Density of instantaneous frequencies in the Kuramoto-Sakaguchi model

We obtain a formula for the statistical distribution of instantaneous frequencies in the Kuramoto-Sakaguchi model. This work is based on the Kuramoto-Sakaguchi's theory of globally coupled phase oscillators, which we review in full detail by discussing its assumptions and showing all steps behind the derivation of its main results. Our formula is a stationary probability density function with a complex mathematical structure, is consistent with numerical simulations and gives a description of the stationary collective states of the Kuramoto-Sakaguchi model

The impact of chaotic saddles on the synchronization of complex networks of discrete-time units

A chaotic saddle is a common nonattracting chaotic set well known for generating finite-time chaotic behavior in low and high-dimensional systems. In general, dynamical systems possessing chaotic saddles in their state-space exhibit irregular behavior with duration lengths following an exponential distribution. However, when these systems are coupled into networks the chaotic saddle plays a role in the long-term dynamics by trapping network trajectories for times that are indefinitely long. This process transforms the network’s high-dimensional state-space by creating an alternative persistent desynchronized state coexisting with the completely synchronized one. Such coexistence threatens the synchronized state with vulnerability to external perturbations. We demonstrate the onset of this phenomenon in complex networks of discrete-time units in which the synchronization manifold is perturbed either in the initial instant of time or in arbitrary states of its asymptotic dynamics. The role of topological asymmetries of Erdös–Rényi and Barabási–Albert graphs are investigated. Besides, the required coupling strength for the occurrence of trapping in the chaotic saddle is unveiled.DFG, 163436311, SFB 910: Kontrolle selbstorganisierender nichtlinearer Systeme: Theoretische Methoden und Anwendungskonzept

An inductor-free realization of the Chua`s circuit based on electronic analogy

Although literature presents several alternatives, an approach based on the electronic analogy was still not considered for the implementation of an inductor-free realization of the double scroll Chua`s circuit. This paper presents a new inductor-free configuration of the Chua`s circuit based on the electronic analogy. This proposal results in a versatile and functional inductorless implementation of the Chua`s circuit that offers new and interesting features for several applications. The analogous circuit is implemented and used to perform an experimental mapping of a large variety of attractors.National Counsel of Technological and Scientific Development (CNPq)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG)State of Minas Gerais Research Foundation (FAPEMIG)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)State of Sao Paulo Research Foundation (FAPESP)Gorceix Foundation and Texas Instruments Inc.,Gorceix Foundation and Texas Instruments Inc.

Experimental characterization of nonlinear systems: a real-time evaluation of the analogous Chua`s circuit behavior

This paper presents an experimental characterization of the behavior of an analogous version of the Chua`s circuit. The electronic circuit signals are captured using a data acquisition board (DAQ) and processed using LabVIEW environment. The following aspects of the time series analysis are analyzed: time waveforms, phase portraits, frequency spectra, Poincar, sections, and bifurcation diagram. The circuit behavior is experimentally mapped with the parameter variations, where are identified equilibrium points, periodic and chaotic attractors, and bifurcations. These analysis techniques are performed in real-time and can be applied to characterize, with precision, several nonlinear systems

Route to chaos and some properties in the boundary crisis of a generalized logistic mapping

A generalization of the logistic map is considered, showing two control parameters a and,8 that can reproduce different logistic mappings, including the traditional second degree logistic map, cubic, quartic and all other degrees. We introduce a parametric perturbation such that the original logistic map control parameter R changes its value periodically according an additional parameter omega = 2/q. The value of q gives this period. For this system, an analytical expression is obtained for the first bifurcation that starts a period-doubling cascade and, using the Feigenbaum Universality, we found numerically the accumulation point R-c where the cascade finishes giving place to chaos. In the second part of the paper we study the death of this chaotic behavior due to a boundary crisis. At the boundary crisis, orbits can reach a maximum value X = X-max, = 1. When it occurs, the trajectory is mapped to a fixed point at X = 0. We show that there exist a general recursive formula for initial conditions that lead to X = X-max. 2017 Elsevier B.V. All rights reserved.Center for Scientific Computing (NCC/GridUNESP) of the Sao Paulo State University (UNESP)UNESP Univ Estadual Paulista, Dept Fis, Av 24A,1515, BR-13506900 Rio Claro, SP, BrazilUNIFESP Univ Fed Sao Paulo, Dept Ciencias Exatas & Terra, Rua Sao Nicolau,210 Ctr, BR-09913030 Diadema, SP, BrazilUNIFESP Univ Fed Sao Paulo, Dept Ciencias Exatas & Terra, Rua Sao Nicolau,210 Ctr, BR-09913030 Diadema, SP, BrazilWeb of Scienc

Coupling-induced periodic windows in networked discrete-time systems

Networked nonlinear systems present a variety of emergent phenomena as a result of the mutual interactions between their units. An interesting feature of these systems is the presence of stable periodic behavior even when each unit oscillates chaotically if in isolation. Surprisingly, the mechanism in which the network interaction replaces chaos by periodicity is still poorly understood. Here, we show that such an onset of regularity can occur via replication of periodic windows. This phenomenon multiplies the stability domains in the system parameter space, not only suppressing chaos but also making the network less vulnerable to external disturbances such as shocks and noise. Moreover, we observe that the network cluster synchronizes for the parameters corresponding to the replica periodic windows. To confirm these observations, we employ the formalism of the master stability function demonstrating that the complete synchronized state is indeed transversally unstable in the replica windows