Class Of Two-dimensional Maps Research Articles

Extremely hidden multi-stability in a class of two-dimensional maps with a cosine memristor

We present a class of two-dimensional memristive maps with a cosine memristor. The memristive maps do not have any fixed points, so they belong to the category of nonlinear maps with hidden attractors. The rich dynamical behaviors of these maps are studied and investigated using different numerical tools, including phase portrait, basins of attraction, bifurcation diagram, and Lyapunov exponents. The two-parameter bifurcation analysis of the memristive map is carried out to reveal the bifurcation mechanism of its dynamical behaviors. Based on our extensive simulation studies, the proposed memristive maps can produce hidden periodic, chaotic, and hyper-chaotic attractors, exhibiting extremely hidden multi-stability, namely the coexistence of infinite hidden attractors, which was rarely observed in memristive maps. Potentially, this work can be used for some real applications in secure communication, such as data and image encryptions.

A novel class of two-dimensional chaotic maps with infinitely many coexisting attractors**Project supported by the National Natural Science Foundation of China (Grant Nos. 11672257, 11632008, 11772306, and 11972173), the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20161314), the 5th 333 High-level Personnel Training Project of Jiangsu Province of China (Grant No. BRA2018324), and the Excellent Scientific and

We study a novel class of two-dimensional maps with infinitely many coexisting attractors. Firstly, the mathematical model of these maps is formulated by introducing a sinusoidal function. The existence and the stability of the fixed points in the model are studied indicating that they are infinitely many and all unstable. In particular, a computer searching program is employed to explore the chaotic attractors in these maps, and a simple map is exemplified to show their complex dynamics. Interestingly, this map contains infinitely many coexisting attractors which has been rarely reported in the literature. Further studies on these coexisting attractors are carried out by investigating their time histories, phase trajectories, basins of attraction, Lyapunov exponents spectrum, and Lyapunov (Kaplan–Yorke) dimension. Bifurcation analysis reveals that the map has periodic and chaotic solutions, and more importantly, exhibits extreme multi-stability.

A New Class of Two-Dimensional Chaotic Maps with Closed Curve Fixed Points

This paper constructs a new class of two-dimensional maps with closed curve fixed points. Firstly, the mathematical model of these maps is formulated by introducing a nonlinear function. Different types of fixed points which form a closed curve are shown by choosing proper parameters of the nonlinear function. The stabilities of these fixed points are studied to show that these fixed points are all nonhyperbolic. Then a computer search program is employed to explore the chaotic attractors in these maps, and several simple maps whose fixed points form different shapes of closed curves are presented. Complex dynamical behaviors of these maps are investigated by using the phase-basin portrait, Lyapunov exponents, and bifurcation diagrams.

KNOT POINTS IN TWO-DIMENSIONAL MAPS AND RELATED PROPERTIES

We consider the class of two-dimensional maps of the plane for which there exists a whole one-dimensional singular set (for example, a straight line) that is mapped into one point, called a "knot point" of the map. The special character of this kind of point has been already observed in maps of this class with at least one of the inverses having a vanishing denominator. In that framework, a knot is the so-called focal point of the inverse map (it is the same point). In this paper, we show that knots may also exist in other families of maps, not related to an inverse having values going to infinity. Some particular properties related to focal points persist, such as the existence of a "point to slope" correspondence between the points of the singular line and the slopes in the knot, lobes issuing from the knot point and loops in infinitely many points of an attracting set or in invariant stable and unstable sets.

Minimal Sets of Antitriangular Maps

We consider the class of two-dimensional maps of the form F(x,y) = (g(y),f(x)), (x,y) ∈ [0,1] × [0,1] = I2, where f and g are continuous interval maps. The paper deals with the structure of minimal sets for this class of maps. We give a complete description of finite minimal sets and prove some partial results concerning the infinite case.

Invariants and chaotic maps

A one-dimensional mapf(x) is called an invariant of a two-dimensional mapg(x, y) ifg(x, f(x))=f(f(x)). The logistic map is an invariant of a class of two-dimensional maps. We construct a class of two-dimensional maps which admit the logistic maps as their invariant. Moreover, we calculate their Lyapunov exponents. We show that the two-dimensional map can show hyperchaotic behavior.