Applied Mathematics Letters

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Applied Mathematics Letters 23 (21 49 416 Contents lists available at ScienceDirect Applied Mathematics Letters jornal homepage: www.elsevier.

f o a b s t r a c t Article history: Received 6 November 29 Received in revised form 9 November 29 Accepted 24 November 29 Keywords: Exponential trichotomy Lyapnov Schmidt method Homoclinic orbit

1 Applied Mathematics Letters 23 ( Contents lists available at ScienceDirect Applied Mathematics Letters jornal homepage: Exponential trichotomy and homoclinic bifrcation with saddle-center eqilibrim Li Xingbo Department of Mathematics, East China Normal University, 2241, Shanghai, China a r t i c l e i n f o a b s t r a c t Article history: Received 6 November 29 Received in revised form 9 November 29 Accepted 24 November 29 Keywords: Exponential trichotomy Lyapnov Schmidt method Homoclinic orbit Saddle center Bifrcation In this paper the bifrcation of a homoclinic orbit is stdied for an ordinary differential eqation with periodic pertrbation. Exponential trichotomy theory with the method of Lyapnov Schmidt is sed to obtain some sfficient conditions to garantee the existence of homoclinic soltions and periodic soltions for this problem. Some known reslts are extended. 29 Elsevier Ltd. All rights reserved. 1. Introdction With the development of nonlinear science, an increasingly large nmber of papers have been devoted to the bifrcation problems of homoclinic orbits with nonhyperbolic eqilibrim (see [1 5] and the references therein. The methods sed in those papers work well in stdying bifrcation problem nder atonomos pertrbation, bt these methods have a large limitation in the discssion of the persistence and bifrcation problem for homoclinic orbits nder periodic pertrbation. In this paper we are interested in the problem of homoclinic bifrcation with nonatonomos pertrbation, more precisely, we will consider the system ẋ = f (x, y + εg x (x, y, t, λ, ε, ẏ = εg y (x, y, t, λ, ε (1.1 where (x, y R n R m, t R, λ R k, < ε 1, g x, g y are 2π periodic with respect to t. When y =, in [6 9] a fnctional analytic approach together with the method of Lyapnov Schmidt is sed to discss the existence of homoclinic soltions and periodic soltions for (1.1 with a hyperbolic eqilibrim. When y, in [1], based on the Poincaré map, the bifrcation of sbharmonic soltions and invariant tori are investigated. Inspired by [6 1], we will se exponential trichotomy theory with the method of Lyapnov Schmidt to discss the problem of homoclinic bifrcation for (1.1 when y. By the Melnikov fnction, we give some sfficient conditions to garantee the existence of homoclinic orbit and the existence of periodic orbit bifrcated from the homoclinic orbit. The main technical difference from the analysis of homoclinic orbits with a hyperbolic eqilibrim is that in the nonhyperbolic sitation the variation eqation along the homoclinic orbit no longer has an exponential dichotomy bt an exponential trichotomy. This reqires a modified approach. This work is spported by NNSFC (No and Shanghai Leading Academic Discipline Project (B47. address: xbli@math.ecn.ed.cn /$ see front matter 29 Elsevier Ltd. All rights reserved. doi:1.116/j.aml

