Narrow Results

The selection and breakup of spiral wave in a coupled network is investigated by imposing Gaussian colored noise on the network, respectively. The dynamics of each node of the network is described by a simplified Chua circuit, and nodes are uniformly placed in a two-dimensional array with nearest-neighbor connection type. The transition of spiral wave is detected by changing the coupling intensity, intensity and correlation time τ in the noise. A statistical variable is used to discern the parameter region for breakup of spiral wave and robustness to external noise. Spiral waves emerge in the network when the network with structure of complex-periodic and chaotic properties. It is found that asymmetric coupling can induce deformation of spiral wave, stronger intensity or smaller correlation time in noise does cause breakup of the spiral wave.

In this paper, a scheme of a bidirectional hyperchaotic communication system is proposed. The hyperchaotic communication system including the transmitter and the receiver is composed of a pair of Chua circuit with coupling characteristics respectively. From the viewpoint of communication security, directly adopting the bidirectional communication system would provide a so-called "theoretical security". This is caused by the coupling relationship between the transmitter and the receiver. With the technique of frequency spectrum, the theoretical security of bidirectional communication systems can be proven. In other words, even if an intruder knows the parameter values of the system, he could not steal information from the system. As a result, it is not necessary for us to use the additional encryption. On the other hand, to achieve the synchronization of bidirectional communications, the method which applies a suitable Lyapunov function and the property of positive definite of a matrix is proposed to design the feedback controllers. Therefore, the message masked by chaotic signal from the transmitter can be perfectly recovered in the receiver. Finally, the simulation results can verify that the proposed method and the scheme of the bidirectional communication are favorable.

This paper develops a theory of global exponential synchronization using two Chua circuits as the platform, for which five theorems are established by constructing new Lyapunov functions and employing the comparison principle in conjunction with the stability theory with respect to partial components.

In previous work, the authors explored the parameter space for Chua's circuit and its generalizations, discovering new routes to chaos, and nearly a thousand new attractors. These were obtained by varying the parameters of the physical circuit and of systems derived from it. Here, we present a novel class of computational system that does not respect the classical constraints in Chua's circuit, and that generates chaotic dynamics via an iterative process based on discrete versions of the equations for Chua's circuit and its variants. We call these systems Chua Machines. After presenting the chaotic dynamics, we provide a formal description of Chua Machines and a Gallery of 222 3D images, illustrating their dynamics. We discuss the method used to discover these systems and the metrics applied in the exploration of their parameter space and offer examples of highly complex bifurcation maps, together with images showing how patterns can evolve with time, or vary significantly changing the values of one of the parameters. Finally, we present a detailed analysis of qualitative changes in a Chua Machine as it traverses the parameter space of the bifurcation map. The evidence suggests that these dynamics are even richer and more complex than their counterpart in the continuous domain.

In this paper, periodic and chaotic behaviors in the Chua circuit system are investigated, and the analytical prediction of periodic flows in such a system is carried out. The solutions of the system in different regions with different parameters are first obtained. The switching boundaries are introduced for systems switching because of different system parameters in different domains. In the vicinity of the switching boundaries, the normal vector-field product is introduced to measure flow switching on the separation boundary, and the conditions for grazing and passable flows to the discontinuous boundary are presented. The basic mappings are defined and periodic responses of such a system are predicted analytically from mapping structures. The local stability and bifurcation analysis are carried out.

The experimental study of in-out intermittency during the incomplete synchronization of two coupled, nonlinear, autonomous, fourth-order, chaotic, circuits is reported. The two circuits are unidirectionally coupled via a linear resistor. The dependence of laminar lengths, on the deviation of the control parameter from its critical value, and the mean laminar length distribution, are studied.

This paper proposes a novel way to look at Chua's circuit and to investigate its chaotic behavior. We propose a systematic method to increase the dimension of Chua's circuit and to choose the element values. Our claims are confirmed by numerical simulations on Chua's circuit and on modified Chua's circuit and also supported by experimental results.

This work discusses the applicability of a method for phase determination of scalar time series from nonlinear systems. We apply the method to detect phase synchronization in different scenarios, and use the phase diffusion coefficient, the Lyapunov spectrum, and the similarity function to characterize synchronization transition in nonidentical coupled Rössler oscillators, both in coherent and non-coherent regimes. We also apply the method to detect phase synchronous regimes in systems with multiple scroll attractors as well as in experimental time series from coupled Chua circuits. The method is of easy implementation, requires no attractor reconstruction, and is particularly convenient in the case of experimental setups with a single time series data output.

