Index Theory and Periodic Solution of Delay Differential Systems

Transcription

1 Index Theory and Periodic Solution of Delay Differential Systems Liu Chungen School of Mathematics, Nankai University workshop on nonlinear PDE and calculus of variation, Chern Mathematical Institute

2 Outline 1 M-boundary problem of a Hamiltonian system 2 Background of delay differential equations First order delay Hamiltonian systems 6 Second order delay Hamiltonian systems

3 1 M-boundary problem of a Hamiltonian system For a skew-symmetric non-degenerate 2N 2N matrix J = (a ij ), it can define a symplectic structure on R 2N by ω(v, w) = v T J 1 w or ω = 1 2 a ij dx i dx j with J 1 = (a ij ). A 2N 2N matrix M is called i,j J -symplectic if there holds M T J 1 M = J 1. We denote the set of all J -symplectic matrices by Sp ( J (2N). The ) usual symplectic group Sp(2N) is the 0 IN special case of J = J N =, i.e., Sp(2N) = Sp I N 0 JN (2N). Here I N is the N N identity matrix. We will write J for J N if the dimension 2N is clear from the text

4 For a J -symplectic matrix M with M k = I 2N, and a function H C 2 (R R 2n ) with H(t + τ, Mz) = H(t, z), we consider kτ-periodic solution of the following M-boundary value problem { ż(t) = J H(t, z(t)) (1) z(τ) = Mz(0). The corresponding functional is defined in E = W 1/2,2 (S 1, R 2n ) with S 1 = R/(kτZ) by ϕ(z) = 1 2 kτ 0 (J 1 ż(t), z(t))dt kτ 0 H(t, z(t))dt. (2) The critical point of ϕ in E is a kτ-periodic solution of the nonlinear Hamiltonian system in (1). In order to solve the problem (1), we define a group action σ on E by σz(t) = Mz(t τ)

5 It is clear that σ k = id and ϕ is σ-invariant, i.e., there holds ϕ(σz) = ϕ(z). (3) Setting E σ = {z E σz = z} = fix(σ), by the well known Palais symmetric principal(see [27]), a critical point of ϕ in E σ is a solution of the boundary problem (1). For z E σ, there holds ϕ(z) = k 2 τ 0 (J 1 ż(t), z(t))dt k τ 0 H(t, z(t))dt

6 The linearized system along a solution z(t) of the nonlinear Hamiltonian system in (1) is the following linear Hamiltonian system ẏ(t) = J H (t, z(t))y(t). Its fundamental solution γ z (t) with γ z (0) = I 2N should satisfy γ z (t) = J H (t, z(t))γ z (t). The following result is well known. Lemma 1.1 γ z is an J -symplectic path, i.e., γ z (t) T J 1 γ z (t) = J 1 for all t R. For simplicity we take τ = 1. For a symmetric continuous matrix function B(t) satisfying M T B(t + 1)M = B(t), suppose γ B (t) is the fundamental solution of the linear Hamiltonian system ẏ(t) = J B(t)y(t)

7 Definition 1.1 The (J, M)-nullity of the symmetric matrix function B is defined by ν J M (B) = dim C ker C (γ B (1) M). For the standard case of J = J N and a matrix P Sp JN (2N), the (J, P )-nullity and Maslov-type index of a symmetric matrix function B was defined first in 2006 by algebra method. We will define the (J, M)-index i J M (B) below via analytic method. So the index pair (i J M (B), νj M (B)) Z {0, 1,, 2N} makes sense for all symmetric continuous matrix function B(t) satisfying M T B(t + 1)M = B(t)

8 Definition 1.2 We define the relative index by I(A M, A M B M ) = m d (P n(a M B M )P n ) m d (P na M P n ), n n, where n > 0 is a constant large enough such that the difference becomes a constant independent of n n. In this case, we define i J M ( B) = I(A M, A M B M ), ν J M ( B) = m 0 (A M B M ). Here we denote by m d (A) the total multiplicity of eigenvalues of the adjointoperator A in the interval (, d], and m 0 (A) = dim ker(a). We recall that P n : E σ En σ is the projection maps and {En} σ is a Galerkin approximation sequence. A M and B M are operators in E σ defined respectively by (A M z, z) = 1 0 (J 1 ż(t), z(t))dt, (B M z, z) = 1 0 (B(t)z(t), z(t))dt. Up to a sign, the relative index I(A M, A M B M ) is exactly the spectral flow of the operator family A M sb M. Precisely there holds I(A M, A M B M ) = sf(a M sb M ). Here sf(a) is the spectral flow of the bounded self-adjoint operator A

