No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Synchronization of uncertain chaotic systems with double strange attractors
Abstract
This paper is devoted to address synchronization between master and slave Newton-Leipnik chaotic systems, each of which has double co-existing strange attractors. Unlike the already existing results, uncertainty was considered in the slaved system here. Synchronization was realized in virtue of sliding mode control, which was designed in an easy going way under auspices of the uncertainty compensation induced by an extended system. On account of which the identity of slaved system model and master system model was facilitated and guaranteed in finite time. Both its speed and invariance together with robustness were discussed theoretically. Simulation results verify the validity of the suggested scheme.
ResearchGate has not been able to resolve any citations for this publication.
It is shown how chaotic systems with more than one strange attractor can be controlled. Issues in controlling multiple (coexisting) strange attractors are stabilizing a desired motion within one attractor as well as taking the system dynamics from one attractor to another. Realization of these control objectives is demonstrated using a numerical example, the Newton–Leipnik system.
Based on the open-plus-closed-loop (OPCL) control method a systematic and comprehensive controller is presented in this paper for a chaotic system, that is, the Newton–Leipnik equation with two strange attractors: the upper attractor (UA) and the lower attractor (LA). Results show that the final structure of the suggested controller for stabilization has a simple linear feedback form. To keep the integrity of the suggested approach, the globality proof of the basins of entrainment is also provided. In virtue of the OPCL technique, three different kinds of chaotic controls of the system are investigated, separately: the original control forcing the chaotic motion to settle down to the origin from an arbitrary position of the phase space; the chaotic intra-attractor control for stabilizing the equilibrium points only belonging to the upper chaotic attractor or the lower chaotic one; and the inter-attractor control for compelling the chaotic oscillation from one basin to another one. Both theoretical analysis and simulation results verify the validity of the proposed means.
Based on the Lyapunov stabilization theory and Gerschgorin theorem, a simple generic criterion is derived for global synchronization of two coupled chaotic systems with a unidirectional linear error feedback coupling. This simple criterion is applicable to a large class of chaotic systems, where only a few algebraic inequalities are involved. To demonstrate the efficiency of design, the suggested approach is applied to some typical chaotic systems with different types of nonlinearities, such as the original Chua’s circuit, the modified Chua’s circuit with a sine function, and the Rössler chaotic system. It is proved that these synchronizations are ensured by suitably designing the coupling parameters.
This paper considers an autonomous nonlinear system of differential equations derived in [Leipnik, 1979]. A criterion for the existence of closed orbits in similar systems is presented. Numerical results are made rigorous by the use of interval analytic techniques in establishing the existence of a periodic solution which is not asymptotically stable. The limitations of the method of locating orbits are considered when a promising candidate for a closed orbit is shown not to intersect itself.
A functional version of LaSalle's invariance principle is derived, i.e., rather than the usual pointwise Lyapunov-like functions it uses specially constructed functionals along system trajectories. This modification enables the principle to handle even nonautonomous systems to which the classical LaSalle's principle is not directly applicable. The new theoretical results are then used to study robust synchronization of general Lie´nard type of systems. The developed technique is finally applied to chaotic oscillators synchronization. Numerical simulation is included to demonstrate the effectiveness of the proposed methodology.
This paper provides a unified method for analyzing chaos synchronization of the generalized Lorenz systems. The considered synchronization scheme consists of identical master and slave generalized Lorenz systems coupled by linear state error variables. A sufficient synchronization criterion for a general linear state error feedback controller is rigorously proven by means of linearization and Lyapunov's direct methods. When a simple linear controller is used in the scheme, some easily implemented algebraic synchronization conditions are derived based on the upper and lower bounds of the master chaotic system. These criteria are further optimized to improve their sharpness. The optimized criteria are then applied to four typical generalized Lorenz systems, i.e. the classical Lorenz system, the Chen system, the Lv system and a unified chaotic system, obtaining precise corresponding synchronization conditions. The advantages of the new criteria are revealed by analytically and numerically comparing their sharpness with that of the known criteria existing in the literature.
