Abstract
We point out the joint occurrence of Pascal triangle patterns and power-law scaling in the standard logistic map, or more generally, in unimodal maps. It is known that these features are present in its two types of bifurcation cascades: period and chaotic-band doubling of attractors. Approximate Pascal triangles are exhibited by the sets of lengths of supercycle diameters and by the sets of widths of opening bands. Additionally, power-law scaling manifests along periodic attractor supercycle positions and chaotic band splitting points. Consequently, the attractor at the mutual accumulation point of the doubling cascades, the onset of chaos, displays both Gaussian and power-law distributions. Their combined existence implies both ordinary and exceptional statistical-mechanical descriptions of dynamical properties.
Highlights
The logistic map has played a prominent role in the development of the field of nonlinear dynamics [1]-[3]
It has served as a standard source for the illustration of nonlinear concepts such as: bifurcations, stable and unstable periodic orbits, periodic windows, ergodic and mixing behaviors, chaotic orbits and universality in the sense of the Renormalization Group (RG) method [1]-[3]
We concisely draw attention to the presence of geometrical and scaling laws in the family of attractors generated by the logistic map and to the consequences that these laws have in the dynamical properties of their most interesting object of study: the period-doubling transition to chaos
Summary
The logistic map has played a prominent role in the development of the field of nonlinear dynamics [1]-[3]. The simplicity of its quadratic expression and the richness and intricacy of the properties that stem from it have captivated a large number of scholars and students over decades It has served as a standard source for the illustration of nonlinear concepts such as: bifurcations, stable and unstable periodic orbits, periodic windows, ergodic and mixing behaviors, chaotic orbits and universality in the sense of the Renormalization Group (RG) method [1]-[3]. It has become a suitable model system for the exploration of statistical-mechanical structures [4]. We provide below a description of how these properties arise in the logistic map and discuss their implications for the mathematical structures in the dynamics of this nonlinear system, itself a convenient numerical laboratory for the study of statistical-mechanical theories [4]
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