To prove Theorem 1.6, it remains to show that generically there does not exist non-wandering elements which are not critical elements and that generically the number of critical elements is finite. We emphasize that the dynamics of (1.1) may have non-trivial non-wandering elements. Indeed, as shown in [57], every two-dimensional flow can be re-alized in the dynamics of (1.1). Thus, one can for example create a sequence of periodic orbits, which piles up on a homoclinic orbit. This orbit is then a non-wandering orbit which is not critical. However, non-trivial non-wandering orbits are generically precluded.
Proposition 6.1. Assume thatf is a non-linearity such that the dynamics of (1.1)satisfy the following properties:
- there exists a compact global attractor for (1.1).
- All the equilibria and periodic orbits of (1.1) are hyperbolic.
- There is no homoclinic orbit and all the heteroclinic orbits are transversal.
Then, the set of non-wandering elements consists in a finite number of equilibrium points and periodic orbits.
As usual in this article, the property stated in Proposition 6.1 has its equivalent for two-dimensional dynamical systems. It mainly relies on Poincar´e-Bendixson property, proved in [14] for (1.1). Proposition 6.1 is the key point to deduce the genericity of Morse-Smale property from the genericity of Kupka-Smale property. The genericity of Morse-Smale property for dynamical systems of orientable surfaces shown in [48] also relies on a similar property, see [43].
We enhance that, if f is such that (1.1) admits a compact global attractor and that any equilibrium point and any periodic orbit are hyperbolic, then there is at most a finite number of equilibrium points. However, as we explained above, there could exist an infinite number of hyperbolic periodic orbits: think of a sequence of hyperbolic periodic orbits
piling up to a homoclinic orbit. One can only ensure that there is a finite number of hyperbolic periodic orbits with a period less than a given number.
We begin the proof of Proposition 6.1 by several lemmas. We assume in the whole section that f has been chosen so that the hypotheses of Proposition 6.1 are satisfied.
LetC be a hyperbolic equilibrium point or periodic orbit of (1.1). We recall that there exists an open neighborhoodB of C inHs(S1) such that each global solutionu(t) of (1.1), satisfying u(t)∈B for all t≤0, belongs to the local unstable manifold Wlocu (C). We refer for example to [18], [51].
Lemma 6.2. Let C be a hyperbolic equilibrium point or periodic orbit of (1.1) and let B be the neighborhood of C as described above. Let (un(t))n∈N be a sequence of solutions of (1.1) such that, for each n∈N, there exist three timesσn < tn< τn such that the following properties hold. For all t ∈(σn, τn), un(t)∈B, un(σn)∈∂B, un(τn)∈∂B and,
d(un(tn),C) := inf
c∈Ckun(tn)−ckHs(S1) −−−−−−→
n−→+∞ 0 .
Then, there exist an extraction ϕ and a globally defined and bounded solution u∞(t) of (1.1) such that u∞(t)∈Wlocu (C), t ≤0, and
∀ T > 0, sup
t∈[−T,T]
°°uϕ(n)(τϕ(n)+t)−u∞(t)°
°Hs(S1)−−−−−−→
n−→+∞ 0 . (6.1)
Proof: First, we claim that τn−tn −→+∞ when n →+∞. Indeed, if this is not true, since C is compact, we can extract a subsequence uψ(n)(tψ(n)) converging to some c ∈ C and such that τψ(n)−tψ(n) converges to some t ≥ 0. Then, by continuity of the Cauchy problem related to (1.1), uψ(n)(τψ(n)) converges to a point ofC, which contradicts the fact that uψ(n)(τψ(n))∈∂B.
We setT = 1. Since (1.1) admits a compact global attractor and that (τn) converges to +∞, the sequence un(τn−T) is precompact in Hs(S1) and there is an extraction ϕ1 such that uϕ1(n)(τϕ1(n) −T) converges to some u∞(−T) ∈ Hs(S1). Let u∞(t) = S(t+T)u∞(−T), t ≥ −T, be the solution of (1.1) associated to u∞(−T). By continuity of the Cauchy problem related to (1.1), uϕ1(n)(τϕ1(n) +t) converges to u∞(t) uniformly with respect to t∈[−T, T]. To achieve the proof of uniform convergence of un(t) to u∞(t) on any compact set of time, it is sufficient to repeat the argument for all T ∈ N and to use the diagonal extractionϕ(n) = ϕn◦...◦ϕ1(n).
Finally, let us notice that u∞(t) belongs to Wlocu (C). Indeed, since un(t) ∈ B for all t∈[tn, τn) and that τn−tn −→+∞, u∞(t)∈B for allt ≤0. Due to the choice ofB, this
implies that u∞(t)∈Wlocu (C). ¤
Let M ∈ R∪ {+∞}. We use the notation [[1, M + 1]] = {k ∈N,1≤ k ≤M + 1}. We say that a sequence of critical elements (Ck)k∈[[1,M+1]] is connected if for any k ∈ [[1, M]],
there exists a heteroclinic orbit uk(t) such that the α−limit set of uk(t) is Ck and its ω−limit set is Ck+1. We recall that a chain of heteroclinic orbits denotes the sequence of heteroclinic orbits corresponding to a connected sequence of critical elements (Ck)k∈[[1,p+1]]
with Cp+1 =C1.
