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[en] Let be rotations on the unit circle and define as for , , where is the shift, and and are rotational angles. It is first proved that the system exhibits maximal distributional chaos for any (no assumption of ), generalizing Theorem 1 in Wu and Chen (Topol. Appl. 162:91–99, 2014). It is also obtained that is cofinitely sensitive and -sensitive and that is densely chaotic if and only if .
[en] We study the dynamics of Laplacian-type coupling induced by logistic family , where , on a periodic lattice, that is the dynamics of maps of the form where determines strength of coupling. Our main objective is to analyze the structure of attractors in such systems and especially detect invariant regions with nontrivial dynamics outside the diagonal. In analytical way, we detect some regions of parameters for which a horseshoe is present; and using simulations global attractors and invariant sets are depicted.
[en] Highlights: • We give an algorithm to compute regularities of SNA’s based on tools of de la Llave–Petrov. • It uses the Keller convergence construction to the attractor. • It uses Daubechies Wavelets with 16 vanishing moments. • The precision is two decimal digits compared with Weierstraß function. • The loss of regularity as parameter changes is observed from wavelet coefficients. We estimate numerically the regularities of a family of Strange Non-Chaotic Attractors related with one of the models studied in (Grebogi et al., 1984) (see also Keller, 1996). To estimate these regularities we use wavelet analysis in the spirit of de la Llave and Petrov (2002) together with some ad-hoc techniques that we develop to overcome the theoretical difficulties that arise in the application of the method to the particular family that we consider. These difficulties are mainly due to the facts that we do not have an explicit formula for the attractor and it is discontinuous almost everywhere for some values of the parameters. Concretely we propose an algorithm based on the Fast Wavelet Transform. Also a quality check of the wavelet coefficients and regularity estimates is done.
[en] Highlights: • Unraveling the bifurcation structure of a single-phase H-bridge inverter. • Demonstration of regular structures formed by persistence border-collision curves. • Detection of qualitatively different regions inside the fixed point stability domain. • Studies of the processes associated with a new route to chaos in switching systems. Recent studies on a power electronic DC/AC converter (inverter) have demonstrated that such systems may undergo a transition from regular dynamics (associated with a globally attracting fixed point of a suitable stroboscopic map) to chaos through an irregular sequence of border-collision events. Chaotic dynamics of an inverter is not suitable for practical purposes. However, the parameter domain in which the stroboscopic map has a globally attracting fixed point has generally been considered to be uniform and suitable for practical use. In the present paper we show that this domain actually has a complicated interior structure formed by boundaries defined by persistence border collisions. We describe a simple approach that is based on symbolic dynamics and makes it possible to detect such boundaries numerically. Using this approach we describe several regions in the parameter space leading to qualitatively different output signals of the inverter although all associated with globally attracting fixed points of the corresponding stroboscopic map.
[en] Highlights: • Entropy of low-significance bits in digital measurements of chaos is examined. • Low-significance bits yield a two-symbol partition with a corrugated structure. • Corrugation at low-significance bits better approximates a generating partition. • Entropy rate estimation using lower-significance bits requires longer block lengths. • Considering only short block lengths can overestimate entropy of physical system. We examine the entropy of low-significance bits in analog-to-digital measurements of chaotic dynamical systems. We find the partition of measurement space corresponding to low-significance bits has a corrugated structure. Using simulated measurements of a map and experimental data from a circuit, we identify two consequences of this corrugated partition. First, entropy rates for sequences of low-significance bits more closely approach the metric entropy of the chaotic system, because the corrugated partition better approximates a generating partition. Second, accurate estimation of the entropy rate using low-significance bits requires long block lengths as the corrugated partition introduces more long-term correlation, and using only short block lengths overestimates the entropy rate. This second phenomenon may explain recent reports of experimental systems producing binary sequences that pass statistical tests of randomness at rates that may be significantly beyond the metric entropy rate of the physical source.
[en] We consider implications of dynamical Borel–Cantelli lemmas for rates of growth of Birkhoff sums of non-integrable observables , k > 0, on ergodic dynamical systems where . Some general results are given as well as some more concrete examples involving non-uniformly expanding maps, intermittent type maps as well as uniformly hyperbolic systems. (paper)
[en] Chaotic dynamics and synchronization of fractional-order systems have attracted much attention recently. Based on stability theory of fractional-order systems and stability theory of integer-order systems, this paper deals with the problem of coexistence of various types of synchronization between different dimensional fractional-order chaotic systems. To illustrate the capabilities of the novel schemes proposed herein, numerical and simulation results are given.
[en] We consider a class of nonideal oscillating (by Sommerfeld and Kononenko) dynamical systems and establish the existence of two types of hyperchaotic attractors in these systems. The scenarios of transitions from regular to chaotic ones attractors and the scenarios of transitions between chaotic attractors of different types are described.
[en] Let , and let G be a locally compact group. Let A be a unital -algebra. We give a sufficient and necessary condition for a sequence of weighted translations on the -algebra-valued Lebesgue space to be topologically transitive in terms of the Haar measure, the weight functions, and an aperiodic sequence in G. Chaos, topological mixing, supercyclicity and dual hypercyclicity for such a sequence are also discussed.
[en] Highlights: • A framework which combines a converter and a chaotic circuit to enhance the security of PUFs is proposed. • The converter is utilized to fast generate the intermediate parameters and the chaotic circuit is integrated to enhance the randomness of PUFs. • A specific construction and simulation experiments for the proposed framework are presented. • The construction uses the arbiter PUF circuit and the Chua's chaotic circuit. The experimental results further validate the feasibility. • Some practical suggestions and specific construction are given. - Abstract: As a new technique for authentication and key generation, physically unclonable function (PUF) has attracted considerable attentions, with extensive research results achieved already. To resist the popular machine learning modeling attacks, a framework to enhance the security of PUFs is proposed. The basic idea is to combine PUFs with a chaotic system of which the response is highly sensitive to initial conditions. For this framework, a specific construction which combines the common arbiter PUF circuit, a converter, and the Chua's circuit is given to implement a more secure PUF. Simulation experiments are presented to further validate the framework. Finally, some practical suggestions for the framework and specific construction are also discussed.