2 41 X. Li / Applied Mathematics Letters 23 ( Persistence of homoclinic orbit nder pertrbation Consider the C r (r 2 system (1.1 and the corresponding npertrbed system ẋ = f (x, y, ẏ =. (2.1 We make the following assmptions: (H 1 There exists y R m, sch that system ẋ = f (x, y (2.2 has a hyperbolic eqilibrim x = x(y and a homoclinic orbit Γ = {γ (t : γ (± = x(y }. The stable manifold W s and nstable manifold W of x(y are n 1 -dimensional and n 2 -dimensional, respectively, with n 1 + n 2 = n. Moreover for any p Γ, dim(w s (x(y W (x(y = dim(t p W s (x(y T p W (x(y = 1. Remark 2.1. From the assmption (H 1, it is easy to see that system (2.1 has an eqilibrim q(x(y, y, which possesses n 1 -dimensional stable manifold W s (q, n 2 -dimensional nstable manifold W (q, and m-dimensional center manifold W c (q respectively. Sppose system (1.1 still has a invariant set q(t, ε nder small pertrbation, which satisfies q(t, ε q = o(ε. For convenience, we assme a transformation has already been made to move the eqilibrim to the origin, then we can assme that g x (,, t, λ, ε =, g y (,, t, λ, ε =. Frthermore, we need the following assmption for (1.1 (H 2 D y g y (,, t, λ, ε = diag(c 1 (ε, C 2 (ε, g y (x, y, t, λ, ε = is odd with respect to t. C 1 (ε is a m 1 m 1 matrix, C 2 (ε is a m 2 m 2 matrix, m 1 + m 2 = m, Re σ (C 1 (ε >, Re σ (C 2 (ε <. Consider the linear variational system of (1.1 ε= U = A(tU (2.3 and its adjoint system V = A (tv (2.4 where A(t = (f + εg x, εg y / (x, y(γ (t, ε=, the sign denotes the transposition. Take p = (γ (t,. Based on (H 1 and ẏ =, the tangent space corresponding to (2.3 can be decomposed into T p R n+m = T 1 T 2 T 3 T 4 T 5 (2.5 which satisfy T 1 = T p W s ( T p W (, T 2 = T p W s (/[T p W s ( T p W (], T 3 = T p W (/[T p W s ( T p W (], T 4 = T p W c (, T 5 = [T p W s ( + T p W (] c [T p W c (] c. From (H 1 and the above decomposition, system (2.3 has exponentially bonded soltion φ(t, and system (2.4 has exponentially bonded soltion ψ(t, sch that φ(t, ψ(t exponentially as t ±. Also we have φ, ψ =, and T 1 = span{φ(t}, T 5 = span{ψ(t}. Now we choose a fndamental soltion matrix X(t, t of (2.3 satisfying X(t, t = Id, then Y(t, t = X 1 (t, t is a fndamental soltion matrix of (2.4.

3 X. Li / Applied Mathematics Letters 23 ( Definition 2.1. We say (2.3 has an exponential trichotomy in J if there exist projections P c (t, P s (t and P (t = I P c P s, t J, and there are constants K 1, and α σ > sch that for t, s J, X(t, sp v (s = P v (tx(t, s, t s, v = c,, s, X(t, sp c (s K e σ t s, X(t, sp s (s K e α(t s, t s, X(s, tp (t K e α(t s, t s. Remark 2.2. If P c (t =, then (2.3 is said to have an exponential dichotomy in J. Similar to [11], we get the following reslt Lemma 2.2. If (H 1 holds, then (2.3 and (2.4 have an exponential trichotomy both in R + and R with the same constants K, α, σ, and the corresponding projections are P i s (t, P i c (t, P i i i i (t and Ps (t, Pc (t, P (t, i = +,, respectively. Moreover RP + s = T p W s, RP = T pw, RP c = RP + c = T p W c, R(P + s + P + = c T pw cs, R(P c + P = T pw c. Remark 2.3. From (2.5 and Lemma 2.2, it is obvios that Denote dim(rp + (t s RP (t = 1, dim(rp + (t RP s (t = 1. E(b, J = {x C : sp{ x(t e b t } < }, t J E(b, r, J = {x C r : x,..., x (r E(b, J}, r N. Then E(b, J and E(b, r, J are Banach spaces with norms x = sp t J { x(t e b t }, x r = r k= x(k, respectively. Now we consider the nonhomogeneos eqation Denote ż = A(tz + h(t. Z = { h(t E( σ, R : (2.6 } ψ (th(tdt =, ψ E(α, 1, R, ψ = A (tψ. (2.7 Lemma 2.3. Sppose that (H 1 is valid, and h(t Z, then system (2.6 has bonded soltion z(t in E( σ, 1, R. Proof. From the discssion above, system (2.4 has exponentially bonded soltion ψ(t E(α, 1, R, then there exists an vector η R n+m sch that { Y(t, P + (η, t, ψ(t = Y(t, P (2.8 s (η, t. It is obvios that P + (η = P (η which is eqivalent to (P + Then we have s s ( + P + c (η = (P ( + P c (η. η [P + ( + s P+ ( c P ( P c (] =. (2.9 Sbstitte (2.8 into (2.7 and notice that Y(t, s = X (s, t, we get [ ] η P + (X(, th(tdt + P (X(, th(t s =. (2.1

