Complete Chaos Synchronization Research Articles

Chaos Synchronization of InGaAsP Lasers

The optical output of a semiconductor laser can fluctuate chaotically by modulating its direct current in limited conditions of the modulated current signal parameters in terms of modulation frequency and modulation index. In this work, single, double, and chaotic pulses of an InGaAsP laser with direct current modulation, are numerically presented through a bifurcation diagram. Numerically, the unidirectional optical coupling system realizes chaotic synchronization between two identical InGaAsP lasers with direct current modulation, as the transmitter/receiver configuration. The transmission time for transmitting light from the transmitted laser to the received laser is essential for controlling the quality of chaos synchronization. The transmission time applies on the order of nanoseconds. Chaos synchronization quality is estimated by a correlation plot and calculated by the cross-correlation coefficient. This study observed the best synchronization quality (complete chaos synchronization) when the two lasers are identical. On the other hand, the chaotic synchronization between two non-identical InGaAsP lasers was investigated. In this case, complete chaos synchronization is not found, and the quality of chaotic synchronization was observed to decrease as the mismatch between the parameters of the two lasers increased.

A theoretical framework for nonlinear dynamics in finance systems: Implications for stability, control and synchronization

It has been widely observed that most deterministic dynamical systems go into chaos for some values of their parameters. One of the most popular and widely used criteria is the conditional Lyapunov exponents, which constitute average measurements of expansion or shrinkage of small displacements along the synchronized trajectory. The Lyapunov characteristic exponents play a crucial role in the description of the behaviour of dynamical systems as they can be used to analyse the stability limit sets and to check sensitive dependence on initial conditions, that is, the presence of chaotic attractors. In this paper, Lyapunov stability theory and linear matrix inequalities (LMI) are employed to design control functions for the respective, control, and synchronization of the chaotic and hyperchaotic finance systems. The designed linear matrix inequalities (LMI) nonlinear controllers are capable of stabilizing the chaotic and hyperchaotic finance systems at any position as well as controlling it to track any trajectory that is a smooth function of time. The respective chaotic attractors were found to have a moderate value of the largest Lyapunov exponents (0.874959 s^(-1) and 0.650847 s^(-1)) with associated (Lyapunov) dimensions of 1.00 and 2.00 for the chaotic and hyperchaotic finance systems respectively. Based on Lyapunov stability theory and linear matrix inequalities (LMI), some necessary and sufficient criteria for stable synchronous behaviour are obtained and an exact analytic estimate of the threshold coupling, k_th, for complete chaos synchronization is derived. Finally, numerical simulation results are presented to validate the feasibility of the theoretical analysis.

Synchronization transitions in a hyperchaotic SQUID trimer.

The phenomena of intermittent and complete synchronization between two out of three identical, magnetically coupled Superconducting QUantum Interference Devices (SQUIDs) are investigated numerically. SQUIDs are highly nonlinear superconducting oscillators/devices that exhibit strong resonant and tunable response to applied magnetic field(s). Single SQUIDs and SQUID arrays are technologically important solid-state devices, and they also serve as a testbed for exploring numerous complex dynamical phenomena. In SQUID oligomers, the dynamic complexity increases considerably with the number of SQUIDs. The SQUID trimer, considered here in a linear geometrical configuration using a realistic model with experimentally accessible control parameters, exhibits chaotic and hyperchaotic behavior in wide parameter regions. Complete chaos synchronization as well as intermittent chaos synchronization between two SQUIDs of the trimer is identified and characterized using the complete Lyapunov spectrum of the system and appropriate measures. The passage from complete to intermittent synchronization seems to be related to chaos-hyperchaos transitions as has been conjectured in the early days of chaos synchronization.

