TRANSVERSALITY FOR TIME-PERIODIC COMPETITIVE-COOPERATIVE TRIDIAGONAL SYSTEMS

. Transversality of the stable and unstable manifolds of hyperbolic periodic solutions is proved for tridiagonal competitive-cooperative time-periodic systems. We further show that such systems admit the Morse-Smale property provided that all the ﬁxed points (of the corresponding Poincar´e map) are hyperbolic. The main tools used here are the integer-valued Lyapunov function, as well as the Floquet theory developed in [1] for general time-dependent tridiagonal linear systems.

1. Introduction. In this paper, we consider the dynamical properties of the following system with tridiagonal structure: x 1 = f 1 (t, x 1 , x 2 ), x n = f n (t, x n−1 , x n ), (1.1) where f = (f 1 , f 2 , · · · , f n ) is a C 1 -function defined on R × R n , it is assumed that there exists T > 0 such that Equations of the form (1.1) usually arise in modelling ecosystems of n species x 1 , x 2 , · · · , x n with a certain hierarchical structure. In such hierarchy, x 1 interacts only with x 2 , x n only with x n−1 , and for i = 2, · · · , n − 1, species x i interacts with x i−1 and x i+1 . Our standard assumption on the tridiagonal system (1.1) is that the variable x i+1 affectsẋ i and x i affectsẋ i+1 monotonically in the same fashion. More precisely, there are δ i ∈ {−1, +1}, i = 1, · · · , n − 1, such that for all (t, x) ∈ R × R n . This is one of the most commonly studied tridiagonal systems, called competitive-cooperative systems, in which individuals either compete or cooperate with their neighboring species. In particular, if δ i = −1 for all i, then (1.1) is called competitive. If δ i = 1 for all i, then (1.1) is called cooperative. Of course, when n = 2, system (1.1) naturally reduces to a two-dimensional competitive or cooperative system.
In the case where f is independent of t, Smillie [10] showed that all bounded trajectories of system (1.1) converge to equilibria. Under the assumption that f is time-periodic with periodic T > 0 (i.e., (1.2)), Hale and Somolinos [4] proved that if n = 2, then all bounded solutions are asymptotic to T -periodic solutions (see also [8] for Lotka-Volterra systems). Later, Smith [11] studied the T -periodic system (1.1) for general n-species and showed that every bounded solution is asymptotic to a T -periodic solution. When f is quasi-periodic or almost-periodic, Wang [12] investigated system (1.1) in the framework of skew-product flows and showed that the ω-limit set of any bounded orbit contains at most two minimal sets, and each minimal set is an almost automorphic extension of the base flow, which is driven by the time-translation for f . Very recently, by developing the theory of Floquet bundles for the associated time-dependent linear system of (1.1), Fang, Gyllenberg and Wang [1] proved that any hyperbolic ω-limit set is a 1-1 extension of the base flow.
Following [11], we now letx Therefore, we can always assume, without loss of generality, that the tridiagonal system (1.1) is cooperative, which means that In the theory of dynamical systems, transversality of stable and unstable manifolds of critical elements plays a central role in connection with structural stability. For the autonomous case (i.e., f is independent of t), Fusco and Oliva [3] have established transversality of the stable and unstable manifolds of hyperbolic equilibria and the structural stability of system (1.1). One can also refer to Hirsch [5] for the lower-dimensional (n ≤ 3) related cases.
In this article, we will focus on transversality of the stable and unstable manifolds, as well as structural stability, for the time-periodic tridiagonal competitivecooperative system (1.1). More precisely, we show that the stable and unstable manifolds of any hyperbolic fixed points for the Poincaré map associated with system (1.1)-(1.3) always intersect transversely (see Theorem 3.1). Based on this, we further prove that the time-periodic system (1.1) admits the Morse-Smale property provided that all the fixed points (of the corresponding Poincaré map) are hyperbolic (see Theorem 4.2). Our results here are a natural generalization of the results of Fusco and Oliva [3] to the time-periodic systems.