In this paper, a new method for robust synchronization between a discrete-time response subsystem and a continuous-time chaotic drive subsystem is proposed. This methodology takes the LMI approach to first design a continuous-time H∞ robust observer that is capable of maintaining synchronization even in the presence of parameter mismatch and noise perturbation, and then uses the prediction-based digital redesign technique to convert it to an equivalent discrete-time observer based on the sampling state-matching principle. The proposed robust hybrid chaos synchronization method is finally validated via numerical simulations on the complex chaotic Chen's system.
The dynamics and bifurcations of the Newton–Leipnik equations are presented. Numerical computations and local stability calculations suggest that the dynamics of the Newton–Leipnik equations are related to the dynamics and bifurcations of a family of odd, symmetric, bimodal maps. The numerically computed dynamics and bifurcations of the Newton–Leipnik equations are compared with the dynamics and bifurcations of a family of odd, symmetric, bimodal maps to motivate the connection between the two systems.
The dynamics and bifurcations of the Newton-Leipnik equations are presented. Numerical computations and local stability calculations suggest that the dynamics of the Newton-Leipnik equations are related to the dynamics and bifurcations of a family of odd, symmetric, bimodal maps. The numerically computed dynamics and bifurcations of the Newton-Leipnik equations are compared with the dynamics and bifurcations of a family of odd, symmetric, bimodal maps to motivate the connection between the two systems.
The dynamics and bifurcations of a family of odd, symmetric, bimodal maps, fα are discussed. We show that for a large class of parameter values the dynamics of fα can be described via an identification with a unimodal map uα. In this parameter regime, a periodic orbit of period 2n + 1 of uα corresponds to a periodic orbit of period 4n + 2 for fα. A periodic orbit of period 2n of uα corresponds to a pair of distinct periodic orbits also of period 2n for fα. In a more general setting we describe the genealogy of periodic orbits in the family fα using symbolic dynamics and kneading theory. We identify which periodic orbits of even periods are born in period-doubling bifurcations and which are born in pitchfork bifurcations and provide a method of describing the "ancestors" and "descendants" of these orbits. We also show that certain periodic orbits of odd periods are born in saddle-node bifurcations.
Recently, a new hyperchaos generator, obtained by controlling a three-dimensional autonomous chaotic system --- Chen's system --- with a periodic driving signal, has been found. In this letter, we formulate and study the hyperchaotic behaviors in the corresponding fractional-order hyperchaotic Chen's system. Through numerical simulations, we found that hyperchaos exists in the fractional-order hyperchaotic Chen's system with order less than 4. The lowest order we found to have hyperchaos in this system is 3.4. Finally, we study the synchronization problem of two fractional-order hyperchaotic Chen's systems.
We review some of the history and early work in the area of synchronization in chaotic systems. We start with our own discovery of the phenomenon, but go on to establish the historical timeline of this topic back to the earliest known paper. The topic of synchronization of chaotic systems has always been intriguing, since chaotic systems are known to resist synchronization because of their positive Lyapunov exponents. The convergence of the two systems to identical trajectories is a surprise. We show how people originally thought about this process and how the concept of synchronization changed over the years to a more geometric view using synchronization manifolds. We also show that building synchronizing systems leads naturally to engineering more complex systems whose constituents are chaotic, but which can be tuned to output various chaotic signals. We finally end up at a topic that is still in very active exploration today and that is synchronization of dynamical systems in networks of oscillators.
In this paper we present a new simple controller for a chaotic system, that is, the Newton–Leipnik equation with two strange attractors: the upper attractor (UA) and the lower attractor (LA). The controller design is based on the passive technique. The final structure of this controller for original stabilization has a simple nonlinear feedback form. Using a passive method, we prove the stability of a closed-loop system. Based on the controller derived from the passive principle, we investigate three different kinds of chaotic control of the system, separately: the original control forcing the chaotic motion to settle down to the origin from an arbitrary position of the phase space; the chaotic intra-attractor control for stabilizing the equilibrium points only belonging to the upper chaotic attractor or the lower chaotic one, and the inter-attractor control for compelling the chaotic oscillation from one basin to another one. Both theoretical analysis and simulation results verify the validity of the suggested method.
