Universal metric properties of bifurcations of endomorphisms

High Density Nodes in the Chaotic Region of 1D Discrete Maps

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.

Long-period clocks from short-period oscillators

We analyze repulsively coupled Kuramoto oscillators, which are exposed to a distribution of natural frequencies. This source of disorder leads to closed orbits of repetitive temporary patterns of phase-locked motion, providing clocks on macroscopic time scales. The periods can be orders of magnitude longer than the periods of individual oscillators. By construction, the attractor space is quite rich. This may cause long transients until the deterministic trajectories find their stationary orbits. The smaller the width of the distribution about the common natural frequency, the longer are the emerging time scales on average. Among the long-period orbits, we find self-similar sequences of temporary phase-locked motion on different time scales. The ratio of time scales is determined by the ratio of widths of the distributions. The results illustrate a mechanism for how simple systems can provide rather flexible dynamics, with a variety of periods even without external entrainment.

Comparison between constant feedback and limiter controllers

Using symbolic dynamics of the one-dimensional unimodal map, the chaos stabilization mechanics of the feedback and limiter control schemes are considered. For feedback control, it is found that the control strength can be efficiently obtained from the superstable parameter of the embedded periodic orbits, and the scaling of the control-space period-doubling bifurcation cascade still obeys the Feigenbaum law. For Sarkovskii orbits, the scaling is also consistent with that of the original chaotic system. For limiter control, a single critical point in the unimodal map is extended to a superstable periodic window and a simple approach for determining the value of the control plateau is found. The scaling in the control space of the period-doubling bifurcation cascade is indeed superexponential. A different scaling for the fine structure of the Sarkovskii sequence is also found. Simple one-dimensional unimodal maps can also be used to generate maximum-length shift-register sequences.

Multifractal Fourier spectra and power-law decay of correlations in random substitution sequences

Binary symbolic sequences produced by randomly alternating substitution rules are considered. Exact expressions for the characteristics of autocorrelations and power spectra are derived. The decay of autocorrelation function obeys the power law. The Fourier spectral measure is either absolutely continuous or a mixture of the absolutely continuous and singular continuous components. For the latter case, the multifractal characteristics of this measure are computed.

Complex periodic orbits, renormalization, and scaling for quasiperiodic golden-mean transition to chaos

At the critical point of the golden-mean quasiperiodic transition to chaos we show the presence of an infinite sequence of unstable orbits in complex domain with periods given by the Fibonacci numbers. The Floquet eigenvalues (multipliers) are found to converge fast to a universal complex constant. We explain this result on the basis of the renormalization group approach and suggest using it for accurate estimates of the location of the golden-mean critical points in parameter space for a class of nonlinear dissipative systems defined analytically. As an example, we obtain data for the golden-mean critical point in the two-dimensional dissipative invertible map of Zaslavsky. We give a set of graphical illustrations for the scaling properties and emphasize that demonstration of self-similarity on two-dimensional diagrams of Arnold tongues requires the use of a properly chosen curvilinear coordinate system. We discuss a procedure of construction of the appropriate local coordinate system in the parameter plane and present the corresponding data for the circle map and Zaslavsky map.

Bicritical behavior of period doublings in unidirectionally coupled maps

We study the scaling behavior of period doublings in two unidirectionally coupled one-dimensional maps near a bicritical point where two critical lines of period-doubling transition to chaos in both subsystems meet. Note that the bicritical point corresponds to a border of chaos in both subsystems. For this bicritical case, the second response subsystem exhibits a type of non-Feigenbaum critical behavior, while the first drive subsystem is in the Feigenbaum critical state. Using two different methods, we make the renormalization-group analysis of the bicritical behavior and find the corresponding fixed point of the renormalization transformation with two relevant eigenvalues. The scaling factors obtained by the renormalization-group analysis agree well with those obtained by a direct numerical method.