By Elhadj Zeraoulia, Julien Clinton Sprott
This e-book relies on study at the rigorous facts of chaos and bifurcations in 2-D quadratic maps, specifically the invertible case similar to the H?©non map, and in 3-D ODE's, specially piecewise linear structures comparable to the Chua's circuit. additionally, the booklet covers a few fresh works within the box of common 2-D quadratic maps, specifically their class into equivalence periods, and discovering areas for chaos, hyperchaos, and non-chaos within the house of bifurcation parameters. Following the most advent to the rigorous instruments used to end up chaos and bifurcations within the consultant platforms, is the examine of the invertible case of the 2-D quadratic map, the place prior works are orientated towards H?©non mapping. 2-D quadratic maps are then labeled into 30 maps with famous formulation. proofs at the areas for chaos, hyperchaos, and non-chaos within the house of the bifurcation parameters are provided utilizing a strategy according to the second-derivative attempt and limits for Lyapunov exponents. additionally incorporated is the facts of chaos within the piecewise linear Chua's approach utilizing equipment, the 1st of that is in line with the development of Poincar?© map, and the second one relies on a computer-assisted evidence. eventually, a rigorous research is equipped at the bifurcational phenomena within the piecewise linear Chua's process utilizing either an analytical 2-D mapping and a 1-D approximated Poincar?© mapping as well as different analytical equipment.
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Extra info for 2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach
Snap-back repellor): Assume that P is an expanding fixed point of f in Br (P ) for some r > 0. Then P is said to be a snap-back repellor of f if there exists a point P0 ∈ Br (p) with P0 = P such that f m (P0 ) = P and the determinant |Df m (P0 )| = 0 for an integer m > 0. 14. , an uncountable set S containing no periodic points of f such that: (b-1) f (S) ⊂ S, (b-2) for every XS ; YS ∈ S with XS = YS , limk−→+∞ sup f k (XS ) − f k (YS ) > 0, (b-3) for every XS ∈ S and any periodic point Yper of f , limk−→+∞ sup f k (XS ) − f k (Yper ) > 0; (c) there is an uncountable subset S0 of S such that for every XS0 ; YS0 ∈ S0 : limk−→+∞ sup f k (XS0 ) − f k (YS0 ) = 0.
Push the subintervals onto the stack 13. endif 14. endif 15. end while 16. print that the condition is proven and stop. 2 Efficacy of the checking routine algorithm Note that each step in the checking routine algorithm needs some additional explanations in view of optimization theory [Dellnitz and Junge (2002)]. Here are some theorems that guarantee the efficacy of the checking routine algorithm [Tibor et al. 16. Assume that the underlying mapping Υ is given by an inclusion mapping T and that the algorithm returns that the checked condition Υ(Q′ ) ⊂ O′ is fulfilled.
1987)]. 22. (a) The invariant manifold of a map f is a set of points X such that f (X) ⊂ X 12 . (b) Given a fixed point Y0 in the invariant manifold, the manifold Y is called stable if ∀y ∈ Y , limn→∞ f n (y) → Y0 . Similarly, a manifold is called unstable if ∀y ∈ Y , limn→∞ f −n (y) → Y0 . (c) A fixed point is hyperbolic if it is the intersection of one or more stable manifolds and one or more unstable manifolds. (c) A homoclinic point is a point x, different from a fixed point, that lies on both a stable manifold and an unstable manifold of the same fixed point Q.