Narrow Results

In this paper, we study the synchronization of three Piecewise Linear (PWL) Rössler systems that exhibit coexistence of two attractors, coupled in linear configuration. In particular, we analyze the four linear configurations of the network motifs. We analyze the interaction of the systems by varying the unidirectional or bidirectional coupling strengths depending on the case under study, in order to characterize the conditions under which complete synchronization between the three PWL Rössler systems are obtained. We have found, by means of the error functions, the range of values of the coupling strengths where the different types of synchronization occurs. The numerical results show that the interaction of various multistable chaotic systems causes different types of synchronization and very complex dynamics between the systems.

A study on the master-slave synchronization scheme between Rayleigh–Duffing and Duffing oscillators is presented. We analyze the elastic and dissipative couplings and a combination of both. We compare the results to explore which coupling is more effective to achieve synchronization between both oscillators. The numerical results demonstrate that for the elastic or dissipative coupling at best there is complete synchronization in only one state of the slave system. However, it was also observed that depending on which oscillator acts as the master system and the coupling used, there may be partial or complete synchronization for large values of the coupling strength. When the combination of both couplings is used, there always exists complete synchronization for the two states of the slave system.

In this work, we study the synchronization in the network motifs of three piecewise Rössler systems that exhibit coexistence of two attractors, coupled in ring configuration, in particular a bidirectional auxiliary coupling. For this configuration, we analyze the interaction of the systems by varying the coupling strengths, in order to characterize the conditions under which complete synchronization between the three systems can be achieved. The numerical results prove that the interaction of various multistable chaotic systems causes different types of synchronization and very complex dynamics between the systems is achieved. We obtain complete synchronization between the three systems and find the range of values of the coupling strengths where the distinct types of synchronization occur.

We study numerically a system of two lasers cross-coupled optoelectronically with a time delay where the output intensity of each laser modulates the pump current of the other laser. We demonstrate control of chaos via variable coupling time delay by converting the laser intensity chaos to the steady-state. We also show that wavelength chaos in an electrically tunable distributed Bragg reflector (DBR) laser diode with a feedback loop that can be controlled via variable feedback time delay.

The famous chaotic circuit known as Chua's oscillator, despite the robustness of the chaotic behavior to parametric mismatches, requires the construction of noncommercial valued inductor. A low cost inductorless version of Chua's oscillator is presented. The chaotic behavior of the circuit is verified by PSpice simulation and also by experimental study on a circuit breadboard. The results lead to excellent agreement with each other and with the results of previous investigators. Experimental results on the possibility of controlling chaos in the modified Chua's oscillator by the inherent feedback mechanism are also reported.

In this paper, subharmonic and chaotic behavior in current controlled DC drive has been investigated. The effects of variation of some chosen parameters on the qualitative behavior of the system have been studied. To avoid occurrence of chaos we present a method for controlling chaos applicable to DC drives. The results of numerical investigation are presented.

The energy-optimal migration of a chaotic oscillator from one attractor to another coexisting attractor is investigated via an analogy between the Hamiltonian theory of fluctuations and Hamiltonian formulation of the control problem. We demonstrate both on physical grounds and rigorously that the Wentzel–Freidlin Hamiltonian arising in the analysis of fluctuations is equivalent to Pontryagin's Hamiltonian in the control problem with an additive linear unrestricted control. The deterministic optimal control function is identified with the optimal fluctuational force. Numerical and analogue experiments undertaken to verify these ideas demonstrate that, in the limit of small noise intensity, fluctuational escape from the chaotic attractor occurs via a unique (optimal) path corresponding to a unique (optimal) fluctuational force. Initial conditions on the chaotic attractor are identified. The solution of the boundary value control problem for the Pontryagin Hamiltonian is found numerically. It is shown that this solution is approximated very accurately by the optimal fluctuational force found using statistical analysis of the escape trajectories. A second series of numerical experiments on the deterministic system (i.e. in the absence of noise) show that a control function of precisely the same shape and magnitude is indeed able to instigate escape. It is demonstrated that this control function minimizes the cost functional and the corresponding energy is found to be smaller than that obtained with some earlier adaptive control algorithms.

This paper presents a new control approach for steering trajectories of three-dimensional nonlinear chaotic systems towards stable stationary states or time-periodic orbits. The proposed method mainly consists in a sliding mode-based control design that is extended by an explicit consideration of system energy as basis for both controller design and system stabilization. The control objective is then to regulate the energy with respect to a shaped nominal representation implicitly related to system trajectories. In this paper, we establish some theoretical results to introduce the control design approach referred to as Energy based Sliding Mode Control (ESMC for short). Then, some capabilities of the proposed approach are illustrated through examples related to the chaotic circuit of Chua.

