RENDICONTI DEL SEMINARIO MATEMATICO

Shadowing in ordinary differential equations 101(ii) the pseudo connecting orbit {w k } +∞k=−∞above is also ε-shadowed by a true orbit{z k } ∞ k=−∞ . Moreover, φt (z 0 ) is hyperbolic and there are real numbers ᾱ and αsuch that ‖φ t (z 0 )−φ t+ᾱ (x 0 )‖ → 0 as k → −∞ and ‖φ t (z 0 )−φ t+α (x 0 )‖ → 0as t → ∞.In the special case of the theorem above when the two periodic orbits coincide,we obtain a transversal homoclinic orbit as hyperbolicity of the connecting orbitimplies its transversality. Now, an additional condition is required to ensure that theconnecting orbit does not coincide with the periodic orbit. The condition given in thetheorem below is that there be a point on the pseudo homoclinic orbit sufficiently distantfrom the pseudo periodic orbit. With the setting as in the previous theorem, westate the following theorem.THEOREM 5. Homoclinic Orbit Shadowing Theorem. Suppose that {y k } +∞k=−∞is a δ pseudo periodic orbit with period N of Eq. (1). Let {w k } +∞k=−∞be a δ pseudohomoclinic orbit of Eq. (1) with associated times {h k } +∞k=−∞ connecting {y k} +∞k=−∞ ={y k+τ } +∞k=−∞ to {y k} +∞k=−∞ , where 0 ≤ τ < N. Suppose that the operator L w is invertiblewithThen if‖L −1w ‖ ≤ K.4Cδ < ε 0 , 4M 1 Cδ ≤ min{2, }, C 2 ( N 1 δ + N 2 δ 2 + N 3 δ 3) < 1,(i) the pseudo periodic orbit {y k } +∞k=−∞above is ε-shadowed by a true periodic orbit{x k } ∞ k=−∞ of period N where ε ≤ 2 ¯Cδ,moreover, φ t (x 0 ) is a hyperbolic (non-equilibrium) periodic orbit;(ii) the pseudo homoclinic orbit {w k } +∞k=−∞above is also ε-shadowed by a true orbit{z k } ∞ k=−∞ . Moreover, φt (z 0 ) is hyperbolic and there are real numbers ᾱ and αsuch that ‖φ t (z 0 )−φ t+ᾱ (x 0 )‖ → 0 as k → −∞ and ‖φ t (z 0 )−φ t+α (x 0 )‖ → 0as t → ∞. Furthermore, provided there exists r with p < r < q such that‖w r − y k ‖ > (‖ f (y k )‖ + 2M 1 ¯Cδ) eM 1(h max +ε 0 )−1+ 4 ¯Cδfor 0 ≤ k ≤ N − 1, then z r does not lie on the orbit of x 0 and so we mayconclude that z 0 is a transversal homoclinic point associated with the periodicorbit φ t (x 0 ).Notes on homoclinic shadowing: We prove the theorems above partly using earliertheorems. First we prove the existence of the periodic orbits using the Periodic ShadowingTheorem. Then we use the Infinite-time Shadowing Lemma to show the existenceof a unique orbit shadowing the pseudo homoclinic orbit. However, here weM 1