9 2 Background of delay differential equations In 1974, Kaplan and Yorke in [Kaplan-Yorke 1974] studied the autonomous delay differential equation and introduced a new technique for establishing the existence of periodic solutions. More precisely, Kaplan and Yorke solved the periodic solutions of the following kinds of delay differential equations and with odd function f. ẋ(t) = f(x(t 1)) ẋ(t) = f(x(t 1)) + f(x(t 2))

10 They turned their problems into the problems of periodic solution of autonomous Hamiltonian system and under some twisted condition on the origin and infinity for the function f, it was proved that there exists at least one periodic solution. Since then many papers (see [G.Fei 1995,2006], [Z.Guo-J.Yu 2005, 2011], [J.Li-X.He 1998,1999] and the references therein) used Kaplan and Y- orke s original idea to search for periodic solutions of more general differential delay equations of the following form ẋ(t) = f(x(t 1)) + f(x(t 2)) + + f(x(t m + 1)). The existence of periodic solutions of above delay differential equation has been investigated by Nussbaum in [Nussbaum 1978] using different techniques

11 3 For simplicity, we first consider the 4τ-periodic solutions of the following delay differential systems x (t) = V (t, x(t τ)), (4) where the function V C 2 (R R n, R) is τ-periodic in variable t and is even in variables x. To find 4τ-periodic solution x(t), we only need to find solution with x(t + 2τ) = x(t). If x(t) is so a solution, let x 1 (t) = x(t), x 2 (t) = x(t τ) and z(t) = (x 1 (t), x 2 (t)) T, then there holds { x 1 (t) = V (t, x 2 (t)) x (5) 2(t) = V (t, x 1 (t))

12 Set H(t, x 1, x 2 ) = V (t, x 1 ) + V (t, x 2 ), then we can rewrite (5) as ( ) 0 In ż(t) = J 1 H(t, z(t)), J 1 =. (6) I n 0 Moreover, if z(t) = (x 1 (t), x 2 (t)) T is a 4τ-periodic solution of (6) with z(t) = σz(t) for the 4-periodic action σz(t) = J 1 z(t + τ), (7) then x(t) = x 1 (t) is a solution of (4) with x(t + 2τ) = x(t). The condition (7) is equivalent to z(τ) = J1 1 z(0). (8) So the problem can be transformed to the problem (1) with J = J 1 and M = J

13 In general, for a function V C 2 (R R n, R) with period τ in variable t and even in variables x, we consider the 2mτ-periodic solutions of the following delay differential system x (t) = V (t, x(t τ))+ V (t, x(t 2τ))+ + V (t, x(t (m 1)τ)). (9)

14 If we get a solution x(t) with x(t mτ) = x(t), then by setting x 1 (t) = x(t), x 2 (t) = x(t τ),, x m (t) = x(t (m 1)τ) and H(t, x 1,, x m ) = V (t, x 1 ) + + V (t, x m ), z = (x 1,, x m ) T, we rewrite the system (9) as ż(t) = A m H(t, z(t)), (10) where the mn mn skew symmetric matrix A m is defined by 0 I n I n A m = I n 0 I n.... I n I n 0 We see that det A m 0 if m 2N and det A m = 0 if m 2N

15 Setting 0 I n 0 0 ( T m = 0 0 I n = I n 0 I n In(m 1) ). If z(t) is a solution of (10) with z(τ) = Tm 1 z(0) then x(t) = x 1 (t) is a 2mτperiodic solution of equation (9) for m 2N. So we also turn our problem into the problem (1) with J = A m and M = Tm 1. In order to understand the case of m 2N + 1, we first recall the notation about

16 4 Any k k skew symmetric matrix A determine a on R k. For any functions F, H C (R k, R), the defined by {F, H} = ( F ) T A H C (R k, R). (11) We recall the definition of a {, } on a manifold M. For any two functions F, H C (M), the Poisson bracket {F, H} C (M) is defined by the following properties (1) {c 1 F 1 + c 2 F 2, H} = c 1 {F 1, H} + c 2 {F 2, H}, c 1, c 2 R, (2) {F, H} = {H, F }, (3) {{F, H}, P } + {{P, F }, H} + {{H, P }, F } = 0, (4) {F, H P } = {F, H}P + H{F, P }

17 A differential manifold M with a is called Poisson manifold. Furthermore if there holds (5) {F, H} = 0 for any function F implies H c, then the {, } determines a symplectic structure, and M is a symplectic manifold. For example, when k = 2m and A = J m = then it defines a on R 2m by m ( F H {F, H} = F ) H. q i p i p i q i i=1 ( 0 Im I m 0 This on R 2m is the standard and it determines the standard symplectic structure on R 2m. ),