We introduce a novel method revealing hidden bifurcations in the multispiral Chua attractor in the case where the parameter of bifurcation $c$ which determines the number of spiral is discrete. This method is based on the core idea of the genuine Leonov and Kuznetsov method for searching hidden attractors (i.e. applying homotopy and numerical continuation) but used in a very different way. Such hidden bifurcations are governed by a homotopy parameter $\epsilon$ whereas $c$ is maintained constant. This additional parameter which is absent from the initial problem is perfectly fitted to unfold the actual structure of the multispiral attractor. We study completely the multispiral Chua attractor, generated via sine function, and check numerically our method for odd and even values of $c$ from 1 to 12. In addition, we compare the shape of the attractors obtained for the same value of parameter $\epsilon$ while varying the parameter $c$.

In this paper, we show that any $n$-dimensional autonomous systems can be regarded as subsystems of $2n$-dimensional Hamiltonian systems. One of the two subsystems is identical to the $n$-dimensional autonomous system, which is called the driving system. Another subsystem, called the response system, can exhibit interesting behaviors in the neighborhood of infinity. That is, the trajectories approach infinity with complicated nonperiodic (chaotic-like) behaviors, or periodic-like behavior. In order to show the above results, we project the trajectories of the Hamiltonian systems onto $n$-dimensional spheres, or $n$-dimensional balls by using the well-known central projection transformation. Another interesting behavior is that the transient regime of the subsystems can exhibit Chua corsage knots. We next show that generic memristors can be used to realize the above Hamiltonian systems. Finally, we show that the internal state of two-element memristor circuits can have the same dynamics as $n$-dimensional autonomous systems.

Recently it was shown that in the dynamical model of Chua circuit both the classical self-excited and hidden chaotic attractors can be found. In this paper the dynamics of the Chua circuit is revisited. The scenario of the chaotic dynamics development and the birth of self-excited and hidden attractors is studied. A pitchfork bifurcation is shown in which a pair of symmetric attractors coexist and merge into one symmetric attractor through an attractor-merging bifurcation and a splitting of a single attractor into two attractors. The scenario relating the subcritical Hopf bifurcation near equilibrium points and the birth of hidden attractors is discussed.

The three-dimensional Muthuswamy–Chua–Ginoux (MCG) circuit model is a generalization of the paradigmatic canonical Muthuswamy–Chua circuit, where a physical memristor assumes the role of a thermistor, and it is connected in series with a linear passive capacitor, a linear passive inductor, and a nonlinear resistor. The physical memristor presents an electrical resistance which is a function of temperature. Nowadays, the MCG circuit model has gained considerable attention due to the lack of extensive numerical explorations and their distinct dynamical properties, exemplified by phenomena such as the transition from torus breakdown to chaos, giving rise to a double spiral attractor associated to independent period-doubling cascades. In this contribution, the complex dynamics of the MCG circuit model is studied in terms of the Lyapunov exponents spectra, Kaplan–Yorke (KY) dimension, and the number of local maxima (LM) computed in one period of oscillation, as two parameters are simultaneously varied. Using the Lyapunov spectra to distinguish different dynamical regimes, KY dimension to estimate the attractors’ dimension, and the number of LM to characterize different periodic attractors, we construct high-resolution two-dimensional stability diagrams considering specific ranges of the parameter pairs $(\alpha ,𝜖)$. These parameters are associated with the inverse of the capacitance in the passive capacitor, and the heat capacitance and dissipation constant of the thermistor, respectively. Unexpectedly, we identify sequences of infinite self-organized generic stable periodic structures (SPSs) and Arnold tongues-like structures (ATSs) merged into chaotic dynamics domains, and the coexistence of different attracting sets (attractors) for the same parameter combinations and different initial conditions (multistability). We explore the multistable dynamics using the stability analysis and computation of Lyapunov coefficients, the inspection of the coexisting attractors, bifurcations diagrams, and basins of attraction. The periods of the ATSs and a particular sequence of shrimp-shaped SPSs obey specific generating and recurrence rules responsible for the bifurcation cascades. As the MCG circuit model has the crucial properties presented by the usual Muthuswamy–Chua circuit model, specific properties explored in our study should be helpful in real problems involving circuits with the presence of physical memristor playing the role of thermistors.

In this paper, we analyze the nonlinear dynamics of the modified Chua circuit system from the viewpoint of Kosambi–Cartan–Chern (KCC) theory. We reformulate the modified Chua circuit system as a set of two second-order nonlinear differential equations and obtain five KCC-invariants which express the intrinsic geometric properties. The deviation tensor and its eigenvalues are obtained, that determine the stability of the system. We also obtain the condition for Jacobi stability and discuss the behavior of deviation vector near equilibrium points.