The Generalized Time-Delayed Hénon Map: Bifurcations and Dynamics

Abstract

We analyze the bifurcations of a family of time-delayed Hénon maps of increasing dimension and determine the regions where the motion is attracted to different dynamical states. As a function of parameters that govern nonlinearity and dissipation, boundaries that confine asymptotic periodic motion are determined analytically, and we examine their dependence on the dimension d. For large d these boundaries converge. In low dimensions both the period-doubling and quasiperiodic routes to chaos coexist in the parameter space, but for high dimensions the latter predominates and prior to the onset of chaos, the systems exhibit multistability. When the nonlinearity parameter is varied, the dimension of chaotic attractors in the systems changes smoothly with increasing number of non-negative Lyapunov exponents.

A higher-dimensional generalization of the Lozi map: Bifurcations and dynamics

“…Such feature are typically of these maps even across different combination of dimensionality parameters (d,k). We should mention that such phenomenon was not found for a sim-ilarly generalized Hénon map [16], and appears to be a result of border collision bifurcations. It is important to note that the theory of bifurcations in smooth dynamical systems does not explain these features [17].…”

“…Unlike the smooth case (i.e. the Hénon map) studied in [16] the bifurcations in the Lozi map can show supercritical bifurcations on either side of the period one boundaries (of course limited upto the saddle node curve), as shown in Fig. 1(b).…”

“…Our interest in the present paper is the generalization of the Lozi map to higher dimensions. One motivation is to compare this piecewise continuous system to a similar high-dimensional Hénon mapping [16]. Of the different ways in which this can be done, we choose to extend the map to d-dimensions by incorporating time-delay feedback while ensuring that the system remains an endomorphism in the absence of dissipation.…”

Lots of Chaotic Behaviors of the Henon Map

“…Obviously, they exist only for (1 -β) 2 + 4α ≥ 0 Liu (2007). To determine the stability of a fixed point, consider a small perturbation from the fixed point by letting x(t) = +∈ 0 .…”

“…Proof. See [2] and [3]. The Iterations of , as | | → ∞ Notations 6.10  For ( ) ∈ ℝ 2 we will denote , by ( ) , ∈ ℤ.…”