Rotating periodic solutions for convex Hamiltonian systems

Accepted Manuscript Rotating periodic solutions for convex Hamiltonian systems Jiamin Xing, Xue Yang, Yong Li

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S0893-9659(18)30342-2 https://doi.org/10.1016/j.aml.2018.10.002 AML 5660

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Applied Mathematics Letters

Received date : 13 July 2018 Revised date : 6 October 2018 Accepted date : 6 October 2018 Please cite this article as: J. Xing, et al., Rotating periodic solutions for convex Hamiltonian systems, Appl. Math. Lett. (2018), https://doi.org/10.1016/j.aml.2018.10.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Rotating Periodic Solutions for Convex Hamiltonian Systems a

Jiamin Xinga 1 , Xue Yanga,b 2 , Yong Lib,a 3 School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, P. R. China. b College of Mathematics, Jilin University, Changchun, 130012, P. R. China.

Abstract The paper concerns the existence of rotating periodic solutions in Hamiltonian systems. This kind of rotating periodic solutions has the form of x(t + T ) = Qx(t) with some symplectic orthogonal matrix Q. When Qk = I for some integer k > 0, it is a symmetric periodic solution and when Qk , I for any k ∈ N+ , it is just a quasi-periodic one corresponding to a rotation. It is proved that if the Hamiltonian is strictly convex, coercive and Q invariant, then there exists a Q-rotating periodic solution on every energy surface. Keywords: Rotating periodic solutions, Hamiltonian systems, Dual method 2010 MSC: 70H05, 70H12 1. Introduction and main results Consider the following Hamiltonian system x′ = J∇H, J =

O −I

I O

!

,

(1.1)

where the Hamiltonian function H ∈ C 1 (R2n , R), O denotes the n × n zero matrix, and I is the identity matrix. The existence of periodic solutions for system (1.1) is a classical problem and has been well developed by many works. Rabinowitz [8] and Weinstein [13] proved the existence of at least one periodic solution on every convex energy surface, see also [1] for another proof. It was conjectured that the number of periodic solutions on a compact convex energy surface in R2n is at least n, and for such multiple works, we refer to [2, 3, 7, 12] and the references therein. It is interesting to consider the symmetric solutions if the Hamiltonian is invariant with some orthogonal matrix Q. Rabinowitz [9] concerned the existence of symmetric periodic solution with Q = diag{−I, I}, Wang [11] proved that there exist two symmetric periodic solutions on any compact convex energy surface with Q = −I, Liu and Zhang [6] proved the existence of n symmetric periodic solutions under a pinched condition with Q = diag{−In−k , Ik , −In−k , Ik }. All these works are about symmetric periodic solutions. Recently, Liu et al [4, 5] concerned the 1 E-mail

address : [email protected]. address : [email protected]. 3 E-mail address : [email protected]. 2 E-mail

Preprint submitted to Applied Mathematics Letters

October 6, 2018

existence of Q-rotating periodic solutions for super-linear second order and resonant Hamiltonian systems. This kind of solutions has the form x(t + T ) = Qx(t) ∀ t ∈ R, where T > 0 and Q is a symplectic orthogonal matrix. Following the structure of Q, it will be seen that a Q-rotating periodic solution x(t) is a special quasi-periodic one corresponding to a rotation, if Qk , I for every k ∈ N+ . When Qk0 = I for some k0 ∈ N+ , x(t) is a subharmonic solution. Obviously, x(t) is T -periodic if Q = I, and anti-periodic if Q = −I. Motivated by these works, in this paper, we consider the rotating periodic solutions for convex Hamiltonian systems. Now we give our main result. Theorem 1.1. Assume H is strictly convex, non-negative, H(0) = 0, and lim H(x) = ∞. |x|→∞

Moreover H is Q invariant for some symplectic orthogonal matrix Q, that is, H(Qx) = H(x) for every x ∈ R2n . Then for every α > 0, there exists a Q-rotating periodic solution x(t) of (1.1) with H(x(t)) = α for all t ∈ R.

