic solutions. This will be made precise and carried out in the next section.
A Symbolic Dynamics of Solutions
To describe our results, the framework of ref. 6 must be recalled. For k β β€2β{0}, let
Viewed on π2, Fk is the class of W1.2 curves of homotopy type k. Let
The elements of Gk are candidates for heteroclinic solutions of (LS) (or homoclinics to 0 of homotopy type k viewed on π2).
For q β Gk and Fk respectively, let
and define
It was shown in refs. 3 and 4 that, if
1 |
there is a v β Fk such that Ik(v) = ck and v is a solution of (LS) (of period T(v) on π2). Moreover, there is a u β Gk such that I(u) = cΜk and u is a solution of (LS) heteroclinic to 0 and k. Let
The elements of Pk are only determined up to a phase shift because, if ΞΈ β β and ΟΞΈq(t) β‘ q(t β ΞΈ), then Ik(q) = Ik(ΟΞΈq) for all ΞΈ β β. Moreover, if p β Fk, so is p + j for all j β β€2. It was shown in ref. 4 that 0 β p(β) for any p β Pk. Therefore, 0 belongs to some component of β2β{p(β)|p β Fk}. This component is bounded by a pair of functions v+, vβ β Pk and will be denoted by β.
The region β will be subdivided as follows. For i β β, set ui = u + (i β 1)k and, for βi β β, set ui = u + ik. Then, U = divides β into β+ and ββ with vΒ±(β) forming a boundary component of βΒ±. Minimizing β«0β L(Ο)dt over the class of Wloc1,2 curves, Ο, with Ο(0) β v+(β) and Ο(β) = 0 yields a C2 solution, z0+ of (LS) in this class, joining v+ and U. Similarly, there is a C2 solution, z0β, of (LS) joining vβ and U with z0β(0) β vβ(β) and z0β(β) = 0. For k β β€, set zkΒ± = z0Β± + ik. The curves U, vΒ±, and ziΒ± divide β in a natural way into βsubrectangles,β βiΒ±, i β β€β{0}. See Fig. 1. Set βi = βi+ βͺ βiβ.
To continue, a stronger version of [1] is needed. Consider the class of W1,2 curves joining v+([0, T(v+)) to vβ([0, T(vβ)). Minimizing β« L(β )dt over this class produces an infimum, b, of the functional. Suppose
2 |
the strengthened geometrical condition. Then, there is a corresponding minimizer, Ο, of the functional that avoids β€2. By using [2], it was shown in ref. 6 that (LS) possesses a pair of solutions, HΒ± with HΒ± homoclinic to vΒ±. Moreover, HΒ± crosses ziΒ± for all i β 0 and also crosses z0β. In fact, HΒ±(0) β z0β(β), and the curves lie in βΒ± except for an interval in which they cross uβ1 and z0β and reenter β1β through u1 (see Fig. 2). The functions HΒ± are also minimal solutions of (LS) in the homotopy class of curves that cross the curves ziΒ± in the above fashion. βMinimalβ means that, for all x < y, HΒ± minimizes β« L(w)dt over the class of W1,2 curves w having the same endpoints and the same crossing (of ziΒ±) properties as HΒ±|xy.
Observe that this minimality property implies that, for any i β j β β€, ΟiH+(β) β© ΟjHβ(β)=Ο.
With the aid of these preliminaries, HΒ± will be used to help construct new homotopy classes of curves and a symbolic dynamics of solutions of (LS). Let
A curve q : β β β will be said to have homotopy type Ο β Ξ£ if q crosses the curves ziΟi, i β β€, in the order given by Ο. Define ΟΒ± β Ξ£ by ΟiΒ± = Β±, i β 0, and Ο0Β± = β. Then, HΒ± the homotopy type ΟΒ±.
Our main result is that, for each Ο β Ξ£, (LS) has a minimal solution of homotopy type Ο. To be more precise, let Ο β Ξ£ and i β β€. Consider ΟiHΟi. It divides β into two subregions. Excise the region between ΟiHΟi and vΟi from β, calling the resulting region β(ΟiHΟi). Associate with Ο the region β©iββ€β(ΟiHΟi) β‘ XΟ. See Fig. 3, where Οi = β, i β€ 0; = +, i > 0 and XΟ is the shaded region.
Now we have
Theorem 3 If (V1) β (V2), (A), and [2] are satisfied, then, for each Ο β Ξ£, there exists a minimal solution QΟ of (LS) of homotopy type Ο lying in XΟ.
Theorem 3 is a consequence of a related result for a subclass of Ξ£. For p, r β {+, β}, let
Theorem 4 If (V1) β (V2), (A), and [2] are satisfied, then, for each Ο β Ξ£pr and p, r β {+, β}, there is a minimal solution QΟ of (LS) of homotopy type Ο lying in XΟ.
Moreover, QΟ is heteroclinic from vp to vr if p β r and is homoclinic to vp if p = r.
