Nonlinear Dynamical Systems Ninth Class

Transcription

1 Nonlinear Dynamical Systems Ninth Class Alexandre Nolasco de Carvalho September 21, 2017

2 Lemma Let {T (t) : t 0} be a dynamically gradient semigroup in a metric space X, with a global attractor A and a disjoint collection of isolated invariant sets Ξ = {Ξ 1,, Ξ n }. Then there exists 1 i n such that Ξ i is a local attractor.

3 Proof: Assume that there are no local attractors in Ξ. Since Ξ 1 is not a local attractor, then for ɛ 0 > 0 suitably small, given ɛ > 0 there exists x 1 O ɛ (Ξ 1 ) A and a real number τ 1 such that T (τ 1 )x 1 / O ɛ0 (Ξ 1 ). By property (G1), there exists t 1 >τ 1 such that T (t 1 )x 1 O ɛ (Ξ j ) for some 1 j n. If j = 1, property (G2) gives us a contradiction. If not, since Ξ j is not a local attractor, we can repeat the argument to construct an ɛ chain, since there are only a finite numbers of elements in Ξ, for each small ɛ > 0, which contradicts (G2) and concludes the result.

4 Next we describe the construction of a Morse decomposition of the attractor of a dynamically gradient semigroup. Let {T (t) : t 0} be a dynamically gradient semigroup with associated disjoint collection of isolated invariant sets Ξ = {Ξ 1,, Ξ n }. If (reordering as needed) Ξ 1 is a local attractor for {T (t):t 0} and Ξ 1 = {a A : ω(a) Ξ 1 = }, then each Ξ i, i >1 is contained in Ξ 1 and that for any a / A\{Ξ 1 Ξ 1 } and global solution φ : R A with φ(0) = a we have that Ξ 1 t φ j (t) t Ξ 1.

5 We already know that Ξ 1 is invariant under the action of {T (t) : t 0} and hence we can consider the restriction T 1 (t) of T (t) to Ξ 1 =: {Ξ 1 } 0. Note that making this restriction, we have removed all possible global solutions φ : R A such that φ(t) Ξ 1 as t ; that is, if φ 1 : R A is a global solution of {T 1 (t) : t 0} and φ 1 (t) Ξ i, then i > 1.

6 Also, it is simple to see that {T 1 (t) : t 0} is a dynamically gradient semigroup in Ξ 1 with disjoint collection of isolated invariant sets {Ξ 2,, Ξ n }, and this implies that if a global t solution φ : R A satisfies Ξ j φ(t) t Ξ 1, then we must have j 1. We may assume, without loss of generality, that Ξ 2 is a local attractor for the semigroup {T 1 (t) : t 0} in Ξ 1.

7 If {Ξ 2 } 1 is the repeller associated to the isolated invariant set Ξ 2 for {T 1 (t) : t 0} in Ξ 1 we may proceed and consider the restriction {T 2 (t) : t 0} of the semigroup {T 1 (t) : t 0} to {Ξ 2 } 1 and {T 2(t) : t 0} is a dynamically gradient semigroup in {Ξ 2 } 1 with associated disjoint collection of isolated invariant sets {Ξ 3,, Ξ n }. Note that is this case, we have removed all the global solutions φ : R X such that φ(t) Ξ 2 as t and therefore, if t Ξ j φ(t) t Ξ 2 then j 2.

8 Proceeding with this until all isolated invariant sets are exhausted we obtain a reordering of {Ξ 1,, Ξ n } in such a way that Ξ j is a local attractor for the restriction of {T (t) : t 0} to {Ξ j 1 } j 2 ({Ξ 0 } 1 = A). With the construction above, if a global solution φ : R A satisfies t Ξ l ξ(t) t Ξ k (1) then l k.

9 This proves that this reordering of Ξ = {Ξ 1,, Ξ n } (which we denote the same) is a Morse decomposition to A. Now we will prove that, for a suitably chosen sequence of local attractors = A 0 A 1 A 2 A n = A, we can construct the Morse decomposition given by Ξ, and to this end, using W u (Ξ i ) = {x X : there exists a global solution ξ : R X with ξ(0) = x such that ξ(t) t Ξ i }, we can define A 0 =, A 1 = Ξ 1 and for j = 2, 3,, n A j = j W u (Ξ i ). (2) i=1

10 It is clear that A n = A. With this construction, we can prove that each A j, j = 0,, n, is a local attractor, and if we consider the correspondent repellers A j, then the sequence of attractor-repeller pairs {(A j, A j )} j=0,,n generates exactly the Morse decomposition given by Ξ; that is, Ξ j = A j A j 1 for all j = 1,, n and n j=0 (A j A j ) = n i=1 Ξ i.