Remark 1.1. If we only assume H is strictly convex and coercive, the conclusion of Theorem 1.1 still holds. Since H is continuous and coercive, it attains the infimum at some x0 ∈ R2n . ˜ Let H(x) = H(x + x0 ) − H(x0 ), then H˜ satisfies all the conditions of Theorem 1.1. For each solution x(t) of H˜ on the energy surface α > 0, x(t) + x0 is also a solution of H on the energy surface α + H(x0 ). Hence the non-negativeness of the Hamiltonian and condition H(0) = 0 can be removed. 2. Proof of main results For every α > 0, denote

Γα := {x : H(x) = α}.

Since H is strictly convex and lim H(x) = ∞, Γα bounds a strictly convex neighbourhood of 0 |x|→∞

in R2n . Then for every x ∈ Γα , there exists a unique ζ ∈ S 2n−1 (the unit sphere in R2n ), and a real number r(ζ) > 0 such that x = r(ζ)ζ. Since H ∈ C 1 (R2n , R), the implicit function theorem implies that r(ζ) is continuously differentiable. Notice that Qx = r(Qζ)Qζ = r(ζ)Qζ ∀x ∈ Γα . Then r(Qζ) = r(ζ) for every ζ ∈ S 2n−1 . Let | · | denote the 2-norm on R2n . Consider the function  −q  αr |x|x |x|q , if x , 0, H(x) = 0, if x = 0,

(2.2)

where 1 < q < 2 is a fixed number. Then H(Qx) = H(x) ∀x ∈ R2n .

(2.3)

It is easy to see that Γα = H −1 (α). Since H and H are both invariant on Γα , there exists a continuous function κ(x) > 0 such that ∇H(x) = κ(x)∇H(x) ∀x ∈ Γα . 2

(2.4)

From H(x) = H(Qx) and (2.3), one has ∇H(x) = Q⊤ ∇H(Qx), ∇H(x) = Q⊤ ∇H(Qx). Then by (2.4), Thus we have

∇H(Qx) = κ(x)∇H(Qx). κ(Qx) = κ(x).

The following lemma shows that the rotating periodic solution of H is also a rotating periodic solution of H after a time transformation. Lemma 2.1. Assume x˜(t) is a Q-rotating periodic solution of Hamiltonian H on Γα , then x(t) = x˜(s(t)) is a Q-rotating periodic solution of Hamiltonian H on Γα , where s is the solution of the equation s′ = κ( x˜(s)).

(2.5)

Proof. Assume T˜ > 0 is a constant, such that x˜(s + T˜ ) = Q x˜(s) ∀s ∈ R. By (2.5), one has h(s) =

Z

a

s

dτ = t + t0 , κ( x˜(τ))

(2.6)

for some constant a and t0 . Then x(t) = x˜(h−1 (t + t0 )), and x′ (t) = x˜′ (h−1 (t + t0 ))

1 h′ (h−1 (t

+ t0 )) = J∇H( x˜(h (t + t0 )))κ( x˜(h−1 (t + t0 ))) −1

= Jκ(x(t))∇H(x(t)) = J∇H(x(t)). Since κ( x˜(t + T˜ )) = κ( x˜(t)), we know that Z s+T˜ dt = T ∀s ∈ R, κ( x˜(t)) s

where T > 0 is a constant. Since κ( x˜(t)) > 0, by (2.6) and (2.7), one has s(t + T ) = s(t) + T˜ ∀t ∈ R. Then for every t ∈ R, x(t + T ) = =

x˜(s(t + T )) = x˜(s(t) + T˜ ) Q x˜(s(t)) = Qx(t). 3

(2.7)

To prove Theorem 1.1, we have to show the followings. Lemma 2.2. H is strictly convex on R2n . Proof. Denote by epi(H) the epigraph of H, that is epi(H) = {(x, γ) ∈ R2n × R; H(x) ≤ γ}. Then the convexity of H is equivalent to the convexity of epi(H). For any point x ∈ R2n , when x , 0, one has H(x) = γ for some γ > 0. It is easy to see that γ e Γγ = {x ∈ R2n ; H(x) = γ} = ( ) q Γα . α 1

Assume x = ρζ with ζ ∈ S 2n−1 , and denote by Ξ the hyperplane that through (x, γ) and spanned by T xe Γγ = T r(ζ)ζ Γα ⊂ R2n × {0} ⊂ R2n+1 and the vector

(x, x · ∇H(x)) = (x, qH(x)) = (x, qγ).

