On the occurrence of chaos via different routes to chaos: period doubling and border-collision bifurcations
One of the most well-known 2D discrete models is the Hénon mapping [\textit{M. Hénon}, Commun. Math. Phys. 50, 69--76 (1976; Zbl 0576.58018)] defined by the relation \[ H(x,y)=\begin{cases} 1-ax^2+y,\\ bx.\end{cases}\tag{1} \] This article introduces and justifies a rarely observed phenomenon shown by a new mapping obtained via a direct modification of the Hénon mapping (1), which is the occurrence of the same chaotic attractor via two different and distinguishable routes to chaos: the standard period doubling and border-collision bifurcations as typical futures. The new proposed system that realized this phenomenon is the following 2D discrete chaotic mapping: \[ f(x,y)=\begin{cases} 1-a|x^2-y|+y,\\ bx,\end{cases}\tag{2} \] where \(a\) and \(b\) are bifurcation parameters. The study is just reduced to some numerical simulations and computations.
2D piecewise smooth discrete-time chaotic mapping
scientific article; zbMATH DE number 5660111