11 Lemma The A j defined in (2) is compact, for each j = 0,, n. Proof: Le {x k } k N be a sequence in A j = j i=1 W u (Ξ i ) such that x k x A. Clearly, there is a 1 i 1 j and subsequence, which we denote the same, such that x k W u (Ξ i1 ). If {x k } converges to j i=1 Ξ i it follows that x A j. Assume that dist({x k : k N}, j i=1 Ξ i) > 0. Through each x k (taking subsequences if needed) there are global solutions ξ k : R A and ξ k (t) t Ξ i1 and ξ k (t) t Ξ i0 t there are a global solutions ζ l :R A with Ξ i some i 0. Then ζ l (t) t Ξ j, i 0 i < j i 1, 1 l i 1 i 0. ζ l0 (0) = x and and therefore x W u (Ξ l0 ) A j.

12 Theorem Let {T (t) : t 0} be a dynamically gradient semigroup with associated disjoint collection of isolated invariant sets Ξ = {Ξ 1,, Ξ n } reordered in such a way that it is a Morse decomposition. Then A j defined in (2) is a local attractor for {T (t) : t 0} in X and Ξ j = A j A j 1, j = 1,, n.

13 Proof: From a previous lemma, it is sufficient to prove that A j = A j 1 W u (Ξ j ) is a local attractor for {T (t) : t 0} restricted to A. It follows from the fact that A j and n i=j+1 Ξ i are compact and disjoint (exercise: prove that A j and Ξ i are disjoint for all i > j) that there exists δ 0 >0 such that O δ0 (A j ) ( n i=j+1 Ξ i)=. If there are δ < δ 0 and δ < δ such that γ + (O δ (A j )) O δ (A j ), then ω(o δ (A j )) attracts O δ (A j ) and (as ω(o δ (A j )) is invariant) is contained in A j proving that A j is a local attractor.

14 If not, there is a sequence {x k } k N in A \ A j with d(x k, A j ) k 0, for each x k a global solution ξ k : R A through x k which converges to n i=j+1 Ξ k i as t and a sequence t k, such that d(ξ k (t), A j ) δ for all t [0, t k ] and d(ξ k (t k ), A j ) = δ. In this way, we construct a global solution ξ : R A such that ξ(0) / A j and d(ξ(t), A j ) δ for all t 0. This and the fact that {T (t): t 0} is a dynamically gradient semigroup gives us a contradiction, since ξ(t) Ξ k as t for some 1 k j, which implies that ξ(0) A j.

15 To prove that Ξ j = A j A j 1 note that A j = j W u (Ξ i ) i=1 and A j 1 = {z A : ω(z) A j 1 = }. Hence, given z A j A j 1 we have that the global solution ξ : R A through z must satisfy that {Ξ i : 1 i j} t ξ(t) t {Ξ i : j i n}.

16 As a consequence, together with the fact that {T (t) : t 0} is a dinamically gradient semigroup with disjoint collection of isolated invariant sets {Ξ 1,, Ξ n }, for which any global solution t ξ : R A satisfies Ξ l ξ(t) t Ξ k with l k, we obtain that z Ξ j. This shows that A j A j 1 Ξ j. The other inclusion is immediate from the definition of A j and A j 1.

17 Proposition Let {T (t) : t 0} be a dinamically gradient semigroup with associated disjoint collection of isolated invariant sets Ξ = {Ξ 1,, Ξ n } reordered in such a way that it is a Morse decomposition. Then, n (A j A j ) = j=0 n i=1 Ξ i

18 Proof: If z n i=1 Ξ i, let k {1, 2,, n} be such that z Ξ k = A k A k 1. Hence z A k A k+1 A n and z A k 1 A k 2 A 0. Thus z n j=k k 1 A j A j j=1 n j=k k 1 (A j A j ) (A j A j ) = j=0 proving the inclusion n i=1 Ξ i n j=0 A j A j. n (A j A j ), j=0

19 Now, let z n j=0 (A j A j ) and I := {i 1, i 2,, i k } J := {j 1, j 2,, j l } such that I J = {0, 1,, n} with I J = and z A i for all i I and z A j for all j J. Clearly, if i := min I, necessarily I = {i, i + 1, i + 2,, n} and J = {0, 1,, i 1}, consequently z A i and z A i 1. So, z A i A i 1 = Ξ i, from which we conclude that n j=0 (A j A j ) n i=1 Ξ i and the proof is completed.