4 412 X. Li / Applied Mathematics Letters 23 ( De to (2.9 and (2.1, there exists a vector ξ R n+m satisfying [P + ( + s P+ ( c P ( P (]ξ = c P + (X(, th(tdt + It follows from the Definition 2.1 that P i (X(, t = X(, j tp i j (t, i = +, ; j = s,. Then we can obtain P s [P + ( + s P+ (]ξ c P + (X(, th(tdt = [P ( + P (]ξ + c P s By variation of constants formla, we can get a fnction z(t E( σ, 1, R sch that (X(, th(t. (2.11 (X(, th(t. (2.12 z(t = X(t, [P + ( + s P+ (]ξ + c X(t, s[p + (s + s P+ c (s]h(sds X(t, sp + (sh(sds, t, z(t = X(t, [P ( + P (]ξ + c X(t, s[p (s + P c (s]h(sds + It is easy to see that z(t is the bonded soltion of (2.7 in E( σ, 1, R. Denote a projection π as follows π : E( σ, R E( σ, R πz(t = ψ(tn 1 ψ (tz(tdt where N = ψ (tψ(tdt. Since Im(I π Z, it follows from Lemma 2.3 that ż = A(tz + (I πh(t has a niqe bonded soltion z(t in E( σ, 1, R. Now we consider the pertrbed system (1.1. For convenience, we rewrite it as follows ż = f (z + εh(z, t, λ, ε where (x, y = z, f (z = (f (x, y,, h = (g x, g y. We now make the change of variable w(t = z(t + t s(t where w(t = (x, y, s(t = (γ (t,. Then system (2.14 can be changed into where ẇ(t = A(tw + G(w, t + t, λ, ε G(w, t + t, λ, ε = f (w + s(t f (s(t A(tw + εh(w + s(t, t + t, λ, ε. Based on the method of Lyapnov Schmidt, (2.15 is eqivalent to the following system t X(t, sp s (sh(sds, t. (2.13 (2.14 (2.15 ẇ(t = A(tw + (I πg(w, t + t, λ, ε, (2.16 πg(w, t + t, λ, ε =. (2.17 Since w(t E( σ, R, it is easy to see G : E( σ, R R R k R E( σ, R. Also for all bonded soltion ψ(t of (2.4 in E(α, 1, R, we have ψ (t(i πg(w, t + t, λ, εdt =. (2.18 Owing to Lemma 2.3, system (2.16 has a niqe nontrivial bonded soltion w = w(t, λ, ε in E( σ, λ, ε satisfying w(t, λ, =. Then if (2.17 holds, namely ψ (tg(w, t + t, λ, εdt =, it follows that system (2.15 has a niqe nontrivial bonded soltion w = w(t, λ, ε in E( σ, λ, ε. (2.19

5 X. Li / Applied Mathematics Letters 23 ( Let H(t, λ, ε = ψ (tg(w, t + t, λ, εdt. (2.2 Based on the definition of G, it is easy to see that G(, t + t, λ, =. Since w(t, λ, =, we get H(t, λ, =, then by simple calclation we obtain H ε = ψ (th(s(t, t + t, λ, dt. Let M(t, λ = H ε (t, λ,, (2.2 can be expressed into the following form H(t, λ, ε = ε[m(t, λ + o(ε]. From (2.14, we know that ψ (tg x (γ (t, t + t, λ, dt M(t, λ = +. ψ (tg y (γ (t, t + t, λ, dt By the implicit fnction theorem, we have the following reslts Theorem 2.4. Sppose that (H 1 holds. If there exists t, λ sch that M( t, λ =, M λ ( t, λ, then there exists a parameter srface λ = λ(t, ε satisfying λ( t, = λ, sch that system (2.15 has a niqe bonded soltion w(t, λ in E( σ, 1, R for λ = λ(t, ε and < ε 1, namely, system (1.1 has a 1-homoclinic orbit Γ (t, ε = w(t + s(t satisfying Γ (t, ε (γ (t, = o( ε. 3. Existence of periodic soltion In this section we consider the existence of periodic soltion bifrcated from homoclinic soltion nder pertrbation. First we consider the following system ż = A(tz + h(t. Let T(ε = π[ 1 ]. It is obvios that T(ε as ε. For t [, T(ε], we se variation of constant and exponential ε trichotomy to constrct the following two soltions z 1 (t = X(t, [P + s ( + P+ c (]ξ X(t, T(εP + (T(εη 1 z 2 (t = X(t, [P ( + P c (]ξ 2 + Denote new sets as follows T(ε t + X(t, P s (η 2 + X(t, s[p + s (s + P+ c (s]h(sds X(t, sp + (sh(sds, t, X(t, s[p (s + P c (s]h(sds X(t, sp s (sh(sds, t. E ε = C 1 ([, T(ε], R n+m E( σ, R, T(ε } Eε {h h = E ε, ψ (th(tdt + ψ (η 2 ψ (T(εη 1 =. Owing to the definition of z 1 (t and z 2 (t, if we can prove that z 1 ( = z 2 (, z 1 (T(ε = z 2 (, (3.1 then we can get a periodic soltion in [, T(ε] for h(t E ε sch that { z1 (t, t, z(t = z 2 (t, t. Next we will prove that ξ i, η i i = 1, 2 can be selected sitably sch that the two eqalities in (3.1 hold. First we consider the variational eqation (2.3 for t [, + ]. Sppose a linear transformation has already been made sch that the coefficient (3.