Then Ξ is a supporting hyperplane for epi(H) at (x, γ). When x = 0, the hyperplane R2n × {0} is a supporting hyperplane at (0, 0), since H(0) = 0 and H(x) > 0 for every x , 0. Thus, we obtain that H is strictly convex. For a function G ∈ C(R2n , R), let G∗ be the Legendre-Fenchel transform of G, that is, G∗ (y) = sup{hx, yi − G(x); x ∈ R2n }. Then we have the following lemma. Lemma 2.3. (See [10]). Assume G ∈ C 1 (R2n , R) is strictly convex and ∇G is strongly monotone, that is, there exists a non-decreasing function u : [0, ∞) → [0, ∞) vanishing only at 0 and satisfying lim u(r) = ∞, such that r→∞

hx1 − x2 , ∇G(x1 ) − ∇G(x2 )i ≥ u(|x1 − x2 |)|x1 − x2 |, for all x1 , x2 ∈ R2n . Then G∗ ∈ C 1 (R2n , R), and ∇G∗ (y) = x for any y = ∇G(x). Since Q is symplectic orthogonal, one has R2n = ker(I − Q) ⊕ Im(I − Q). Let LP = (I − Q) |Im(I−Q) and H ∗ be the Legendre-Fenchel transform of H. Denote Z 1 τ y(t)dt = 0}. B = {y ∈ L p ([0, T ], R2n) : y(t + T ) = Qy(t), lim τ→∞ τ 0 B1 = {y ∈ W 1,p ([0, T ], R2n) : y(t + T ) = Qy(t) ∀t ∈ R}.

Clearly, B and B1 are Banach spaces. We introduce the operator K : B → B1 , Z t Z T −1 Jy(s)ds − LP (Ky)(t) = Jy(s)ds, 0

4

0

(2.8)

and consider the functional E on B, E(y) =

Z

T

0

! 1 H (y) − hy, Kyi dt. 2 ∗

(2.9)

Then we have the following lemma. Lemma 2.4. Assume y is a critical point of E on B. Then x(t) = ∇H ∗ (y(t)) is a Q-rotating periodic solution of (1.1) with Hamiltonian H. Proof. From the expression of H we see H ∗ is finite everywhere, H ∗ (0) = 0 and H ∗ ≥ 0. Since H is strictly convex and homogeneous on rays of degree q > 1, ∇H is strong monotone. By q Lemma 2.3, we know that H ∗ ∈ C 1 (R2n , R). Denote by p = q−1 the conjugate exponent of q. Then p > 2 and we have ) (

x y  H(x) H ∗ (y) 2n − = sup ; x∈R , |y| p |y| p |y| p−1 |y| ( )  x  x y 2n = sup , − H p−1 ; x ∈ R |y| p−1 |y| |y| ! y . (2.10) = H∗ |y| Hence ∇H ∗ is also strong monotone and by Lemma 2.3, y = ∇H(x) is equivalent to the system x = ∇H ∗ (y). If y is a critical point of E on B, then one has Z T h(∇H ∗ (y) − Ky)(t), ψ(t)i dt = 0 ∀ψ ∈ B. (2.11) 0

It follows that ∇H ∗ (y(t)) − (Ky)(t) = ξ,

(2.12)

for some constant vector ξ ∈ ker(I − Q). Let x(t) = ∇H ∗ (y(t)). Then x(t) = (Ky)(t) + ξ ∀t ∈ R.

(2.13)

By Lemma 2.3 and (2.8), one has 

y(t) = ∇H(x(t)) y(t) = −Jx′ (t).