Theorem 4 will be proved first and then Theorem 3 follows from it by an approximation argument. As in refs. 4β6, the proof of Theorem 4 involves finding QΟ as the minimizer of an appropriately renormalized functional over a class of curves lying in XΟ. Renormalization is necessary because the natural functional is infinite on the class of curves in XΟ. The first step in the proof is to introduce an appropriate class of curves. Let Ο β Ξ£pr and
where
Because Ο β Ξ£pr, there is a smallest ββ, β+ β β such that Οi = p for all i β€ βββ and Οi = r for all i β₯ β+. Define si = si(q) via q(ti) = zir(si(q)), i β₯ β+ and q(ti) = zip(si(q)), i β€ βββ. Then we require that
Remark 5 The sequence (ti(q)) need not be unique.
If (ti) and (tΜi) are two such sequences, by (Ξ³4), q(t) β ziΒ±(β+) for t β [ti, tΜi].
The renormalized functional on ΞΟ will be defined as follows. Let q β ΞΟ. Set
for i β₯ 1 and
for i β€ β1 where Ξ±i = 0 if βββ β€ i β€ β+ and Ξ±i = 1 otherwise. Now define
Because there may be more than one possible choice of (ti(q)), it must be shown that J(q) is independent of the choice of (ti(q)). Thus, suppose that J(q) < β. Then ai(q) β 0 as |i| β β, so
6 |
for large |i|. By a simpler version of the proof of Proposition 3.12 of ref. 4,
7 |
and
8 |
Hence, si(q) β 0 as |i| β β. As in ref. 6, set
where J corresponds to ti(q)) and JΜ to (tΜi(q)), with both (ti(q)) and (tΜi(q)) satisfying (Ξ³3). Then, [8] and si(q) β 0 imply
9 |
Because q|tβtΜβ lies on zβr (β+), [7] and [8] show |tβ(q) β tΜβ(q)| β 0 as β β β and similarly for ββ. Hence, [6], the right hand side of [9] β 0 as β β β. Consequently, J(q) = JΜ(q), and J is well defined.
Now define
10 |
Theorem 4 will be proved by showing there is a QΟ β ΞΟ such that J(QΟ) = cΟ. Moreover, QΟ is a minimal solution of (LS). Note that, by [8], QΟ β ΞΟ and J(QΟ) < β implies that QΟ is asymptotic to vp as t β ββ and to vr as t β β. The minimization argument is related to that of ref. 6, and, therefore, ref. 6 will be referred to for details when appropriate.
If, for example, i β₯ β+, gluing βk) to produces an element of Fk. Hence, by the definition of ck,
11 |
Combining these estimates shows
12 |
that is, J is bounded from below on ΞΟ. An upper bound for cΟ is provided by gluing a curve in XΟ, joining vp(βββT(vp)) and vr(β+T(vr)) to and , yielding H β ΞΟ with cΟ β€ J(H) < β.
Let (qm) be a minimizing sequence for [10]. Consider Οβ+Hr(t). Now, qm(tβ+(qm)) lies on zβ+r(β+) between Οβ+β1Hr(tβ+(Hr)) and Οβ+Hβr(tβ+(Hβr)) and a fortiori between Οβ+(Hr(tβ+(Hr)) and Οβ+β1Hr(tβ+(Hr)). It can be assumed that lies between Οβ+β1Hr|βtβ+(Hr) and tβ+Hr|tβ+(Hr)β. Indeed, suppose qm((x1, x2)) is outside of this region and qm(xi) = Οβ+Hr(yi), i = 1, 2. Replacing qm|x1x2 by Οβ+Hr|y1y2 yields qΜm β ΞΟ with J(qΜm) < J(qm) via the minimality property of HΒ±. If qm|x1β lies outside the region, replace Hr|y1β by qm|x1β, calling the resulting function Δ€. Because Οβ+Hr is the minimizer of J in an associated class of curves (6) (containing Δ€), J(Δ€) > J(Οβ+Hr), which implies
13 |
Therefore, by [13] gluing qm|ββx1 to Οβ+Hr|y1β yields qΜm β ΞΟ with J(qΜm) < J(qm). Similar reasoning shows that qm|βββtββ(qm) lies between and .
As in refs. 4β6, (qm) is bounded in Wloc1,2 and therefore, along a subsequence, converges weakly in Wloc1,2 and strongly in Llocβ to Q = QΟ β Wloc1,2, with Q satisfying (Ξ³1) β (Ξ³2) as well as the constraints on (qm) of the previous paragraph. As in ref. 6, there are numbers Ai > 0 such that
14 |
By [14], it can be assumed that ti(qm) β tΜi for all i β β€. It remains to show that (Ξ³3) β (Ξ³5) hold for Q. The convergence already established shows for all i β β€, as m β β.
15 |
Therefore, by [15] and (Ξ³3) for qm
16 |
and (Ξ³3) β (Ξ³4) holds for Q with ti(Q) = tΜi. Finally, as m β β,
17 |
The latter equality defines sΜi and implies
18 |
as m β β. Now, (Ξ³5) for qm and [18] gives (Ξ³5) for Q so Q β ΞΟ.
