On the dynamics of the q-deformed logistic map

On <i>q</i>-deformed logistic maps

“…The phenomenon of bistability appears for suitable q. This was already pointed out in [4] and [10] for single q-deformations. We point out that this phenomenon is known for population models with the so-called Allee effect given by unimodal maps (see [20] or [26]).…”

“…It allows us to compute the topological entropy of unimodal maps by comparing with maps of the family T of strictly piecewise continuous maps with constant slope ±s, maximum at 1/2, and such that the topological entropy is known and equal to log s. The comparison is made by means of the images of turning points of f and the members of T . It reminds the bisection method (see [10] for a schematic algorithm description).…”

“…In this section we consider the q-deformed logistic family when a single q-deformation is applied. Some results of this section were partially studied in [16] and [10], although with a slightly different approach focussed on local stability and chaos. Recall that f a , a ∈ (0, 4], denotes the logistic family, φ q , q ∈ (−∞, 2), denotes the q-deformation and F a,q = f a • φ q .…”

“…On the other hand, in [10], we studied the parameter region for which the qdeformed system is chaotic and checked that the so-called Parrondo's paradox (see e.g. [9]) appears in this system by finding parameter values of a and q such that f a has a simple dynamics while the dynamics of Φ q,a is complicated.…”

The Dynamics of a Four-Step Feedback Procedure to Control Chaos

“…Mathematical examples includes interval dynamics (see [8] and [5]), dynamics of complex maps (see [2] and [3]), local stability problems ( [14] and [16]) among others. In addition, we can find applications of this paradox to Physics (see [12] and [19]), Biology (see [23], [25] and [26]) and Social Sciences (see [13] and [20]). It is stated in [18] that "the range of stability of the dynamical system can be increased by using the switching strategy".…”