Papri Saha

On the application of adaptive control and phase synchronization in non-linear fluid dynamics

In this paper we study certain aspects of non-linear dynamical systems arising in atmospheric physics from the viewpoint of their synchronization. Apart from phase synchronization, a new method using an adaptive controller is formulated. The model under investigation was originally obtained by Stenflo (Phys. Scr. 53 (1996) 83) in case of acoustic gravity waves and its chaotic properties were studied by Liu (Phys. Scr. 61 (2000) 526), Zhou (J. Math. Phys. 38 (1997) 5225) and Banerjee et al. (Phys. Scripta 63 (2001) 177). Here, we have shown how an adaptive controller can be constructed for the synchronization of two such systems by analysing the error equation and utilizing the properties of the corresponding Hurwitz matrix. As a result, suitable criteria are obtained for synchronization. On the other hand, the phase of such a chaotic time series is defined from the view of Hilbert transform and is shown that the technique of phase synchronization can be equally applied to such systems.

Some aspects of synchronization and chaos in a coupled laser system

Abstract Some dynamical properties and synchronization are studied in a coupled system of solid state lasers without external modulation. It is observed that two different situations may arise – when the individual systems are operating in the stable region and when both of them are chaotic. In both the situations the coupling leads to a chaotic scenario. These two events are analysed with respect to the coupling parameter. The synchronization of two lasers is studied by using the notion of parameter variation, drive decomposable system and adaptive control mechanism.

Chaotic aspects of lasers with host-induced nonlinearity and its control

Abstract Recently, absolute instabilities were analysed in lasers that contain a dispersive host material with cubic nonlinearities, which leads to self-phase modulation (SPM) and intensity-dependent absorption (IDA). Nonlinear equations governing the operation of these new kind of lasers are known to possess certain new kind of features not described by the usual Lorenz–Haken dynamics. These nonlinear equations were derived from the Maxwell–Bloch equations and the stability equations were investigated by performing a linear stability analysis by Tartwijk and Agarwal. We try to explore the whole nonlinear region of the system which shows a period doubling route to chaos. The chaotic region is described through phase space diagrams, Lyapunov exponents, temporal variation of predictability and local divergence behaviour. Different aspects of the chaotic attractor reconstructed by the method of time delays, proposed by Takens, is discussed in the light of recurrence plot analysis leading to explicit information about embedding dimension and entropy of line distribution. We implement adaptive and threshold control mechanisms to revert back to a stable configuration of the system.

On the study of control and anti-control in magnetoconvection

Abstract Instability and the onset of chaos is analyzed in the phenomenon of magnetoconvection with the help of phase space analysis, Lyapunov exponents and temporal variation of predictability. In contrast to the previous analysis in the literature, we have explored the period doubling route to chaos, which is shown to be equally effective as Hopf bifurcation. Initially the physical system of magnetoconvection with positive Chandrasekhar number is analyzed and next a prototype of a new dynamical system given for q

Normal form analysis and chaotic scenario in a Schrödinger–Boussinesq system

Abstract The coupled Schrodinger–Boussinesq system is known to describe various physical processes in plasma. On the one hand it describes high-frequency upper-hybrid waves coupled to appropriate low-frequency ones such as magnetosonic modes and on the other hand it also describes the Langmuir oscillations in plasma. The system has been seen to be integrable for very specialised values of the parameters. Here in this communication we have analysed the same set near the resonant region M →1 (where M stands for the Mach number) in a different way. A parameter p = eg 2 / σ has been identified, whose variation exhibits the change in the nature of the fixed point and defines the critical region best suited for normal form analysis. This reduces the four-dimensional system to a two-dimensional one. The nature of this reduced system is analysed graphically. In a certain region of p the system showed chaoticity and it was studied through Lyapunov exponent, Poincare section and phase space analysis.

Control of chaos in laser plasma interaction

Abstract The chaotic dynamics originating from the equation governing the laser plasma interaction is studied. Our motivation is to show that it is possible to control this chaotic scenario either to a periodic state or to a totally steady state by adopting two different modes of control – one is the sinusoidal time variation of one parameter of the system and the other is the proportional pulse approach. Extensive use is made of Poincare section, power spectrum analysis and phase space plot to prove the assertions. The observations can be of practical use in the simulation of plasma experiments.

Optically injected laser system: characterization of chaos, bifurcation, and control.

A single mode semiconductor laser subjected to optical injection, described by a set of three coupled nonlinear ordinary differential equations, exhibiting chaos is considered. By means of a recurrence analysis, quantification of the strange attractor is made. Analytical studies of the system using asymptotic averaging technique, derive certain conditions describing the prediction of 1-->2 bifurcation, which have subsequently been verified on numerical simulation. Furthermore, the locus of points on the parameter phase space representing Hopf bifurcation has been derived. The problem of control of chaos by a new procedure based on adaptive stabilization is also addressed. The results of such control are shown explicitly. Though this analysis deals with a very specific set of equations, the overall features that come out of the study remains valid for almost all laser systems.

Bifurcation and chaos in a system simulating magnetic field configurations

Abstract In the present article, the behaviour of a nonlinear dynamical system has been analysed using the approach of bifurcation theory. The system is important due to the fact that it can simulate the magnetic field configurations in various situations. The nature of bifurcation has been explored in the parameter space with the help of continuation algorithm. The various limit and bifurcation points (BPs) are classified. In the second part, we have studied the temporal evolution of the system which also shows a chaotic behaviour. The system under consideration shows instability both with respect to parameter variation and evolution of time. Lastly, some mechanisms have been studied to control such chaotic scenario.

ON THE STUDY OF DELAY FEEDBACK CONTROL AND ADAPTIVE SYNCHRONIZATION NEAR SUB-CRITICAL HOPF BIFURCATION

The effect of delay feedback control and adaptive synchronization is studied near sub-critical Hopf bifurcation of a nonlinear dynamical system. Previously, these methods targeted the nonlinear systems near their chaotic regime but it is shown here that they are equally applicable near the branch of unstable solutions. The system is first analyzed from the view point of bifurcation, and the existence of Hopf bifurcation is established through normal form analysis. Hopf bifurcation can be either sub-critical or super-critical, and in the former case, unstable periodic orbits are formed. Our aim is to control them through a delay feedback approach so that the system stabilizes to its nearest stable periodic orbit. At the vicinity of the sub-critical Hopf point, adaptive synchronization is studied and the effect of the coupling parameter and the speed factor is analyzed in detail. Adaptive synchronization is also studied when the system is in the chaotic regime.

Modified projective synchronisation of different order chaotic systems with adaptive scaling factor

In this paper, modified projective synchronisation between two nonlinear systems with dissimilar order is studied. A new adaptive controller is designed with the help of Lyapunov stability theory. It is observed that the different order systems synchronise and the method is quite effective when the difference in order is one and two. The numerical results are supported by analytical calculations. It is also seen that the method is equally applicable to variable scaling factor and the scaling factor can be estimated from the process of functional projective synchronisation.