18 Let M be a Poisson manifold and H : M R be a smooth function. The Hamiltonian vector field associated with H is the unique smooth vector field V H on M satisfying V H (F ) = {F, H} for every smooth function F : M R. If the on R k is defined by (11), the Hamiltonian vector field is V H (x) = A H(x). The Hamiltonian equation becomes ẋ(t) = A H(x(t)). For a time depending function H t (x) = H(t, x), the Hamiltonian vector field makes sense as V Ht (x) = A x H(t, x) and the Hamiltonian equation becomes ẋ(t) = A x H(t, x(t)). The following Darboux Theorem will be very useful

19 Theorem 4.1 (Darboux Theorem) Suppose the rank of the k k matrix A in (11) is 2m with k = 2m + l, then there is a coordinates transformation y = Bx such that m ( f h {F, H}(x) = f ) h (y) = f(y) T J h(y), q i p i p i q i i=1 where f(y) = F (B 1 y), h(y) = H(B 1 y) and ( y ) = (p 1,, p m, q 1,, q m, z 1,, z l ) T. That is to say J = BAB T Jm 0 =

20 Particularly, for the matrix A m defined in (10) with m 2N + 1, and the matrix B m defined by I n I n 0 0 B m = I n 0, I n I n I n I n there holds B m A m B T m = ( ) Am So by choosing matrix ( C m 1 satisfying ) C m 1 A m 1 Cm 1 T = J nl with ( l = m 1 2) and taking B m Cm 1 0 = there holds B 0 I m mb m A m (B mb m ) T Jnl 0 =. ( ) 0 0 ỹ By taking the coordinates transformation y = = B m z and h(t, y) = H(t, z), the system (10) becomes y m ỹ(t) = A m 1 ỹh(t, y), y m c. (12)

21 From the coordinates transformation, we see that y i = x i for 1 i m 1 and y m = x 1 x 2 + x m 1 + x m. So system (12) becomes as z(t) = A m 1 H(t, z(t)), (13) where H(t, z) = H(t, z, c x 1 +x 2 +x m 1 ) and z = (x 1,, x m 1 ). When choosing c = 0, any solution z(t) of (13) with z(0) = B m 1 z(1) determines a solution of system (10) with x m (t) = x 1 (t) + x 2 (t) + x m 1 (t), so x(t) = x 1 (t) is a solution of the delay system (9), where 0 I n I n 0 B m 1 = I n. I n I n I n I n So in this case, we also transform the problem into the problem (1) with J = 1 A m 1 and M = B m

22 Now by some arguments of index theory and variational method, we have the following statement: Theorem 4.2 Suppose V satisfies the following conditions (V1) There exists a constant C > 0 such that V (t, x) C, (t, x) [0, 1] R n. (V ) There exists a continuous symmetric matrix function C (t) such that V (t, x) = C (t)x + o( x ) uniformly in t R as x. Let C 0 (t) = V (t, 0). For m 2N C 0 (t) 0 0 B 0 (t) = 0 C 0 (t) 0..., 0 0 C 0 (t) and C (t) 0 0 B (t) = 0 C (t) C (t) Set (i 0, ν 0 ) = (i J M (B 0), ν J M (B 0)), (i, ν ) = (i J M (B ), ν J M (B )) with J = A m, M = T 1 m

23 For m 2N + 1, 2C 0 (t) C 0 (t) C 0 (t) C 0 (t) C 0 (t) B 0 (t) = C 0 (t) 2C 0 (t) C 0 (t)... C 0 (t). C 0 (t). C 0 (t) C 0 (t) C 0 (t) C 0 (t) 2C 0 (t) and B (t) = 2C (t) C (t) C (t) C (t) C (t) C (t) 2C (t) C (t) C (t) C (t)..... C (t) C (t) C (t) C (t) 2C (t). Set (i 0, ν 0 ) = (i J M (B 0), ν J M (B 0)), (i, ν ) = (i J M (B ), ν J M (B )) with J = 1 A m 1, M = B m

24 If i / [i 0, i 0 + ν 0 ], the delay differential system (9) possesses at least one nontrivial 2mτ-periodic solution x with x(t mτ) = x(t). Furthermore, if the trivial solution z 0 = 0 of problem (10), with H(t, z) = V (t, x 1 ) + + V (t, x m ) for even m and H(t, z) = V (t, x 1 ) + + V (t, x m 1 ) + V (t, x 1 + x 2 + x m 1 ) for odd m, is not pseudo-degenerated, and i / [i 0 2N, i 0 + ν 0 + 2N], the delay differential system (9) possesses at least two nontrivial 2mτ-periodic solutions as above

