Abstract
We report on the definition and characteristics of nodes in the chaotic region of bifurcation diagrams in the case of 1D mono-parametrical and S-unimodal maps, using as guiding example the logistic map. We examine the arrangement of critical curves, the identification and arrangement of nodes, and the connection between the periodic windows and nodes in the chaotic zone. We finally present several characteristic features of nodes, which involve their convergence and entropy.
Highlights
Let the one-dimensional, mono-parametrical and S-unimodal map, xt+1 = f, where xt = f (t) ( x0 ; p) is the tth iterated map, x0 is the initial value, and p is the nonlinear parameter
A bifurcation diagram is separated into two parts, the zone of Order, p < p∞, [3,4,5,17,18,19,20,21,22], where only periodic orbits may occur, and the zone of Chaos, p > p∞, [3,6,7,13,17,18,23,24,25,26,27,28,29,30], where the chaos appears, and both chaotic and periodic orbits may occur; p∞ is the Feigenbaum’s point [4,9,31], which defines the boundary point of the two zones
As the nonlinear parameter p increases beyond the node of order 1, we find a node of order 2 at p = 3.982570733172925
Summary
Aside from the main zones of order and chaos, which of periodic orbits inside each WOM forms a number of complete bifurcation diagrams in miniature have already been referred, secondary zones of order and chaos exist for p > p∞, inside each WOM that equals the WOM period (Figure 2). 3, we examine the identification in the chaotic zone, which are interwoven with the location and period of WOMs. In our analysis, arrangement of similar nodes infeatures the chaotic zone, separating in two2,types: primarythe andcritical secondary weand focus on SCB, but characterize all CBs.those. The similarities between the main chaotic zone in (b) and the miniature chaotic zone in (e) are arrangements of WOMs, critical curves, and nodes, are some of the common features of the main and remarkable. The diagrams are computed for 106 iterations.). (Taken from [16])
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