Chaos synchronization by variable strength linear coupling and Lyapunov function derivative in series form

A new general strategy to achieve chaos synchronization by variable strength linear coupling without another active control is proposed. They give the criteria of chaos synchronization for two identical chaotic systems and two different chaotic dynamic systems with variable strength linear coupling. In this method, the time derivative of Lyapunov function in series form is firstly used. Lorenz system, Duffing system, Rössler system and Hyper-Rössler system are presented as simulated examples.     

Synchronization transition in degenerate optical parametric oscillators induced by nonlinear coupling

In this paper, the synchronization of certain degenerate optical parametric oscillators is investigated in detail. Complete and/or partial synchronization can be reached when linear controller, constructed by the real part or imaginary part of the subharmonic mode, is imposed on the chaotic degenerate optical parametric oscillators with appropriate coupling intensity. The Lyapunov exponents under different coupling coefficients are calculated to find the critical condition for complete synchronization. Transition from complete synchronization to partial synchronization is observed when nonlinear coupling is introduced into the two chaotic oscillators. It is found that synchronization of chaotic oscillators keeps robust when the intensity of the nonlinear coupling is less than the intensity of the linear coupling; the complete synchronization state is destructed and transient period for partial synchronization is in certain delay when the intensity of the nonlinear coupling is beyond the intensity of the linear coupling.     

Chaos synchronization for master slave piecewise linear systems: Application to Chua’s circuit

This paper deals with chaos synchronization for master slave piecewise linear systems. The synchronization problem is formulated as a global stability problem of error synchronization dynamics. New sufficient conditions are provided using a Lyapunov approach and the so-called S-procedure. We show that the synchronization problem can be solved as an optimization problem subject to a set of Linear Matrix Inequalities (LMI) for which a state feedback controller is designed efficiently. The effectiveness of the proposed solution is verified via simulation results using the original Chua’s circuit model. Furthermore, it will be proven that the new sufficient conditions relaxed the conservatism of previous existing works.     

Chaos in fractional-order Genesio-Tesi system and its synchronization

In this paper we study the chaotic dynamics of fractional-order Genesio-Tesi system. Theoretically, a necessary condition for occurrence of chaos is obtained. Numerical investigations on the dynamics of this system have been carried out and properties of the system have been analyzed by means of Lyapunov exponents. It is shown that in case of commensurate system the lowest order of fractional-order Genesio-Tesi system to yield chaos is 2.79. Further, chaos synchronization of fractional-order Genesio-Tesi system is investigated via two different control strategies. Active control and sliding mode control are proposed and the stability of the controllers are studied. Numerical simulations have been carried out to verify the effectiveness of controllers.     

Generalized synchronization of chaotic systems by pure error dynamics and elaborate Lyapunov function

The generalized synchronization is studied by applying pure error dynamics and elaborate Lyapunov function in this paper. Generalized synchronization can be obtained by pure error dynamics without auxiliary numerical simulation, instead of current mixed error dynamics in which master state variables and slave state variables are presented. The elaborate Lyapunov function is applied rather than the current plain square sum Lyapunov function, deeply weakening the power of Lyapunov direct method. The scheme is successfully applied to both autonomous and nonautonomous double Mathieu systems with numerical simulations.     

Effects of time delay and coupling strength on synchronization transitions in excitable homogeneous random network

In this paper we numerically investigate the effects of time delay and coupling strength on synchronization transitions in excitable homogeneous random network. Different roles of time delay and coupling strength have been discovered by synchronization parameter and space–time plots. Specifically, we have found three distinct parameter regions, i.e., asynchronous region (domain I for small time delay), transition region (domain II for moderate time delay) and synchronous region (domain III for large time delay) as time delay is increased. The phenomenon of multi-stability is observed in the transition region. While coupling strength can enhance synchronization in the transition region and can reduce synchronization time in the synchronous region. All these results are independence on the system size.     

Parameter identification and synchronization of fractional-order chaotic systems

The knowledge about parameters and order is very important for synchronization of fractional-order chaotic systems. In this article, identification of parameters and order of fractional-order chaotic systems is converted to an optimization problem. Particle swarm optimization algorithm is used to solve this optimization problem. Based on the above parameter identification, synchronization of the fractional-order Lorenz, Chen and a novel system (commensurate or incommensurate order) is derived using active control method. The new fractional-order chaotic system has four-scroll chaotic attractors. The existence and uniqueness of solutions for the new fractional-order system are also investigated theoretically. Simulation results signify the performance of the work.     