(2.14)

Hence x(t) is a Q-rotating periodic solution of system (1.1) with Hamiltonian H. Proof of Theorem 1.1. By (2.10), one has lim E(y) = ∞, where k · k is the norm on B. Since kyk→∞

H ∗ is continuous and convex, we see E is weakly lower semi-continuous. Hence E attains its infimum at y∗ ∈ B. Since Q is orthogonal and JQ = QJ, there exists an orthogonal matrix P, such that ( ! !) 0 1 0 1 P⊤ JP = diag ,··· , , −1 0 −1 0 5

P⊤ QP = diag

(

cos θ1 − sin θ1

where 0 < θi ≤ 2π for 1 ≤ i ≤ n. Let ( Λ = P diag

0 − θT1

Then eΛT = Q. Take

sin θ1 cos θ1

θ1 T

0

!

!

,··· ,

,··· ,

cos θn − sin θn

0 − θTn

θn T

0

!)

sin θn cos θn

!)

,

P⊤ .

y˜ (t) = eΛt a,

where a = Pη with η = (η1 , η2 , 0, · · · , 0), η1 , η2 ∈ R\{0}. Then y˜ ∈ B and by a simple calculation, one has h˜y, K y˜i > 0. It follows from (2.10) that E(˜y) < 0 for a small enough. Thus, there exists a y∗ ∈ B, such that E(y∗ ) = inf E < 0. ∗

∗

B

Since y , 0, by Lemma 2.4, we have x = ∇H ∗ (y∗ ) is a Q-rotating periodic solution of system q−2 (1.1). Assume H(x∗ (t)) = ρ > 0. By the homogeneity of H, x(t) = ( αρ )1/q x∗ (( αρ ) q t) is a Q-rotating periodic solution of system (1.1) and H(x(t)) = α. By Lemma 2.1, the proof is completed. Acknowledgment The authors thank the referee for helpful suggestions which improved this paper. This work was supported by National Basic Research Program of China (grant No. 2013CB834102), NSFC (grant No. 11571065, 11171132, 11201173), Science and Technology Developing Plan of Jilin Province (No. 20180101220JC), JilinDRC (No. 2017C028-1) and the Fundamental Research Funds for the Central Universities (No. 2412018QD036). References References [1] Clarke, F; Ekeland, I. Hamiltonian trajectories having prescribed minimal period. Comm. Pure Appl. Math. 33 (1980), 103-116. [2] Ekeland, I; Lasry, J. On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface. Ann. Math. 112 (1980), 283-319. [3] Liu, C; Long, Y; Zhu, C. Multiplicity of closed characteristics on symmetric convex hypersurfaces in R2n . Math. Ann. 323 (2002), 201-215. [4] Liu, G; Li, Y; Yang, X. Rotating periodic solutions for super-linear second order Hamiltonian systems. Appl. Math. Lett. 79 (2018), 73-79. [5] Liu, G; Li, Y; Yang, X. Existence and multiplicity of rotating periodic solutions for resonant Hamiltonian systems. J. Differential Equations 265 (2018), 1324-1352. [6] Liu, H; Zhang, D. On the number of P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in R2n . Sci. China Math. 58 (2015), 1771-1778. [7] Long, Y; Zhu, C. Closed characteristics on compact convex hypersurfaces in R2n . Ann. of Math. 155 (2002), 317368. [8] Rabinowitz. P. Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), 157-184. [9] Rabinowitz, P. On the existence of periodic solutions for a class of symmetric Hamiltonian systems. Nonlinear. Anal., Theory Methods Appl. 11 (1987), 599-611. [10] Struwe, M. Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Fourth edition. Springer-Verlag, Berlin, 2008.

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[11] Wang W. Closed trajectories on symmetric convex Hamiltonian energy surfaces. Discrete Contin. Dyn. Syst. 32 (2012) 679-701. [12] Wang, W; Hu, X; Long, Y. Resonance identity, stability and multiplicity of closed characteristics on compact convex hypersurfaces. Duke Math. J. 139 (2007), 411-462. [13] Weinstein, A. Periodic orbits for convex Hamiltonian systems. Ann. of Math. 108 (1978), 507-518.

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