6 414 X. Li / Applied Mathematics Letters 23 ( matrix of linear variational system of (2.1 in origin is given by diag(a 1, A 2,, where Re σ (A 1 <, Re σ (A 2 >. Take γ (t = (x 1 (t, T γ (t W s for t large enogh, it is easy to see ( A1 + o(x 1 (t O(x 1 (t O(x 1 (t A(t = A 2 + o(x 1 (t. Since x 1 (t exponentially as t +, then A(+ = A = diag(a 1, A 2,. Similarly, we can get A( = A. The next lemma and its proof are a slightly extension of the Lemma 1 in [8]. Lemma 3.1. Given A(t, A and its Jordan form A, there exists a constant matrix C and a constant a >, sch that lim X(t, t e atā = C. t ± In order to give a precise analysis, we decompose the projections P + s, P+, P s, P, respectively, into the following sch that P ss + P s = P + s, P ss + P s = P, P s + P = P s, P s + P = P + RP ss = span{φ(t}, RP = span{ψ(t}. From (2.5, we can take z i ( satisfying P ss z i ( =, P + c z 1( =, P c z 2( =. (3.2 Then from the expression of z 1 (t, z 2 (t, we know that z 1 ( = z 2 ( is eqivalent to P s ξ 1 X(, P s η 2 X(, sp s h(sds =, (3.3 P s ξ 2 X(, T(εP s η 1 X(, sp s h(sds =, (3.4 T(ε P X(, η 2 P X(, T(εη 1 + T(ε We solve (3.3 and (3.4 for ξ 1, ξ 2 respectively and sbstitte them into the eqation z 1 (T(ε = z 2 ( P X(, sh(sds =. (3.5 and rearrange terms to get the eqation [ ] T(ε X(T(ε, X(, P s η 2 + X(, sp s h(sds + X(T(ε, s(p + s + P + c h(sds + (P s + P η 1 [ ] = X(, X(, T(εP s η 1 + X(, sp s h(sds + X(, s(p + P c h(sds + (P s + P η 2 T(ε which can be simplified into [ X(T(ε, P s + P X(, T(ε + X(, P s X(, T(ε]η 1 + [ X(T(ε, P s X(, + X(, (P s + P X(, ]η 2 T(ε T(ε = X(T(ε, s(p ss + P s + P + c h(sds + X(, sp s h(sds + X(T(ε, sp s h(sds + X(, s(p ss + P s + P c h(sds L(h, ε. (3.6 Notice that ẏ in (2.1. Based on assmption (H 2 and the constrction of fndamental soltion matrix X(t, s in [2], we know that T(ε X(T(ε, sp + c h(sds = X(, sp c h(sds.