Criterion of globally complete chaos synchronization for diverse three-node VCSEL networks with coupling delays

For diverse three node vertical-cavity surface-emitting laser (VCSEL) networks with uniform time-delayed coupling, we propose a new criterion of the globally complete chaos synchronization (GCCS) among all node lasers by using the mater-stability function (MSF). It is found that GCCS can be achieved for an arbitrarily given three-node VCSEL networks when two points determined by constant row sum and two transversal eigenvalues fall into the region of stability where maximum Lyapunov exponent (MLE) is negative and the MLE as a function of constant row sum and the eigenvalue associated to perturbations within the synchronization manifold is positive.

Chaos Synchronization in Time‐Dependent Duplex Networks

Though the complete chaos synchronization on single‐layer networks has been well understood, it is still a challenge on multiplex networks. In this work, we study the complete chaos synchronization on time‐dependent duplex networks in which interaction pattern among oscillators alternates periodically between two single‐layer networks. The alternations between two layers are characterized by the offset strength A and the switching frequency ω. We find that there are two dynamical regimes depending on ω. For high ω, the critical A for the stable complete synchronization is independent of ω and the fast‐switching approximation suggests that the time‐dependent duplex networks can be approximated by the time‐independent duplex networks with effective coupling strength. For low ω, the critical A depends on ω nonmonotonically. At extremely low ω, the estimation of the critical A can be obtained by a single‐mode approximation taking one dominant transversal network mode to complete synchronization into considerations.

Phase diagram of coupled thermoacoustic chaotic oscillators

Linear analysis of thermoacoustic oscillations have been extensively made, but there is still a lack of knowledge on nonlinear phenomena such as synchronization and chaotic oscillations. Thermoacoustic chaotic oscillations have been reported in Taconis and combustion oscillators but these systems do not allow a well-controlled experiment because of the extreme temperature conditions. By making the system acoustically dissonant, we recently succeeded in generating thermoacoustic chaotic oscillations with modest temperatures. In this study, we present spatiotemporal evolution of thermoacoustic chaos from pressure measurements, and report an experimental bifurcation diagram when two thermoacoustic chaotic oscillators are dissipatively coupled to each other. The two-parameter bifurcation diagram is constructed by varying the frequency mismatch and the coupling strength. Complete chaos synchronization is observed in the region with a frequency mismatch of less than 1% of the uncoupled oscillator, with on-off intermittency phenomenon for low coupling strength. In other regions, synchronization between quasiperiodic oscillations and that between limit-cycle oscillations and oscillation death are observed as well as asynchronous states.

Bifurcation diagram of coupled thermoacoustic chaotic oscillators.

A thermoacoustic chaotic oscillator is a fluid system that presents thermally induced chaotic oscillations of a gas column. This study experimentally reports a bifurcation diagram when two thermoacoustic chaotic oscillators are dissipatively coupled to each other. The two-parameter bifurcation diagram is constructed by varying the frequency mismatch and the coupling strength. Complete chaos synchronization is observed in the region with a frequency mismatch of less than 1% of the uncoupled oscillator. In other regions, synchronization between quasiperiodic oscillations and that between limit-cycle oscillations and amplitude death are observed as well as asynchronous states.

Chaotic Synchronization in Models of Impulsive Power Systems with Delay

Global complete chaos synchronization in impulsive power systems with delay is studied. The sufficient conditions of exponential synchronization are established based on the theory of stability of impulsive differential equations with delay. Illustrative examples that demonstrate the applicability and effectiveness of the obtained results are given.

Real-time multi-target ranging based on chaotic polarization laser radars in the drive-response VCSELs.

According to the principle of complete chaos synchronization and the theory of Hilbert phase transformation, we propose a novel real-time multi-target ranging scheme by using chaotic polarization laser radar in the drive-response vertical-cavity surface-emitting lasers (VCSELs). In the scheme, to ensure each polarization component (PC) of the master VCSEL (MVCSEL) to be synchronized steadily with that of the slave VCSEL, the output x-PC and y-PC from the MVCSEL in the drive system and those in the response system are modulated by the linear electro-optic effect simultaneously. Under this condition, by simulating the influences of some key parameters of the system on the synchronization quality and the relative errors of the two-target ranging, related operating parameters can be optimized. The x-PC and the y-PC, as two chaotic radar sources, are used to implement the real-time ranging for two targets. It is found that the measured distances of the two targets at arbitrary position exhibit strong real-time stability and only slight jitter. Their resolutions are up to millimeters, and their relative errors are very small and less than 2.7%.

