Pulsive feedback control for stabilizing unstable periodic orbits in a nonlinear oscillator with a non-symmetric potential

We examine a strange chaotic attractor and its unstable periodic orbits in case of one degree of freedom nonlinear oscillator with non symmetric potential. We propose an efficient method of chaos control stabilizing these orbits by a pulsive feedback technique. Discrete set of pulses enable us to transfer the system from one periodic state to another.Comment: 11 pages, 4 figure

Bifurcations, Chaos, Controlling and Synchronization of Certain Nonlinear Oscillators

In this set of lectures, we review briefly some of the recent developments in the study of the chaotic dynamics of nonlinear oscillators, particularly of damped and driven type. By taking a representative set of examples such as the Duffing, Bonhoeffer-van der Pol and MLC circuit oscillators, we briefly explain the various bifurcations and chaos phenomena associated with these systems. We use numerical and analytical as well as analogue simulation methods to study these systems. Then we point out how controlling of chaotic motions can be effected by algorithmic procedures requiring minimal perturbations. Finally we briefly discuss how synchronization of identically evolving chaotic systems can be achieved and how they can be used in secure communications.Comment: 31 pages (24 figures) LaTeX. To appear Springer Lecture Notes in Physics Please Lakshmanan for figures (e-mail: [email protected]

Revisiting linear augmentation for stabilizing stationary solutions: potential pitfalls and their application

Linear augmentation has recently been shown to be effective in targeting desired stationary solutions, suppressing bistablity, in regulating the dynamics of drive response systems and in controlling the dynamics of hidden attractors. The simplicity of the procedure is the highlight of this scheme but at the same time questions related to its general applicability still need to be addressed. Focusing on the issue of targeting stationary solutions, this work demonstrates instances where the scheme fails to stabilize the required solutions and leads to other complicated dynamical scenarios. Appropriate examples from conservative as well as dissipative systems are presented in this regard and potential applications for relevant observations in dissipative predator--prey systems are also discussed.Comment: updated version with title change, additional figures, text and explanation

Amplitude Death: The emergence of stationarity in coupled nonlinear systems

When nonlinear dynamical systems are coupled, depending on the intrinsic dynamics and the manner in which the coupling is organized, a host of novel phenomena can arise. In this context, an important emergent phenomenon is the complete suppression of oscillations, formally termed amplitude death (AD). Oscillations of the entire system cease as a consequence of the interaction, leading to stationary behavior. The fixed points that the coupling stabilizes can be the otherwise unstable fixed points of the uncoupled system or can correspond to novel stationary points. Such behaviour is of relevance in areas ranging from laser physics to the dynamics of biological systems. In this review we discuss the characteristics of the different coupling strategies and scenarios that lead to AD in a variety of different situations, and draw attention to several open issues and challenging problems for further study.Comment: Physics Reports (2012

Feedback Control of Traveling Wave Solutions of the Complex Ginzburg Landau Equation

Through a linear stability analysis, we investigate the effectiveness of a noninvasive feedback control scheme aimed at stabilizing traveling wave solutions of the one-dimensional complex Ginzburg Landau equation (CGLE) in the Benjamin-Feir unstable regime. The feedback control is a generalization of the time-delay method of Pyragas, which was proposed by Lu, Yu and Harrison in the setting of nonlinear optics. It involves both spatial shifts, by the wavelength of the targeted traveling wave, and a time delay that coincides with the temporal period of the traveling wave. We derive a single necessary and sufficient stability criterion which determines whether a traveling wave is stable to all perturbation wavenumbers. This criterion has the benefit that it determines an optimal value for the time-delay feedback parameter. For various coefficients in the CGLE we use this algebraic stability criterion to numerically determine stable regions in the (K,rho) parameter plane, where rho is the feedback parameter associated with the spatial translation and K is the wavenumber of the traveling wave. We find that the combination of the two feedbacks greatly enlarges the parameter regime where stabilization is possible, and that the stability regions take the form of stability tongues in the (K,rho)--plane. We discuss possible resonance mechanisms that could account for the spacing with K of the stability tongues.Comment: 33 pages, 12 figure

Exact Floquet states of a driven condensate and their stabilities

We investigate the Gross-Pitaevskii equation for a classically chaotic system, which describes an atomic Bose-Einstein condensate confined in an optical lattice and driven by a spatiotemporal periodic laser field. It is demonstrated that the exact Floquet states appear when the external time-dependent potential is balanced by the nonlinear mean-field interaction. The balance region of parameters is divided into a phase-continuing region and a phase-jumping one. In the latter region, the Floquet states are spatiotemporal vortices of nontrivial phase structures and zero-density cores. Due to the velocity singularities of vortex cores and the blowing-up of perturbed solutions, the spatiotemporal vortices are unstable periodic states embedded in chaos. The stability and instability of these Floquet states are numerically explored by the time evolution of fidelity between the exact and numerical solutions. It is numerically illustrated that the stable Floquet states could be prepared from the uniformly initial states by slow growth of the external potential.Comment: 14 pages, 3 eps figures, final version accepted for publication in J. Phys.