By Elhadj Zeraoulia, Julien Clinton Sprott
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Extra resources for 2-D Quadratic Maps and 3-D ODE Systems. A Rigorous Approach
The A1. There exist two subsets of Q, ˜ ˜ sets Q1 and Q2 are disjoint and compact, ˜ intersects both Q ˜ 1 and Q ˜ 2, A2. 16. (a) A connection Γ ˜ that intersects both Q ˜ 1 and Q ˜ 2. of Q ˜ for which (b) A preconnection γ˜ is a compact connected subset of Q f (˜ γ ) is a connection. (c) The cross number m is the largest number such that every connection contains at least m mutually disjoint preconnections. 8. Let f be the map satisfying Assumption A. Then there ex˜I ⊂ Q ˜ for which f ˜ 1 is semiconjugate to an ists a compact invariant set Q Q m-shift map10 .
2. 3). 1. 1). If e1 = −1, e2 = 0, l1 = 0, l2 = −1, and si = 0, i = 1, 2, then the resulting attractors are rotated through 180 degrees, and through 270 degrees if e1 = 0, e2 = −1, l1 = −1, l2 = 0, and si = 0, i = 1, 2, . More generally, if h(x, y) is a rotation by angle θ, then the attractors are also rotated by the same angle θ. 11) as shown in Fig. 6(a). 12) May 27, 2010 12:25 58 World Scientific Book - 9in x 6in 2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach (a) (b) Fig. 3. 59948 and a1 = 1.
13. Let ξ be a continuous piecewise linear vector field associated with a third-order autonomous system x′ = ξ (x), where x ∈ R3 . Assume the origin is an equilibrium point with a pair of complex eigenvalues (ρ + iω, ρ < 0, ω = 0) and a real eigenvalue γ > 0 satisfying |ρ| < γ. If in addition, ξ has a homoclinic orbit through the origin, then ξ can be infinitesimally perturbed into a nearby vector field ξ ′ with a countable set of horseshoes. Using this version, [Chua et al. 13 is satisfied. In particular, for proving the existence of a homoclinic orbit, they proved that there exist parameters such that the trajectory along the unstable real eigenvector associated with the origin enters the complex stable eigenspace, and hence returns to the origin as shown in Sec.
2-D Quadratic Maps and 3-D ODE Systems. A Rigorous Approach by Elhadj Zeraoulia, Julien Clinton Sprott