Adaptive backstepping control, synchronization and circuit simulation of a 3-D novel jerk chaotic system with two hyperbolic sinusoidal nonlinearities

Abstract In this research work, a six-term 3-D novel jerk chaotic system with two hyperbolic sinusoidal nonlinearities has been proposed, and its qualitative properties have been detailed. The Lyapunov exponents of the novel jerk system are obtained as L1 = 0.07765,L2 = 0, and L3 = −0.87912. The Kaplan-Yorke dimension of the novel jerk system is obtained as DKY = 2.08833. Next, an adaptive backstepping controller is designed to stabilize the novel jerk chaotic system with two unknown parameters. Moreover, an adaptive backstepping controller is designed to achieve complete chaos synchronization of the identical novel jerk chaotic systems with two unknown parameters. Finally, an electronic circuit realization of the novel jerk chaotic system using Spice is presented in detail to confirm the feasibility of the theoretical model

Manipulation of the complete chaos synchronization in dual-channel encryption system based on polarization-division-multiplexing

For the dual-channel encryption system, based on polarization-division-multiplexing, we put forward a new control scheme for complete chaos synchronization by means of linear electro-optic (EO) effect. In the scheme, the chaotic synchronization quality of each linear polarization (LP) mode component varies periodically with the applied electric field. The variation regulation is as follows: Complete chaos synchronization ↔ acute oscillation. With the applied electric field fixed at a certain value, the robustness of the complete chaotic synchronization quality due to the bias current and the feedback strength is improved greatly by EO modulation. Each LP mode can obtain the complete chaos synchronization in a large range of the bias current and the feedback strength. And the encoding message modulated to each LP mode can be almost re-established.

Ultimate Bound of a 3D Chaotic System and Its Application in Chaos Synchronization

Two ellipsoidal ultimate boundary regions of a special three-dimensional (3D) chaotic system are proposed. To this chaotic system, the linear coefficient of theith state variable in theith state equation has the same sign; it also has two one-order terms and one quadratic cross-product term in each equation. A numerical solution and an analytical expression of the ultimate bounds are received. To get the analytical expression of the ultimate boundary region, a new result of one maximum optimization question is proved. The corresponding ultimate boundary regions are demonstrated through numerical simulations. Utilizing the bounds obtained, a linear controller is proposed to achieve the complete chaos synchronization. Numerical simulation exhibits the feasibility of the designed scheme.

Sliding Mode Controller Design for Global Chaos Synchronization of Coullet Systems

This paper derives new results for the design of sliding mode controller for the global chaos synchronization of identical Coullet systems (1981). The synchronizer results derived in this paper for the complete chaos synchronization of identical hyperchaotic systems are established using sliding control theory and Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Coullet systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Coullet systems.

Investigation of chaos synchronization in photonic crystal lasers

Chaotic dynamics and chaos synchronization in photonic crystal (PC) lasers with optical feedback are investigated numerically. The effect of various system parameters such as amplitude reflectivity of the external mirror “r”, external cavity length “Le”, and injection current “I” on system dynamics is addressed in detail. Simulation results are presented using MATLAB to address system behavior. The parameters r, Le, and I are varied over the ranges (0.05–0.25), (2.8–3.2mm), and (1.1Ith–2Ith), respectively. The results indicate that very small parameter mismatches between the transmitter laser and receiver laser affect strongly complete chaos synchronization between them.

