By Elhadj Zeraoulia, Julien Clinton Sprott

ISBN-10: 9814307742

ISBN-13: 9789814307741

This ebook relies on study at the rigorous facts of chaos and bifurcations in 2-D quadratic maps, specifically the invertible case resembling the H?©non map, and in 3-D ODE's, in particular piecewise linear platforms reminiscent of the Chua's circuit. additionally, the booklet covers a few contemporary works within the box of common 2-D quadratic maps, in particular 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 turn out chaos and bifurcations within the consultant structures, is the learn of the invertible case of the 2-D quadratic map, the place past works are orientated towards H?©non mapping. 2-D quadratic maps are then categorized into 30 maps with famous formulation. proofs at the areas for chaos, hyperchaos, and non-chaos within the house of the bifurcation parameters are offered utilizing a method in line with the second-derivative try out and boundaries 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 is predicated on a computer-assisted facts. ultimately, a rigorous research is equipped at the bifurcational phenomena within the piecewise linear Chua's approach utilizing either an analytical 2-D mapping and a 1-D approximated Poincar?© mapping as well as different analytical equipment.

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**Example text**

13. , P er (f ) = Ω(f ), the closure is the nonwandering set itself. Note that the “Axiom A” diffeomorphism serves as a model for the general behavior at a transverse homoclinic point where the stable and unstable manifolds of a periodic point intersect, and it plays a crucial role in the study of homoclinic bifurcations. 7. (a) The horseshoe map f is a diffeomorphism defined from the unit square D of the plane into itself. (b) The horseshoe map is one-to-one. (c) The domain of f −1 is f (D). (d) The horseshoe map f is an Axiom A diffeomorphism.

24 given below) is applied to prove the existence of a true transversal homoclinic orbit to a true periodic orbit near these pseudo-orbits of the H´enon map [H´enon (1976)]14 . See Sec. 6. 23. (Connecting Orbit Shadowing Theorem) Suppose f : Rn → Rn is a C 2 diffeomorphism, and {xk , k ∈ Z} to {yk , k ∈ Z} are two ǫ-pseudo-periodic-orbits with periods N and N ′ , respectively, of f . Let z = {zk , k ∈ Z} be an ǫ-pseudo-connecting-orbit of f connecting {xk , k ∈ Z} to {yk , k ∈ Z}. 57) note that it was shown in [Coomes et al.

2001)] in the discrete case. The resulting chaos is called horseshoe type or Sil’nikov chaos. These horseshoes gives extremely complicated behavior typically observed in chaotic systems [Guckenheimer and Holmes (1983)] as shown in the previous section. A more detailed discussion of homoclinic bifurcations is given in Sec. 3 in connection with the H´enon map [H´enon (1976)]. 42) where the vector field f : R3 −→ R3 belongs to class C r (r ≥ 1), x ∈ R3 is the state variable of the system, and t ∈ R is the time.

### 2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach by Elhadj Zeraoulia, Julien Clinton Sprott

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