25 5 First order delay Hamiltonian systems For a function G C 2 (R R 2n, R) with G(t + τ, x) = G(t, x), we consider the 2τ periodic solutions of following first order delay Hamiltonian system ẋ(t) = J n G(t, x(t τ)). (14) For a 2τ period solution x(t) of (14), by setting x 1 (t) = x(t), x 2 (t) = x(t τ) and z = (x 1, x 2 ) T, the delay Hamiltonian system (14) is read as ż(t) = J 2n H(t, z(t)) (15) ( ) with H(t, z) = H(t, x 1, x 2 ) = G(t, x 1 ) + G(t, x 2 ) and J 0 Jn 2n =. ( J ) n 0 0 In If z(t) is a solution of (15) with z(0) = P 2n z(τ), P 2n =, then x(t) = I n 0 x 1 (t) is a 2τ-periodic solution of (14)

26 In general, we consider the following first order delay Hamiltonian systems ẋ(t) = J n ( G(t, x(t τ))+ G(t, x(t 2τ))+ + G(t, x(t (m 1)τ))). (16) For an mτ-periodic solution x(t) of (16), by setting x 1 (t) = x(t), x 2 (t) = x(t τ),, x m (t) = x(t (m 1)τ) and z = (x 1, x 2,, x m ) T, the delay Hamiltonian system (16) is read as ż(t) = J n,m H(t, z(t)) (17) with H(t, z) = G(t, x 1 ) + + G(t, x m ) and 0 J n J n J n J n,m = J n 0 J n J n..... J n J n J n

27 Conversely, if z(t) is a solution of (17) with z(0) = P n,m z(τ), where 0 I n 0 0 ( ) P n,m = 0 0 I n = In(m 1), I n 0 I n then x(t) = x 1 (t) is a mτ-periodic solution of (16)

28 Theorem 5.1 Suppose G satisfies the conditions (V1) and (V ) in Theorem4.2. Then the system (16) possesses an mτ-periodic solution x 0. Suppose z 0 (t) is the solution of (17) corresponding to x 0. Let B 0 (t) = H (t, z 0 (t)) and B (t) = C (t) C (t) C (t) Set (i 0, ν 0 ) = (i J M (B 0), ν J M (B 0)), (i, ν ) = (i J M (B ), ν J M (B )) with J = J n,m, M = Pn,m. 1 If i / [i 0, i 0 + ν 0 ] holds, the system (16) possesses at least two mτ-periodic solutions. Furthermore, if z 0 is not pseudo-degenerated, and i / [i 0 2N, i 0 + ν 0 + 2N] holds, then the system (16) possesses at least three an mτ-periodic solutions

29 6 Second order delay Hamiltonian systems For a function V C 2 (R R n, R) with V (t + τ, x) = V (t, x), we consider the periodic solutions of following second order delay Hamiltonian system ẍ(t) + V (t, x(t 2τ)) = 0. We can turn it into a first order delay Hamiltonian system as (14) with H(t, x, y) = y 2 2 V (t, x) and y(t) = ẋ(t + τ). In general, we consider the mτ-periodic solutions of the following second order delay system ẍ(t) = [ V (t, x(t τ)) + V (t, x(t 2τ)) + + V (t, x(t (m 1)τ))]. (18)

30 Let x 1 (t) = x(t), x 2 (t) = x(t τ), x m (t) = x(t (m 1)τ) and z(t) = (x 1 (t),, x m (t)) T, then by x(t + mτ) = x(t), there holds z(t) = A m H(t, z(t)), where H(t, z) = V (t, x 1 ) + + V (t, x m ) and 0 I n I n A m = I n 0 I n.... I n I n 0 It is easy to see that det A m 0. By taking y(t) = A 1 m ż(t) and w = (z, y) T, there holds ẇ(t) = J mn H(t, w(t)), (19) where H(t, w) = 1 2 (A my, y) H(t, z)

31 A solution ( of this Hamiltonian ) system with the boundary value condition Pn,m 0 w(0) = 0 A 1 w(τ) determine an mτ-periodic solution x(t) = m P n,m A m x 1 (t) of the second order delay system. We also turn our problem into the problem (1) with J = J mn and M = Pn,m. 1 By the same arguments, we also obtained an existence and multiplicity result as above theorems for this case

32 Thank you! liucg@nankai.edu.cn