Adaptive synchronization of Chua＇s system with uncertain inputs

The synchronization of Chua's system, whose inputs include an unknown constant parameter, is studied in this paper. A constructive method is applied to designing an adaptive controller, in which only one variable information of the master system is needed. With the action of control signals, the parameter of the slave system will approach the corresponding unknown parameter in the master system. At the same time, the synchronization errors will also converge to zero asymptotically. Numerical simulations show that the proposed theoretical approach is very effective.     

An active control synchronization for two modified Chua circuits

From modern control theory, an active control method to synchronize two modified Chua circuits with each other,which exhibit chaos, is presented. Some sufficient conditions of linear stability of the chaotic synchronization are obtained from rigorous mathematic justification. On the basis of the state-observer, the controller is analytically deduced using the active control. It is shown that this technique can be applied to achieve synchronization of the two systems with each other, whether they are identical or not. Finally, numerical simulations show the effectiveness of the proposed control scheme.     

Adaptive synchronization of a critical chaotic system

This paper further investigates the synchronization problem of a new chaotic system with known or unknown system parameters. Based on the Lyapunov stability theory, a novel adaptive control law is derived for the synchronization of a new chaotic system with known or unknown system parameters. Theoretical analysis and numerical simulations show the effectiveness and feasibility of the proposed schemes.     

Global synchronization of Chua＇s chaotic delay network by using linear matrix inequality

Global synchronization of Chua‘s chaotic dynamical networks with coupling delays is investigated in this paper.Unlike other approaches, where only local results were obtained, the network is found to be not linearized in this paper.Insteat, the global synchronization is obtained by using the linear matrix inequality theory. Moreover, some quite simple linear-state-error feedback controllers for global synchronization are derived, which can be easily constructed based on the minimum eigenvalue of the coupling matrix. A simulation of Chua‘s chaotic network with global coupling delays in nodes is finally given, which is used to verify the theoretical results of the proposed global synchron izationscheme.     

Adaptive synchronization with nonlinear input

This paper considers the adaptive synchronization problem of the drive—response type chaotic systems. Using a transmitted scalar signal with an unknown time-delay, a response system is constructed. By appropriately selecting the observer gain and designing the controller, synchronization can be achieved in the presence of the drive system's disturbances and unknown parameters. A well-known chaotic system, Chua's circuit, is considered as an illustrative example to demonstrate the effectiveness of the proposed approach.     

Adaptive synchronization of hyperchaotic Lü system with uncertainty

This paper presents a novel adaptive control scheme for synchronization of the latest hyperchaotic Lü system. Based on the Lyapunov stability theory, a feedback controller and a parameter update law are designed for the synchronization of hyperchaotic L\"{u} systems with uncertainty. Numerical simulations are given to demonstrate the validity of the synchronization technique.     

Adaptive coupled synchronization of non-autonomous systems in ring networks

The adaptive coupled synchronization method for non-autonomous systems is proposed. This method can avoid estimating the value of coupling coefficient. Under the uniform Lipschitz assumption, we derive the asymptotical synchronization for a general coupling ring network with N identical non-autonomous systems~ even when N is large enough. Strict theoretical proofs are given. Numerical simulations illustrate the effectiveness of the present method.     

Adaptive synchronization of uncertain Liu system via nonlinear input

This paper addresses the adaptive synchronization for uncertain Liu system via a nonlinear input. By using a single nonlinear controller, the approach is utilized to implement the synchronization of Liu system with total parameters unknown. This method is simple and can be easily designed. What is more, it improves the existing conclusions in Ref [12]. Simulation results prove that the controller is effective and feasible in the end.     

Birth of strange nonchaotic attractors through formation and merging of bubbles in a quasiperiodically forced Chuaʼs oscillator

We have constructed a quasiperiodically forced Chua?s circuit and report the occurrence of strange nonchaos in it. The birth of SNA is not through the well known mechanisms reported in literature, but through a novel route which we christen as “Formation and merging of bubbles route”. This is so named because the Poincaré strands undergo a series of splittings and mergings in certain regions, giving bubbles like appearance. These bubbles coalesce as the control parameter is varied, resulting in the birth of SNA. This has been confirmed numerically using various statistical measures.     

Adaptive synchronization of hyperchaotic Lü system with uncertainty

This paper presents a novel adaptive control scheme for synchronization of the latest hyperchaotic Lü system. Based on the Lyapunov stability theory, a feedback controller and a parameter update law are designed for the synchronization of hyperchaotic Lfi systems with uncertainty. Numerical simulations are given to demonstrate the validity of the synchronization technique.     

A novel mixed-synchronization phenomenon in coupled Chua’s circuits via non-fragile linear control

Dynamical variables of coupled nonlinear oscillators can exhibit different synchronization patterns depending on the designed coupling scheme.In this paper,a non-fragile linear feedback control strategy with multiplicative controller gain uncertainties is proposed for realizing the mixed-synchronization of Chua’s circuits connected in a drive-response configuration.In particular,in the mixed-synchronization regime,different state variables of the response system can evolve into complete synchronization,anti-synchronization and even amplitude death simultaneously with the drive variables for an appropriate choice of scaling matrix.Using Lyapunov stability theory,we derive some sufficient criteria for achieving global mixed-synchronization.It is shown that the desired non-fragile state feedback controller can be constructed by solving a set of linear matrix inequalities (LMIs).Numerical simulations are also provided to demonstrate the effectiveness of the proposed control approach.     