In this paper, synchronization of two hyperchaotic oscillators via a single variable’s unidirectional coupling is studied. First, the synchronizability of the coupled hyperchaotic oscillators is proved mathematically. Then, the convergence speed of this synchronization scheme is analyzed. In order to speed up the response with a relatively large coupling strength, two kinds of chaotic coupling synchronization schemes are proposed. In terms of numerical simulations and the numerical calculation of the largest conditional Lyapunov exponent, it is shown that in a given range of coupling strengths, chaotic-coupling synchronization is quicker than the typical continuous-coupling synchronization. Furthermore, A circuit realization based on the chaotic synchronization scheme is designed and Pspice circuit simulation validates the simulated hyperchaos synchronization mechanism.
Based on the asymptotic stability theory of dynamical systems and matrix theory, a general criterion of synchronization stability of N coupled neurons with symmetric configurations is established in this Letter. Especially, three types of connection styles (that is, chain, ring and global connections) are considered. As an illustration, complete synchronization of four coupled identical chaotic Chay neurons is investigated. The maximal conditional Lyapunov exponent is calculated and used to determine complete synchronization. As a result, complete synchronization of four coupled identical chaotic Chay neurons can be achieved when the coupling strength is above a critical value, which is dependent on the specific connection style. Numerical simulation is in good agreement with the theoretical analysis.
Euler's rigid body equations are modified with the addition of a linear feedback. A system of three quadratic differential equations is obtained which for certain feedback gains has two strange attractors. The attractor for an orbit is determined by the location of the initial point for that orbit.
In this paper, we study a sort of chaotic system—Newton–Leipnik system which possesses two strange attractors. The static and dynamic bifurcations of the system are studied. The chaos controlling is performed by a simpler linear controller, and numerical simulation of the control is supplied. At the same time, Lyapunov exponents of the system show that the result of the chaos controlling is right.
This paper studies the effect of parameter mismatch on the impulsive synchronization of a class of coupled chaotic systems. A new definition for global quasisynchronization is introduced and used to analyze the synchronous behavior of coupled chaotic systems in the presence of parameter mismatch. Using the linear decomposition and comparison-system methods, a global synchronization error bound together with a sufficient condition is derived. Numerical simulations on the chaotic Chua's circuit are presented to verify the theoretical results.
In this paper, a new scheme based on integral observer approach is designed for a class of chaotic systems to achieve synchronization. Unlike the proportional observer approach, the proposed scheme is demonstrated to be effective under a noisy environment in the transmission channel. Based on the Lyapunov stability theory, a sufficient condition for synchronization is derived in the form of a Lyapunov inequality. This Lyapunov inequality is further transformed into a linear matrix inequality (LMI) form by using the Schur theorem and some matrix operation techniques, which can be easily solved by the LMI toolboxes for the design of suitable control gains. It is demonstrated with the Murali-Lakshmanan-Chua system that a better noise suppression and a faster convergence speed can be achieved for chaos synchronization by using this integral observer scheme, as compared with the traditional proportional observer approach.
Dual-wavelength chaos is generated in one erbium-doped fiber laser. Tunable optical filters (TOF) are used to select the wavelengths. The generation of chaos is achieved by either modulating the pump laser diode near the relaxation oscillation frequency of the fiber laser or by introducing the modulation through an electrooptic modulator built into each wavelength branch. The receiver is fabricated with identical parameters as the transmitter except for an open loop structure. By tuning the TOF in the receiver, synchronization of chaos is observed in either of the two wavelengths. Thus, it can pave the way for wavelength-division-multiplexing chaos communications.