Next, it must be shown that J(Q) < β and J(Q) = cΟ. There is an M > 0 such that
19 |
For q = qm, write
20 |
where J+(q) denotes the sum over those ai(q) such that ai(q) β₯ 0. Note that the definition of ai(q) implies ai(q) < 0 is only possible when i β₯ β+ + 1 or i β€ βββ β 1. By [11] and [19],
21 |
and therefore
22 |
Hence, for any n β β with, e.g., n > β+ + ββ,
23 |
This implies
or equivalently
24 |
Hence, J(Q) < β via [24].
A variant of arguments from refs. 4β6 now shows J(Q) = cΟ. Indeed, let Ι > 0. There is an m0 = m0(Ι) such that m β₯ m0,
25 |
Further, choose j = j(Ι) so that
26 |
It can also be assumed that, for m β₯ m0,
27 |
Therefore, by [25β27] and [11],
28 |
The constraints on qm established in the paragraph containing [13] imply
29 |
for j sufficiently large. Now, [28β29] yield J(Q) = cΟ.
That Q is a solution of (LS) follows from simple local minimization and comparison arguments as in Proposition 5.4 of ref. 4.
Remark 30 For i β₯ β+, si+1(Q) < si(Q); for i β€ βββ, siβ1(Q) < si(Q).
Indeed, if equality holds in the + case, excising Q|titi+1 from Q and gluing Q|ββti to (Q β k)|ti+1β yields Q* β ΞΟ with J(Q*) < J(Q), a contradiction, unless Q|titi+1 coincides with vr. But, because Q is a solution of (LS), this is impossible.
To complete the proof of Theorem 4, it must be shown that Q is a minimal solution of (LS). Suppose x < y. We claim Q|xy minimizes β« L(β )dt over the class of W1,2 curves with the same end points as Q|xy and that cross the ziΒ± in the order given by Ο. Indeed, let w denote the minimizer of this variational problem. It suffices to prove that Q*, the curve obtained by replacing Q|xy by w, belongs to ΞΟ, for then
31 |
Therefore, there must be equality in [31], and Q* is a solution of (LS). But Q and Q* coincide on an open set, so uniqueness of solutions of (LS) implies Q β‘ Q*.
To verify that Q* satisfies (Ξ³1) β (Ξ³5), note that the range of w lies in XΟ via the minimality properties of the boundary curves of XΟ. Hence, (Ξ³1) holds. Parametrizing Q* appropriately gives (Ξ³2). There is a finite set of zjΒ± that Q|xy intersects zjΒ±. Because w is a solution of (LS), there is a natural corresponding set of tj(w), namely tj(w) is the unique (via the minimality of zjΒ±) value of t at which w intersects zjΒ±. Thus, Q* satisfies (Ξ³3), and minimality arguments imply (Ξ³4). Suppose (Ξ³5) fails, e.g., for i > 0. Then, si+1(Q*) > si(Q*) for some smallest i. Because sj(Q*) β 0 as j β β, there is a smallest j > i + 1 such that sj(Q*) β€ si(Q*). If sj = si, excise Q*|titj(Q*)(Q*) from Q* and glue Q*|ββti to (Q* β k)|tjβ, obtaining QΜ β ΞΟ with J(QΜ) < J(Q*) β€ J(Q), a contradiction. If sj < si, define P(t) = Q*(t) + k, t β₯ ti(Q*). Suppose for convenience that r = +. Because si+1 > si, P(ti) lies between Q*(β) and v+(β) while P(tjβ1) lies between Q*(β) and the portion of βXΟ given by appropriate segments of {ΟβHβ}. Therefore, there is a t* β (ti, tjβ1) such that P(t*) β Q*(β); i.e., Q*(t*) + k = Q(tΜ). Excising Q*|t*tΜ from Q* and arguing as for sj = si yields Q such that J(Q) < J(Q*). Possibly, (Ξ³5) still fails for QΜ, but, repeating the above argument a finite number of times yields Q β ΞΟ such that J(QΜ) < J(Q), a contradiction. This Q is a minimal solution of (LS), and Theorem 4 is proved.
Proof of Theorem 3.
Let Ο β Ξ£. Define dm = (dmi)iββ€ β Ξ£ as follows: dmi = Οi, |i| β€ m; dmi = Οm, i β₯ m; dmi = Οβm, i β€ β m. Then, dm β Ξ£Οβm,Οβm, so, by Theorem 4, there is a minimal solution Qm β Ξdm of (LS). The form of dmand Xdm together with (LS) imply the functions Qm are bounded in Cloc2 and therefore converge in Cloc2 to QΟ β XΟ. It readily follows that QΟ is a minimal solution of (LS) of homotopy type Ο, and the proof is complete.
Acknowledgments
This work is dedicated to JΓΌrgen Moser for his 70th birthday. I acknowledge with thanks helpful conversations with Sergey Bolotin.
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