By Jack K. Hale, Luis T. Magalhaes, Waldyr Oliva,

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**Extra resources for Dynamics in Infinite Dimensions, 2nd Edition**

**Example text**

For any set S ⊂ C 0 , one can deﬁne C Φt ϕ. ω(S) = τ ≥0 t≥τ ϕ∈S In a similar way, if x(t, ϕ) is a solution of the RFDE(F ) for t ∈ (−∞, 0], x0 (·, ϕ) = ϕ, one can deﬁne the α-limit set of the negative orbit {xt (·, ϕ), −∞ < t ≤ 0}. Since the map Φt may not be one-to-one, there may be other negative orbits through ϕ and, thus, other α-limit points. To take into account this possibility, we deﬁne the α-limit set of ϕ in the following way. For any ϕ ∈ C 0 and any t ≥ 0, let H(t, ϕ) = ψ ∈ C 0 : there is a solution x(t, ϕ) of the RFDE(F ) on (−∞, 0], x0 (·, ϕ) = ϕ, x−t (·, ϕ) = ψ and deﬁne the α-limit set α(ϕ) of ϕ as C α(ϕ) = τ ≥0 H(t, ϕ).

Perturbing α and β while maintaining α/β = constant we obtain hyperbolicity. 1, by the use of spherical coordinates y1 = cos ψ cos ϕ, y2 = cos ψ sin ϕ, y3 = sin ψ, the equation π(A) on S 2 can be written ˙ ψ(t) = − α2 + β 2 ϕ(t) ˙ = α2 + β 2 1/2 1/2 sin ψ(t) cos ψ(t − 1) cos ϕ0 − ϕ(t) + ϕ(t − 1) cos ψ(t − 1) sin ϕ0 − ϕ(t) + ϕ(t − 1) cos ψ(t) where 0 < ϕ0 < π satisﬁes cos ϕ0 = α2 + β 2 2 sin ϕ0 = α + β In the equator of S 2 , we have −1/2 2 −1/2 α β. 2 Examples of RFDE on manifolds ϕ(t) ˙ = α2 + β 2 1/2 sin ϕ0 − ϕ(t) + ϕ(t − 1) .

Then (k, s) ∈ l, (ln s)−1 = t − r and so f (ϕ) = i(k, s) = −[t − r + r]−2 exp[t − r + r]−1 = −t−2 exp[t]−1 = u (t) = ϕ (0). c) For (ln h)−1 ≤ t ≤ (ln ε)−1 and ϕ = ut ∈ C, ε ≤ k = ρ0 (ϕ) = u(t) = exp[t]−1 ≤ h, t = (ln k)−1 , ϕ(−r) = u(t − r) = exp[t − r]−1 , s = S(ϕ) = z(k)k+(1−z(k)) exp(t−r)−1 , and so s = z(k)[k−exp(t−r)−1 ]+exp(t−r)−1 or s= exp j(k)−1 −exp(t − r)−1 [k−exp(t − r)−1 ]+exp(t − r)−1 = exp j(k)−1 . k − exp(t−r)−1 j(k) = (ln s)−1 = t + Ψ (k), (k, s) ∈ l, then f (ϕ) = i(k, s) = −[t + Ψ (k) − Ψ (k)]−2 exp[t + Ψ (k) − Ψ (k)]−1 = − t12 exp t−1 = u (t) = ϕ (0).