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By Elhadj Zeraoulia, Julien Clinton Sprott

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Thus f (x) is either a homoclinic point or the fixed point Q. Since f takes a point on the unstable manifold W u (Q) away from the fixed point Q, then f (x) cannot be the fixed point. Thus it is a homoclinic point. For the same reason, it is not one of the n considered homoclinic points. Thus there exist n + 1 homoclinic points. 11 The homoclinic tangency is the tangled intersection of such invariant manifolds with a homoclinic point as shown in Fig. 5. 12 Of course every fixed point, with the exception of centers, will be an element of some invariant manifold.

42) if the eigenvalues of the Jacobian A = Df (P ) are γ, ρ + iω, where ρ, γ < 0, and ω = 0. , limt−→+∞ γ (t) =limt−→−∞ γ (t) = P . , limt−→+∞ δ (t) = P1 , and limt−→−∞ γ (t) = P2 . 11. Assume the following: April 27, 2010 14:29 World Scientific Book - 9in x 6in ws-book9x6 27 Tools for the rigorous proof of chaos and bifurcations (i) The equilibrium point P is a saddle focus, and |γ| > |ρ| . (ii) There exists a homoclinic orbit based at P . Then (1) The Sil’nikov map, defined in a neighborhood of the homoclinic orbit of the system, possesses a countable number of Smale horseshoes in its discrete dynamics.

If there exists a single homoclinic point on a stable and an unstable invariant manifold corresponding to a particular hyperbolic fixed point, then there exist an infinite number of homoclinic points on the same invariant manifolds. Proof. The proof is done by induction on the number of homoclinic points. Assume that there exist n homoclinic points for these invariant manifolds. Let W s (Q) be the stable manifold and W u (Q) be the unstable manifold, and let x be the homoclinic point farthest from the fixed point Q along the unstable manifold W u (Q).

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