On the boundedness of solutions to the Lorenz-like family of chaotic systems

This paper deals with a class of three-dimensional autonomous nonlinear systems which have potential applications in secure communications, and investigates the localization problem of compact invariant sets of a class of Lorenz-like chaotic systems which contain T system with the help of iterative theorem and Lyapunov function theorem. Since the Lorenz-like chaotic system does not have y in the second equation, the approach used to the Lorenz system cannot be applied to the Lorenz-like chaotic system. We overcome this difficulty by introducing a cross term and get an interesting result, which includes the most interesting case of the chaotic attractor of the Lorenz-like systems. Furthermore, the results obtained in this paper are applied to study complete chaos synchronization. Finally, numerical simulations show the effectiveness of the proposed scheme.

Spectrally efficient multiplexing of chaotic light

We numerically demonstrate multiplexing and demultiplexing of two distinct chaotic optical signals with strongly overlapped spectra, which are generated by mutually coupled external-cavity semiconductor lasers. Demultiplexing is performed by complete chaos synchronization, while the lasers at the receiving end are optically injected with the same multiplexed signal that couples the emitters together. Such a configuration could lead to spectrally efficient chaos-based optical encryption and decryption of multiple data streams.

Synchronization and suppression of chaos in non-locally coupled map lattices

We considered coupled map lattices with long-range interactions to study the spatiotemporal behaviour of spatially extended dynamical systems. Coupled map lattices have been intensively investigated as models to understand many spatiotemporal phenomena observed in extended system, and consequently spatiotemporal chaos. We used the complex order parameter to quantify chaos synchronization for a one-dimensional chain of coupled logistic maps with a coupling strength which varies with the lattice in a power-law fashion. Depending on the range of the interactions, complete chaos synchronization and chaos suppression may be attained. Furthermore, we also calculated the Lyapunov dimension and the transversal distance to the synchronization manifold.

Spatial and polarization entanglement of lasing patterns and related dynamic behaviors in laser-diode-pumped solid-state lasers

To provide the underlying physical mechanism for formations of spatial- and polarization-entangled lasing patterns (namely, SPEPs), we performed experiments using a c-cut Nd:GdVO(4) microchip laser with off-axis laser-diode pumping. This extends recent work on entangled lasing pattern generation from an isotropic laser, where such a pattern was explained only in terms of generalized coherent states (GCSs) formed by mathematical manipulation. Here, we show that polarization-resolved transverse patterns can be well explained by the transverse mode-locking of distinct orthogonal linearly polarized Ince-Gauss (IG) mode pairs rather than GCSs. Dynamic properties of SPEPs were experimentally examined in both free-running and modulated conditions to identify long-term correlations of IG mode pairs over time. The complete chaos synchronization among IG mode pairs subjected to external perturbation is also demonstrated.

Synchronization criteria for non-autonomous chaotic systems via sinusoidal state error feedback control

A sufficient criterion for synchronizing two identical non-autonomous chaotic systems coupled by sinusoidal state error feedback control is firstly derived in the form of a matrix inequality by the Lyapunov direct method. The matrix inequality is then simplified into some algebraic inequalities based on the Gerschgorin disc theorem. In order to obtain simpler algebraic criteria, the state variables are restricted in a sub-region. The property that similar matrices have the same eigenvalues is further applied to obtain more flexible synchronization criteria. Numerical examples are simulated to show that complete chaos synchronization can be achieved under these criteria by three practical non-autonomous chaotic systems: the Mathieu–Duffing equation of two dimensions, a gyrostat system of three dimensions and a loudspeaker system of four dimensions.

Complete and generalized chaotic synchronization in a system of three interacting maps

The bifurcation decay of complete chaos synchronization and the mechanisms of transition to partial synchronization of regular and chaotic oscillations are considered in a ring of three logistic maps with symmetric diffusion coupling. The modes of partial chaos synchronization are shown to be at the same time the modes of generalized chaotic synchronization.