This paper is meant to serve as a tutorial describing the link between symbolic dynamics as a description of a chaotic attractor, and how to use control of chaos to manipulate the corresponding symbolic dynamics to transmit an information bearing signal. We use the Lorenz attractor, in the form of the discrete successive maxima map of the z-variable time-series, as our main example. For the first time, here, we use this oscillator as a chaotic signal carrier. We review the many previously developed issues necessary to create a working control of symbol dynamics system. These include a brief review of the theory of symbol dynamics, and how they arise from the flow of a differential equation. We also discuss the role of the (symbol dynamics) generating partition, the difficulty of finding such partitions, which is an open problem for most dynamical systems, and a newly developed algorithm to find the generating partition which relies just on knowing a large set of periodic orbits. We also discuss the importance of using a generating partition in terms of considering the possibility of using some other arbitrary partition, with discussion of consequences both generally to characterizing the system, and also specifically to communicating on chaotic signal carriers. Also, of practical importance, we review the necessary feedback-control issues to force the flow of a chaotic differential equation to carry a desired message.

This paper deals with the control of a class of n-dimensional chaotic systems. The proposed method consists in a Variable Structure Control approach based on system energy consideration for both controller design and system stabilization. First, we present some theoretical results related to the stabilization of global invariant sets included in a selected two-dimensional subspace of the state space. Then, we define some conditions, involving both system definition and control law design, under which the stabilized orbits can be maintained in a neighborhood of an invariant, nondegenerate, closed conic section (i.e. an ellipse or a circle). Finally, an example related to the chaotic circuit of Chua is given.

We study control of chaos by periodic modulation without feedback in discrete maps. We show that period-p forcing shifts and splits the naturally occurring period-p windows (and their multiples) and leads to the stabilization of chaos, and that for multiplicative sinusoidal perturbations of the logistic map stabilization can be done at a desired value of the control parameter and at a desired periodicity, simply by varying the amplitude of forcing, provided the modulated system remains bounded.

A technique of using nonlinear approximations to control chaotic dynamical systems is extended so it can be used to control such systems when only data generated can be observed.

We suggest a link between coding theory (iterative algorithms) and chaos theory. A whole range of phenomena known to occur in nonlinear systems, like the existence of multiple fixed points, oscillatory behavior, bifurcations, chaos and transient chaos are found in turbo-decoding algorithms. We develop a simple technique to control transient chaos in turbo-decoding algorithms and improve the performance of the classical turbo codes.

This letter presents an experimental confirmation of controlling the chaotic behavior of a target unstable periodic orbit when the periodically switched nonlinear circuit has a chaotic attractor. The pole assignment for the corresponding discrete system derived from such a nonautonomous system via Poincaré mapping works effectively, and the control unit is easily realized by the window comparator, sample-hold circuits, and so on.

A method for controlling onto saddle-type fixed points developed by Yagasaki and Uozumi is extended so as to make the capture region many times larger than that of the original method.

In this paper, a chaotic switched arrival system with N buffers and controlled internal connections is considered. An unstable N periodic orbit embedded in the chaotic attractor is theoretically derived. A novel method for the stabilization of the unstable N periodic orbit using the flow rates of the internal connections as control variables is presented. It is showed that the switched arrival system can be stabilized by using N - 2 internal connections. Numerical simulation results are also provided to demonstrate the theoretical analysis.

Duffing-type electrical oscillator is a second-order nonlinear electric circuit driven by a sinusoidal voltage source. The nonlinear element is a nonlinear inductor. We have studied the dynamics of two resistively coupled oscillators of this type in two cases. The first, when the oscillators are identical having chaotic dynamics, and the second, when the oscillators are in different dynamic states (periodic and chaotic, respectively). In the first case, chaotic synchronization is observed, while in the second case control of the chaotic behavior is achieved.

We consider a one-dimensional chain of identical sites, appropriate for colonization by a biological species. The dynamics at each site is subjected to the demographic Allee effect. We consider nonzero probability p of dispersal to the nearby sites and we prove, for small values of p, the existence of asymptotically stable time-periodic and space-localized solutions, such that the central site carries the vast majority of the metapopulation, while the populations at nearby sites attain very small values. We study numerically a chain of three sites, both for the case of open ends or periodic boundary conditions. We study the bifurcations leading to transition from chaotic to periodic behavior and vice-versa and note that the increase of the dispersal probability in both cases controls the chaotic behavior of the metapopulation.

An optimal chaos control procedure is proposed. The aim of using this method is to stabilize the chaotic behavior of forced continuous-time nonlinear systems by using an approximation sequence technique and linear optimal control. The idea of the approximation technique is to use a sequence of linear, time-varying equations to approximate the solution of nonlinear systems. In each of these equations, the linear-quadratic optimal tracking control is applied. The purpose is to find a linear time-varying feedback controller which produces an optimized trajectory that converges to a desired signal. This controller is then used in the original nonlinear system. As an example, the procedure is proved to work in the Duffing oscillator and the chaotic pendulum, in which the goal is to direct chaotic trajectories of these systems to a period-n orbit.

A method of greatly decreasing the activation time of a control method based on stable manifold information is proposed.