Chaos generalized synchronization of an inertial tachometer with new Mathieu-Van der Pol systems as functional system by GYC partial region stability theory

A new strategy to achieve generalized chaos synchronization by GYC partial region stability theory is proposed. By using the GYC partial region stability theory the Lyapunov function is a simple linear homogeneous function of error states and the controllers are more simple and introduce less simulation error because they are in lower order than that of traditional controllers. In simulation examples, an inertial tachometer system and Mathieu-Van der Pol system are used.     

Chaos and complexity in mine grade distribution series detected by nonlinear approaches

In this article, the underlying dynamics of treating grade distribution is interpreted as a chaotic system instead of a stochastic system for a better understanding. Here, we study the behavior of grade distribution spatial series acquired at the Chadormalu mine in Bafgh city of Iran to distinguish the possible existence of low‐dimensional deterministic chaos. This work applies a variety of nonlinear techniques for detecting the chaotic nature of the grade distribution spatial series and adopts a nonlinear prediction method for predicting the future of the grade distributions. First, the delay time dimension is computed using auto mutual information function to reconstruct the strange attractors. Then, the dimensionality of the trajectories is obtained using Cao's method and, correspondingly, the correlation dimension method is adopted to quantify the embedding dimension. The low embedding dimensions achieved from these methods show the existence of low dimensional chaos in the mining data. Next, the high sensitivity to initial conditions is evaluated using the maximal Lyapunov exponent criterion. Positive Lyapunov exponents obtained demonstrate the exponential divergence of the trajectories and hence the unpredictability of the data. Afterward, the nonlinear surrogate data test is done to further verify the nonlinear structure of the grade distribution series. This analysis provides considerable evidence for the being of low‐dimensional chaotic dynamics underlying the mining spatial series. Lastly, a nonlinear prediction scheme is carried out to predict the grade distribution series. Some computer simulations are presented to illustrate the efficiency of the applied nonlinear tools. © 2016 Wiley Periodicals, Inc. Complexity 21: 355–369, 2016     

New Taylor series approach to state-space analysis and optimal control of linear systems

A new Taylor series approach is presented which reduces the problem of determining the state vector coefficient matrixX for time-invariant systems to an expression involving multiplications of matrices of small dimensions. This approach is numerically superior to known techniques and is extended to cover the time-varying case, wherein analogous expressions are derived. Furthermore, the optimal control problem is solved using the same technique. Finally, an expression is derived for the computation of the approximation error involved in computingX, prior to determiningX.This work was partially supported by the Greek State Scholarship Foundation (IKY).     

Group synchronization in complex dynamical networks with different types of oscillators and adaptive coupling schemes

This paper focus on schemes and corresponding criteria for group synchronization in complex dynamical networks consisted of different group of chaotic oscillators. The global asymptotically stable criteria for a linearly or adaptively coupled network are derived to ensure each group of oscillators synchronize to the same behavior. Theoretical analysis and numerical simulation results show that the group synchronization can be guaranteed by enhancing the external coupling strength whenever there are connections or not within the groups under the “same input” condition. All of the results are proved rigorously. Finally, a network with three groups, a scale-free sub-network, a small-world sub-network and a ring sub-network, is illustrated, and the corresponding numerical simulations verify the theoretical analysis.     

Bifurcations, chaos and synchronization in ADVP circuit with parallel resistor

In this work we investigate the dynamical behaviors of Van der Pol-Duffing circuit (ADVP) with parallel resistor. The model is described by a continuous-time three dimensional autonomous system. The stability conditions of the equilibria are analyzed. The existence of periodic solutions and their stabilities about the node equilibrium point of the system are studied by using Hopf's theorem and Hsü and Kazarinoff theorem. Lyapunov spectrum is calculated for the proposed system. Adaptive synchronization using backstepping design is applied successfully to the system. Chaotic behaviors and the efficiency of the synchronization method are verified by numerical simulations.     

