Chaotic Dynamics in Two-Dimensional Noninvertible Maps

Dynamical Behavior in a DPCM System with an Order 2 Predictor. Case of a Constant Input

“…4. A chaotic attractor; that means an invariant area (A) where dynamics presents no regularity from numerical simulations; in this paper chaos will be considered in a nonstrict sense [30], chaos in a strict sense meaning that it is possible to prove that the set (A) is made up of points, the orbits of which are sensitive with respect to initial conditions; chaos in a strict sense has to be proved, which has not been done in the case of our DPCM system.…”

“…An important tool used to study noninvertible maps is that of the critical manifold (see [29], [30] for more details). A noninvertible map is characterized by the fact that a point in the state space can possess a different number of rank-1 preimages, depending on where it is located in the state space.…”

“…Finally, the chaotic attractor will disappear by a contact bifurcation with its basin boundary (Figure 8(i)). This kind of bifurcation is very classical (see [30], [21], [13]) and corresponds to chaotic attractor disappearance in nearly all cases for the DPCM system.…”

“…Then, the ICC crosses through a curve LC −1 , which creates oscillations on it (Figure 9(a)). The oscillations increase and the ICC becomes a weakly chaotic ring (WCR) (Figures 9(c) and (d)) [24], [30]. Such an attractor has been called a WCR in [30] because its Lyapunov exponent, even greater than 1, is very close to 1.…”

“…The oscillations increase and the ICC becomes a weakly chaotic ring (WCR) (Figures 9(c) and (d)) [24], [30]. Such an attractor has been called a WCR in [30] because its Lyapunov exponent, even greater than 1, is very close to 1. Such phenomena have been more precisely described for other values of a DPCM system in [15].…”