Localization of hidden Chua?s attractors

The classical attractors of Lorenz, Rossler, Chua, Chen, and other widely-known attractors are those excited from unstable equilibria. From computational point of view this allows one to use numerical method, in which after transient process a trajectory, started from a point of unstable manifold in the neighborhood of equilibrium, reaches an attractor and identifies it. However there are attractors of another type: hidden attractors, a basin of attraction of which does not contain neighborhoods of equilibria. In the present Letter for localization of hidden attractors of Chua?s circuit it is suggested to use a special analytical-numerical algorithm.     

How similar is the performance of the cubic and the piecewise-linear circuits of Chua?

This Letter reports phase diagrams quantifying and contrasting the dynamical performance of the paradigmatic piecewise-linear and cubic circuits of Chua. Although both circuits may be regarded as macroscopically isomorphic over wide regions in control parameter space, we show that their microscopic structure displays a myriad of rather distinctive intrinsic features making them unique. Inhomogeneities embedded in periodic and chaotic phases complicate some applications of the circuits but may also adequately act as realistic noise proxies in synchronization problems. In addition, infinite cascades of spirals and hubs observed experimentally very recently in a related dissipative flow are shown to be also present in both circuits of Chua, emerging however in a rather distinctive asymmetric way. Thus Chua's circuits may be used to study experimentally elusive and theoretically intricate phenomena generating periodicity hubs.     

Adaptive synchronization of a hyperchaotic L system based on extended passive control

This paper investigates the adaptive synchronization of hyperchaotic Lü systems based on the method of extended passive control. By combining the feedback control, the extended passive control method with two output variables is developed, which can synchronize hyperchaotic Lü systems asymptotically and globally more easily without knowing the bound of state of the hyperchaotic system. Adaptive laws are introduced to estimate the unknown parameters as well. Simulation results show the effectiveness and flexibility of the proposed control scheme.     

Dynamical behaviors of a system with switches between the Rssler oscillator and Chua circuits

The behaviors of a system that alternates between the R¨ossler oscillator and Chua’s circuit is investigated to explore the influence of the switches on the dynamical evolution.Switches related to the state variables are introduced,upon which a typical switching dynamical model is established.Bifurcation sets of the subsystems are derived via analysis of the related equilibrium points,which divide the parameters into several regions corresponding to different types of attractors.The dynamics behave typically in period orbits with the variation of the parameters.The focus/cycle periodic switching phenomenon is explored in detail to present the mechanism of the movement.The period-doubling bifurcation to chaos can be observed via the doubling increase of the turning points related to the switches.Furthermore,period-decreasing sequences have been obtained,which can be explained by the variation of the eigenvalues associated with the equilibrium points of the subsystems.     

Dynamical behaviors of a system with switches between the Rssler oscillator and Chua circuits

The behaviors of a system that alternates between the R¨ossler oscillator and Chua＇s circuit is investigated to explore the influence of the switches on the dynamical evolution.Switches related to the state variables are introduced,upon which a typical switching dynamical model is established.Bifurcation sets of the subsystems are derived via analysis of the related equilibrium points,which divide the parameters into several regions corresponding to different types of attractors.The dynamics behave typically in period orbits with the variation of the parameters.The focus/cycle periodic switching phenomenon is explored in detail to present the mechanism of the movement.The period-doubling bifurcation to chaos can be observed via the doubling increase of the turning points related to the switches.Furthermore,period-decreasing sequences have been obtained,which can be explained by the variation of the eigenvalues associated with the equilibrium points of the subsystems.     

Prediction-based control of chaos and a dynamic Parrondoʼs paradox

We analyze the stabilization of an unstable periodic orbit (UPO) by periodic prediction-based control (PBC). We rigorously prove that, for 2-periodic orbits, a pulse strategy reduces the necessary control strength to stabilize the UPO. Moreover, we find that in some cases the periodic control prevents some undesirable effects induced by the PBC method. In this way, we provide an example of a dynamic Parrondo?s paradox: the switching between two undesirable dynamics results in a nicely periodic dynamic behavior.     

Electronic structure of disordered graphene with Greenʼs function approach

The Green functions play a big role in the calculation of the local density of states of the carbon nanostructures. We investigate their nature for the variously oriented and disclinated graphene-like surface. Next, we investigate the case of a small perturbation generated by two heptagonal defects and from the character of the local density of states in the border sites of these defects we derive their minimal and maximal distances on the perturbed cylindrical surface. For this purpose, we transform the given surface into a chain using the Haydock recursion method. We will suppose only the nearest-neighbor interactions between the atom orbitals, in other words, the calculations suppose the short-range potential.     

Study on eigenvalue space of hyperchaotic canonical four-dimensional Chua’s circuit

The eigenvalue space of the canonical four-dimensional Chua's circuit which can realize every eigenvalue for four-dimensional system is studied in this paper. First, the analytical relations between the circuit parameters and the eigenvalues of the system are established, and therefore all the circuit parameters can be determined explicitly by any given set of eigenvalues. Then, the eigenvalue space of the circuit is investigated in two cases by the nonlinear elements used. According to the types of the eigenvalues, some novel hyperchaotic attractors are presented. Further, the dynamic behaviours of the circuit are studied by the bifurcation diagrams and the Lyapunov spectra of the eigenvalues.