7 X. Li / Applied Mathematics Letters 23 ( Also based on Lemma 3.1 and [8], we have ( In1 n2 sp t X(, (P s + P X(, C O n2 n 2 ( On1 n2 X(T(ε, (P s + P X(, T(ε C I n2 n 2 sp t O m m O m m C 1 e 2Mt <, C 1 e 2Mt <. Then (3.6 can be changed into {( ( In1 n In1 1 + C 1 n 1 [X(, (P s + P X(, C } { X(T(ε, P s X(, ]C X(, T(ε C ( I n2 n 2 C 1 η 2 + Owing to Lemma 3.1, we have [( In1 n 1 ] [ + P 1 (ε C 1 η 2 + I n2 n 2 ( I n2 n 2 C 1 + X(, P s X(, T(ε]C + P 2 (ε where P 1 (ε = o(e k1t(ε, P 2 (ε = o(e k2t(ε, k 1, k 2 are positive constants. If we take ( ( z1 C 1 η 1 = z 2, C 1 η 2 =, then (3.7 becomes [( In1 n 1 In2 n2 + P(ε ] C 1 C 1 [X(T(ε, (P s + P ] } C 1 η 1 = C 1 L(h, ε. C 1 η 1 = C 1 L(h, ε (3.7 η = C 1 L(h, ε (3.8 where η = (z 1, z 2,, P(ε = O(e kt(ε, k = min{k 1, k 2 }. Take ε > small enogh, then for < ε ε, (3.8 has soltion η(h for given h(t E ε, ths we get η 1, η 2 sch that z 1 (T(ε = z 2 (. We have now solved (3.3, (3.4 and z 1 (T(ε = z 2 (. It is obvios that if eqality (3.5 holds for given η 1, η 2, namely, h(t Eε h, we have the following reslts Lemma 3.2. If h(t E h ε, and assmption (H 1, (H 2 hold, system (3. has a 2T(ε-periodic soltion z ε (t for < ε 1. Now we discss the existence of periodic soltions to (1.1, and seek periodic soltions of very large period which in some sense are near γ (t. For convenience, we consider (2.14 and make the change of variable w(t = z(t + t s(t 1 2T(ε e(εt, where e(ε = s( s(t(ε. Now (2.14 becomes where ẇ(t = A(tw + Q (w, t + t, λ, ε, Q (w, t + t, λ, ε = f ( w(t + s(t + 1 2T(ε e(εt f (s(t A(tw 1 e(ε + εh. 2T(ε From (3.9, we know that system (2.14 has 2T(ε-periodic soltion if and only if (3.1 has 2T(ε-periodic soltion w(t satisfying w( = w(t(ε. The condition for this is Q Eε h which is eqivalent to the following bifrcation eqation T(ε ψ (tq (w, t + t, λ, εdt + ψ (η 2 ψ (T(εη 1 =. (3.9 (3.1

8 416 X. Li / Applied Mathematics Letters 23 ( Denote H(t, λ, ε = T(ε We obtain the following Theorem ψ (tq (w, t + t, λ, εdt + ψ (η 2 ψ (T(εη 1. (3.11 Theorem 3.3. If (H 1, (H 2 hold, sppose we have a point ( t, λ sch that H( t, λ, =, Hλ ( t, λ,, then there exists a parameter srface λ = λ(t, ε satisfying λ( t, = λ, sch that (1.1 has a 2T(ε-periodic soltion for λ = λ(t, ε, < ε 1. References [1] B. Deng, Homoclinic bifrcations with nonhyperbolic eqilibrim, SIAM. J. Math. Anal. 14 (3 ( [2] D.M. Zh, M.A. Han, Bifrcation of homoclinic orbits in fast variable space, Chin. Ann. Math. 23A (4 ( (in Chinese. [3] A.J. Hombrg, Singlar heteroclinic cycles, Jornal of Differential Eqations 161 ( [4] X.B. Li, D.M. Zh, Homoclinic bifrcation with nonhyperbolic eqilibria, Nonlinear Anal. 66 ( [5] K. Yagasaki, T. Wagenknecht, Detection of symmetric homoclinic orbits to saddle-center in reversible systems, Physica D 214 ( [6] F. Battelli, Saddle-node bifrcation of homoclinic orbits in singlar systems, Discrete Contin. Dyn. Syst. 7 ( [7] J. Grendler, Homoclinic soltions for atonomos O.D.E with nonatonomos pertrbation, Jornal of Differential Eqations 122 ( [8] M. Feckan, J. Grendler, Bifrcation from homoclinic to periodic soltion in singlar ordinary differential eqations, J. Math. Anal. Appl. 246 ( [9] C.R. Zh, The coexistence of sbharmonics bifrcated from homoclinic orbits in singlar systems, Nonlinearity 21 ( [1] Z.Y. Ye, M.A. Han, Bifrcations of sbharmonic soltions and invariant tori for a singlarly pertrbed system, Chin. Ann. Math. Ser. B 28 ( [11] D.M. Zh, M. X, Exponential trichotomy, othogonality condition and their application, Chin. Ann. Math., Ser. A 18 ( (in Chinese.

Conditions for Approaching the Origin without Intersecting the x-axis in the Liénard Plane

Conditions for Approaching the Origin without Intersecting the x-axis in the Liénard Plane Filomat 3:2 (27), 376 377 https://doi.org/.2298/fil7276a Pblished by Faclty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Conditions for Approaching