The main tool we used here is the integer-valued Lyapunov function σ, first defined by Smillie [10] (or Fusco and Oliva [2,3], see also similar forms by Mallet-Paret and Smith [7], and Mallet-Paret and Sell [6]). Besides, we further utilize the approach of the skew-product flows to generalize a technical lemma (by Fusco and Oliva [3, Lemma 9]) from the asymptotically autonomous cases to the asymptotically time-periodic cases (see Lemma 3.2). We then accomplished our approach by combining this lemma with the Floquet theory (which relates the values of σ to the Floquet solutions) developed in [1] for the general time-dependent linear tridiagonal systems.
The paper is organized as follows. In section 2, we introduce an integer-valued Lyapunov function and recall results from Floquet theory (established in [1]) for time-dependent linear systems that will play an important role in the proof of our main results. In section 3, we focus on transversality of the stable and unstable of hyperbolic periodic solutions of system (1.1). Finally, we discuss the Morse-Smale property of system (1.1).
2. Linear systems. In this section, we consider the linear tridiagonal system in the formẋ x n = a n,n−1 (t)x n−1 + a nn (t)x n , with all the coefficient functions being bounded and uniformly continuous on R.
We further assume that there is an ε 0 > 0 such that a i,i+1 (t) ≥ ε 0 , a i+1,i (t) ≥ ε 0 , for all t ∈ R and 1 ≤ i ≤ n − 1. Hereafter, we also write the corresponding matrix A(t) = (a ij (t)) n×n . We will introduce an integer-valued Lyapunov function σ associated with (2.1). Following [10], we define a continuous map σ : where # denotes the cardinality of the set. Note that Λ is open and dense in R n and it is the maximal domain on which σ is continuous. Besides, it is also useful to define two integer-valued functions (see, e.g. [3]) σ m : R n → {0, 1, · · · , n − 1}, σ M : R n → {0, 1, · · · , n − 1}, by letting σ m , σ M be the minimum and maximum value of σ(x ) for x ∈ U ∩ Λ, where U is a small neighborhood of x. Note that x ∈ Λ is equivalent to σ m (x) = σ(x) = σ M (x). We are now collecting certain properties of σ, which are stated in the following lemma. When A(t) is periodic in time t, we have the following lemma concerning the Floquet multipliers of (2.1) and the corresponding eigenspaces.
For the general time-dependent system (2.1) without periodicity assumption on A(t), we now introduce the so called "Floquet space" of A(t), which will be useful in the forthcoming sections.
For any fixed integers m and l satisfying 0 ≤ m ≤ l ≤ n − 1, we define the set, called the Floquet space of A, When m = l, we write W m,l (A) as W m (A) for brevity. The following proposition was proved in [1]. 3. Transversality. In this section, we shall prove that the stable and unstable manifolds of hyperbolic T -periodic solutions of (1.1)-(1.3) intersect transversely.
Let p(t) be a T -periodic solution of (1.1)-(1.3). Consider the linearized equation of (1.1) along p(t):ż = Df (t, p(t))z, t ∈ R, z ∈ R n , (3.1) which is a T -periodic linear equation in the form of (2.1). p(t) is called hyperbolic if none of its Floquet multipliers is on the unit-circle S 1 ⊂ C.
Motivated by [3], for any given integer −1 ≤ h ≤ n − 1, let K h and K h be the sets In particular, we set K −1 = {0} and K −1 = R n . The sets K h and K h so defined are cones. Moreover, K h and K h are open sets, Lemma 3.3. Let A(t) be the coefficient matrix of (2.1) and K h , K h be the corresponding cones. Then: Proof. For (i), we only prove that W m,l (A) ⊂ K m−1 , since the other is similar. It is obvious that W 0,l ⊂ K −1 = R n . For any m > 0, if x ∈ W m,l (A) ∩ Λ, then it follows from the definition of W m,l (A) that σ m (x) = σ(x) > m − 1, so we only need to consider the case that a nonzero vector x ∈ W m,l (A) \ Λ. Choose a small t 0 > 0 so that x(t 0 ) ∈ Λ. By the definition of W m,l (A), one has σ(x(t 0 )) > m − 1. Suppose now that σ m (x) ≤ m − 1, then it follows from the definition of σ m that there exists a sequence y n ∈ Λ with σ(y n ) ≤ m − 1 such that y n → x as n → +∞. Then the continuity of σ implies that σ(y n (t 0 )) = σ(x(t 0 )) > m − 1, for n sufficiently large. On the other hand, clearly σ ( y n (t 0 )) ≤ σ(y n ) ≤ m − 1, a contradiction. This implies that W m,l ⊂ K m−1 . Thus (i) has been proved. See [3,Lemma 8] for (ii) and (iii).