Limit cycles of discontinuous piecewise differential systems formed by linear centers in R2 and separated by two circles

We show that discontinuous planar piecewise differential systems formed by linear centers and separated by two concentric circles can have at most three limit cycles. Usually is a difficult problem to provide the exact upper bound that a class of differential systems can exhibit. Here we also provide examples of such systems with zero, one, two, or three limit cycles.     

A new result in the theory and computation of the least-norm solution of a linear program

By perturbing properly a linear program to a separable quadratic program, it is possible to solve the latter in its dual variable space by iterative techniques such as sparsity-preserving SOR (successive overrelaxation) algorithms. The main result of this paper gives an effective computational criterion to check whether the solutions of the perturbed quadratic programs provide the least-norm solution of the original linear program.This research was sponsored by the United States Army under Contract No. DAAG29-80-C-0041. This material is based upon work supported by the National Science Foundation, Grant Nos. DCR-84-20963 and DMS-82-109050, and by the Italian National Research Council (CNR).The author wishes to thank Professor O. L. Mangasarian for his helpful comments which helped to improve the paper.     

Strict monotonicity and improved complexity in the standard form projective algorithm for linear programming

In a recent paper, Shaw and Goldfarb show that a version of the standard form projective algorithm can achieve step complexity, as opposed to the O(nL) step complexity originally demonstrated for the algorithm. The analysis of Shaw and Goldfarb shows that the algorithm, using a constant, fixed steplength, approximately follows the central trajectory. In this paper we show that simple modifications of the projective algorithm obtain the same complexity improvement, while permitting a linesearch of the potential function on each step. An essential component is the addition of a single constraint, motivated by Shaw and Goldfarb's analysis, which makes the standard form algorithm strictly monotone in the true objective.This paper was written while the author was a research fellow at the Center for Operations Research and Econometrics, Université Catholique de Louvain, Louvain-la-Neuve, Belgium. Research supported by CORE, and the Center for Advanced Studies, University of Iowa.     

Feedback stabilization and adaptive stabilization of stochastic nonlinear systems by the control Lyapunov function

Our aims of this paper are twofold: On one hand, we study the asymptotic stability in probability of stochastic differential system, when both the drift and diffusion terms are affine in the control. We derive sufficient conditions for the existence of control Lyapunov functions (CLFs) leading to the existence of stabilizing feedback laws which are smooth, except possibly at the equilibrium state. On the other hand, we consider the previous systems with an unknown constant parameters in the drift and introduce the concept of an adaptive CLF for stochastic system and use the stochastic version of Florchinger's control law to design an adaptive controller. In this framework, the problem of adaptive stabilization of nonlinear stochastic system is reduced to the problem of non-adaptive stabilization of a modified system.     

Instrumental variable estimation for stochastic differential equations linear in drift parameter and driven by a sub-fractional Brownian motion

We investigate the asymptotic properties of instrumental variable estimators of the drift parameter for stochastic processes satisfying linear stochastic differential equations driven by a sub-fractional Brownian motion.     

Cluster mixed synchronization via pinning control and adaptive coupling strength in community networks with nonidentical nodes

In this paper, complex networks with community structure and nonidentical nodes are investigated. The cluster mixed synchronization of these networks is studied by using some linear pinning control schemes. Only the nodes in one community which have direct connections to the nodes in other communities are controlled. Adaptive coupling strength method is adopted to achieve the synchronization as well. According to Lyapunov stability theory, several sufficient conditions for the network to achieve cluster mixed synchronization are derived. Numerical simulations are provided to verify the correctness and the effectiveness of the theoretical results.     

Existence of periodic orbits and chaos in a class of three-dimensional piecewise linear systems with two virtual stable node-foci

In this paper, we consider a new class of piecewise linear (PWL) systems with two virtual stable node-foci (the meaning of “virtual” is from Bernardo et al. (2008)) which exhibits periodic orbits and chaos. This fact that PWL systems have no unstable equilibria but has chaos will unavoidably make the exploration of this chaos more complicated. Particular values for bifurcation diagram are provided. Based on mathematical analysis and Poincaré map, periodic orbits of this kind of system without unstable equilibrium points are derived, the corresponding existence theorems are given, and the obtained results are applied to specific examples.