Lemma 6.3. Assume thatf is as in Proposition 6.1. Then, there is no connected sequence of critical elements with infinite length. As a consequence, there is no chain of heteroclinic orbits and, every ω−limit set and every non-empty α−limit set of trajectories of (1.1) consist exactly of one critical element
Proof: LetM ∈N∪ {+∞}and let (Ck)k∈[[1,M+1]] be a connected sequence of closed orbits with heteroclinic connections (uk(t))k∈[[1,M]]. We consider the Morse indices i(Ck) of the closed orbits. We have several cases:
- if Ck and Ck+1 are both periodic orbits, then Theorem 1.2 shows that i(Ck)> i(Ck+1).
- if Ck is an equilibrium point and ifCk+1 is an equilibrium point or a periodic orbit, then dim(Wu(Ck))=i(Ck) and codim(Ws(Ck+1))=i(Ck+1). Thus, since the intersection of Wu(Ck) and Ws(Ck+1) is non-empty and transversal, one must have i(Ck)> i(Ck+1).
- if Ck is a periodic orbit and Ck+1 is an equilibrium, then dim(Wu(Ck))=i(Ck)+1 and codim(Ws(Ck+1))=i(Ck+1). Therefore, i(Ck)≥i(Ck+1).
Hence, the Morse index of Ck is non-increasing and decreases except ifuk goes from a peri-odic orbit to an equilibrium point. However, a sequence cannot consist only of connections from a periodic orbit to an equilibrium and the Morse index must decrease at least every two steps. Therefore, since the Morse indices are non-negative, M is bounded by 2i(C0).
The non-existence of connected sequence of critical elements of infinite length precludes the existence of chains of heteroclinic orbits since every chain (Ck)k∈[[1,p+1]] with Cp+1 = C1 can be extended to a periodic connected sequence of critical elements and thus to a connected sequence of infinite length.
Let u0 ∈ Hs(S1) be chosen such that its ω−limit set ω(u0) is not a unique periodic orbit. Then, the Poincar´e-Bendixson property stated in Theorem 1.1 shows that ω(u0) consists of equilibrium points and homoclinic or heteroclinic orbits connecting them. We know that homoclinic orbits are precluded. Moreover, since there is no connected sequence of equilibria of infinite length, there exists an equilibrium point e where no connected sequence can be extended, that is, such that Wu(e)∩ω(u0) = {e}. Let Be be a small neighborhood of e in Hs(S1) such that any solution u(t) of (1.1) satisfying u(t) ∈ Be for any t ≤ 0, belongs to the unstable manifold Wu(e). If ω(u0) 6= {e}, then one easily constructs three sequences of times (σn), (tn) and (τn) going to +∞ and satisfying the hypotheses of Lemma 6.2 with C ={e} and un(t) =S(t)u0. But then Lemma 6.2 implies that there exists a solution u∞(t) of (1.1) belonging to the unstable manifold Wu(e) and with u(τϕ(n)) convergingu∞(0). Thereforeu∞(0)∈ω(u0)∩∂Be and Wu(e)∩ω(u0)6={e}, which leads to a contradiction. Therefore, ω(u0) ={e}. ¤
Proof of Proposition 6.1: Let ˜u0 ∈Hs(S1) be a non-wandering element and let ˜u(t) = S(t)˜u0. Using the definition of a non-wandering element, we easily construct a sequence of trajectories un(t) such that un(0) converges to ˜u0 and, for any sequence (tn), there exists a sequence (t0n) such that t0n > tn and un(t0n) converges to ˜u0. By Lemma 6.3, there exists a hyperbolic critical element C1 such that ω(˜u0) = {C1}. Let B1 be a neighborhood of C1 as in Lemma 6.2. Assume that ˜u0 6∈ C1. Replacing B1 by a smaller neighborhood if needed, we may assume that ˜u0 6∈B1. There is a sequence of times (tn) and a pointc∈ C1 such that ˜u(tn) → c. By continuity of the Cauchy problem, we may assume without loss of generality that un(tn) → c. As we can find a sequence of times t0n such that t0n > tn and un(t0n) → u˜0 6∈ B1, there exists a sequence of times τn1 such that un(t) ∈ B1 for t ∈ [tn, τn1) and un(τn1) ∈ ∂B1. By Lemma 6.2, we may assume without loss of generality that un(τn1 +t) converges to some ˜u1(t) ∈ Wu(C1). Now, we can repeat the arguments:
there exist a critical elementC2 such that ω(˜u1(t)) = {C2} and a sequence of timesτn2 such that, up to an extraction, un(τn2 +t) converges to some solution ˜u2(t) ∈ Wu(C2) and so on... Thus, we are constructing a connected sequence of critical elements of infinite length, which is precluded by Lemma 6.3. This means that ˜u0 belongs to C1, that is, that our non-wandering element either is an equilibrium point or belongs to a periodic orbit.
To finish the proof of Proposition 6.1, it suffices to show that the number of critical elements is finite. First, as noticed earlier, due to the compactness of the global attractor and the hyperbolicity of the equilibrium points and the periodic orbits, the number of equilibrium points and periodic orbits of smallest period bounded by a given number is finite. Thus we only need to show that there does not exist an infinite sequence of periodic orbits γn(x, t) of smallest period pn, where pn tends to infinity when n goes to infinity. If we had such a sequence, we would be able to repeat the arguments of the first part of the proof and, by using lemmas 6.2 and 6.3, construct a connected sequence of critical elements
of infinite length, which leads to a contradiction. ¤