By Elhadj Zeraoulia, Julien Clinton Sprott
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Additional resources for 2-D Quadratic Maps and 3-D ODE Systems. A Rigorous Approach
N }. For i = 1 to N − 1 value (i) = 1 2 2 (x (i) − xmax ) + (x (i + 1) − xmin ) . 79) (5) Plot a histogram of the value array23. 1) modelling a threedimensional Smale-Williams attractor. (b) Determine the corresponding stable and unstable manifolds for each equilibrium point. (c) Choose an appropriate Poincar´e section S, and draw the corresponding bifurcation diagram for a chosen set of parameters showing a route to chaos. (d) Search for a hyperchaotic attractor. (e) Compare the behavior of Eq.
The distance between s and s is defined as ∞ 1 |si − si | d (s, s) = . 6. The space Σm is compact, totally disconnected, and perfect. 6 is a Cantor set, which frequently appears in the characterization of the complex structure of an invariant set in a chaotic dynamical system. 7. (a) σ(Σm ) = Σm , and σ is continuous. (b) The shift map σ as a dynamical system defined on Σm has the following properties: (b-1) σ has a countable infinity of periodic orbits consisting of orbits of all periods; (b-2) σ has an uncountable infinity of nonperiodic orbits; and (b-3) σ has a dense orbit.
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. 5.
2-D Quadratic Maps and 3-D ODE Systems. A Rigorous Approach by Elhadj Zeraoulia, Julien Clinton Sprott