Now we are ready to prove Theorem 3.1.
Proof of Theorem 3.1. Without loss of generality, we only prove W u (p − (0)) W s (p + (0)), because the general "τ " case can be deduced similarly. And also, for brevity, we just let the period T = 1.
Let m ± = dimW u (p ± (0)), then m + ≤ m − − 1. It then follows that the n − m − vectors q + m − +1 , · · · , q + n are in being the eigenspace associated with j-th Floquet multiplier ofẋ = A + (t)x for j = m − + 1, · · · , n. Let Σ = span{q + m − +1 , · · · , q + n } = ⊕ n j=m − +1 E α + j , then by is open and W s (p + (0)) is a smooth manifold. Then for any integer z, it follows from Σ ⊂ K m − ∩ T p + (0) W s (p + (0)) that there exists some z 0 > z sufficiently large such that T φ(z 0 ) W s (p + (0)) contains an (n − m − ) dimensional linear space Σ 0 , which is also contained in K m − . Let Σ z be the image of Σ 0 under the linear equation (in backward time)ẏ = D x f (t, φ(t, 0, x 0 ))y. Then Σ z ⊂ T φ(z) W s (p + (0)), and moreover, Lemma 3.3(iii) implies that dimΣ z = n − m − and Σ z ⊂ K m − . By the similar arguments and Lemma 3.3(ii), one yields that T φ(z) W u (p − (0)) contains an m − dimensional linear subspace Σ z ⊂ K m − . Noticing that dimΣ z + dimΣ z = n and K m − ∩ K m − = {0}, one obtains that R n = Σ z ⊕ Σ z . Thus, we have completed the proof. For any x ∈ R n , we define the Poincaré map P associated with φ(t, 0, x) by letting P (x) = φ(T, 0, x). P is said to be point dissipative, if there exists a bounded set B in R n , for any x ∈ R n there exists a m 0 ∈ N such that P m (x) ∈ B for all m > m 0 .
x ∈ R n is said to be the periodic point of period m of P if P m (x) = x, P j (x) = x, j = 1, 2, · · · , m − 1. A periodic point of period m is hyperbolic if the spectrum of DP m (x) does not lie on the unit-circle S 1 ⊂ C. In particular, if m = 1, we call that x is a fixed point of P , moreover, if the spectrum of DP (x) does not lie on the unit-circle S 1 ⊂ C we call x a hyperbolic fixed point of P .
It follows from Lemma 4.3 that P is point dissipative. In order to prove (M1), it suffices to show that N (P ) ⊂ F ix(P ). Here N (P ) is the set of the nonwandering points of P and F ix(P ) the set of the fixed points of P .
It then follows from Lemma 2.1 that ψ l (t) ∈ Λ for any l ≥ K and t ∈ [N, k l + N ]. So, ψ l 1 (t) = 0 for any t ∈ [N, k l + N ] and l ≥ K. Noticing that ψ l 1 (N ) → ψ 1 (N ) as l → ∞, one has ψ l 1 (t) > 0 for any N ≤ t ≤ k l + N , with l ≥ K. As a consequence, φ 1 (k l + N, x l ) > φ 1 (N + 1, x l ), for any l ≥ K. Letting l → ∞, we conclude that φ 1 (N, x 0 ) ≥ φ 1 (N + 1, x 0 ), which contradicts the assumption that ψ 1 (N ) > 0. Thus, we have proved that N (P ) ⊂ F ix(P ). Moreover, since P is satisfied for the condition (H) and F ix(P ) is a closed set in R n , F ix(P ) ⊂ A is compact. Due to the fact that all the fixed points P are hyperbolic, F ix(P ) is a discrete set. So, N (P ) = F ix(P ) is a finite set. Thus, (M1) is satisfied. As for (M2), Theorem 3.1 already guarantees transverality of the stable and unstable manifolds between hyperbolic T -periodic solutions. Thus, we have completed the proof.