The effect of temperature on generic stable periodic structures in the parameter space of dissipative relativistic standard map

In this work, we have characterized changes in the dynamics of a two-dimensional relativistic standard map in the presence of dissipation and specially when it is submitted to thermal effects modeled by a Gaussian noise reservoir. By the addition of thermal noise in the dissipative relativistic standard map (DRSM) it is possible to suppress typical stable periodic structures (SPSs) embedded in the chaotic domains of parameter space for large enough temperature strengths. Smaller SPSs are first affected by thermal effects, starting from their borders, as a function of temperature. To estimate the necessary temperature strength capable to destroy those SPSs we use the largest Lyapunov exponent to obtain the critical temperature ($T_C$) diagrams. For critical temperatures the chaotic behavior takes place with the suppression of periodic motion, although, the temperature strengths considered in this work are not so large to convert the deterministic features of the underlying system into a stochastic ones.Comment: 8 pages and 7 figures, accepted to publication in EPJ

Bifurcation structures and transient chaos in a four-dimensional Chua model

A four-dimensional four-parameter Chua model with cubic nonlinearity is studied applying numerical continuation and numerical solutions methods. Regarding numerical solution methods, its dynamics is characterized on Lyapunov and isoperiodic diagrams and regarding numerical continuation method, the bifurcation curves are obtained. Combining both methods the bifurcation structures of the model were obtained with the possibility to describe the {\it shrimp}-shaped domains and their endoskeletons. We study the effect of a parameter that controls the dimension of the system leading the model to present transient chaos with its corresponding basin of attraction being riddled.Comment: 9 figures, to appear in PL

Parameter space of experimental chaotic circuits with high-precision control parameters

ACKNOWLEDGMENTS The authors thank Professor Iberê Luiz Caldas for the suggestions and encouragement. The authors F.F.G.d.S., R.M.R., J.C.S., and H.A.A. acknowledge the Brazilian agency CNPq and state agencies FAPEMIG, FAPESP, and FAPESC, and M.S.B. also acknowledges the EPSRC Grant Ref. No. EP/I032606/1.Peer reviewedPublisher PD

Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators

The behavior of neurons can be modeled by the FitzHugh-Nagumo oscillator model, consisting of two nonlinear differential equations, which simulates the behavior of nerve impulse conduction through the neuronal membrane. In this work, we numerically study the dynamical behavior of two coupled FitzHugh-Nagumo oscillators. We consider unidirectional and bidirectional couplings, for which Lyapunov and isoperiodic diagrams were constructed calculating the Lyapunov exponents and the number of the local maxima of a variable in one period interval of the time-series, respectively. By numerical continuation method the bifurcation curves are also obtained for both couplings. The dynamics of the networks here investigated are presented in terms of the variation between the coupling strength of the oscillators and other parameters of the system. For the network of two oscillators unidirectionally coupled, the results show the existence of Arnold tongues, self-organized sequentially in a branch of a Stern-Brocot tree and by the bifurcation curves it became evident the connection between these Arnold tongues with other periodic structures in Lyapunov diagrams. That system also presents multistability shown in the planes of the basin of attractions

Exploring an experimental analog Chua’s circuit

In this work we carry out experimental studies of the paradigmatic Chua’s circuit using an approach of the analog computation instead of performing experiments in the canonical circuit. This means that we have built an electronic circuit that integrates (analog computation), in continuous time, the equations of motion of the canonical Chua’s circuit. The equations of motion of the analogical circuit are equivalent to the canonical circuit, so that the dynamical behaviour is the same. With this approach, we successfully obtain an experimental parameter plane using the largest Lyapunov exponent (here named Lyapunov diagram), directly calculated from the experimental time series, with a good precision, so that different types of dynamical behaviours were characterized in this diagram. Results are in very good agreement with numerical simulation with an additional Gaussian noise. The approach by analog computation used here can be extended to a wide range of dynamical systems, once that the analog circuit simulates, by circuitry implementation, the dynamics of these systems from an experimental point of view

Tracking multistability in the parameter space of a Chua’s circuit model

The effect of applying an asymmetric periodic continuous-feedback signal F(x), which is a function of voltage x, across one of the capacitors in the classical Chua’s circuit model is examined. We have performed a numerical investigation on the dynamics of the Chua’s circuit model to discuss a procedure of distorting, suppressing and moving attractors and stable periodic structures (SPSs) in phase and parameter spaces, respectively, by accordingly choosing the control parameters in the function F(x). Increasing the asymmetry strength in F(x), we are able to track and to move overlapped SPSs related to multistability phenomenon in phase space, and to suppress SPSs related to one single attractor. Our results indicate that the methodology used here will have a wide spectrum of applications in generic nonlinear complex dynamical systems presenting multistability, related to both theoretical and experimental issues