Onset And Characterisation Of Chaos In A Two Dimensional Nonlinear Deterministic Model

Nature is full of phenomena which we call "chaotic", the weather being a prime example. What we mean by this is that we cannot predict it to any significant accuracy, either because the system is inherently complex, or because some of the governing factors are not deterministic. However, during recent years it has become clear that random behaviour can occur even in very simple systems with very few number of degrees of freedom, without any need for complexity or indeterminacy. The discovery that chaos can be generated even with the help of systems having completely deterministic rules - often models of natural phenomena - has stimulated a lo; of research interest recently. Not that this chaos has no underlying order, but it is of a subtle kind, that has taken a great deal of ingenuity to unravel. In the present thesis, the author introduce a new nonlinear model, a ‘modulated’ logistic map, and analyse it from the view point of ‘deterministic chaos‘.

STUDIES ON THE STRESS FLUCTUATIONS IN SHEARED STOKESIAN SUSPENSIONS USING CHAOS THEORY AND NONLINEAR DYNAMICS

The thesis report results obtained from a detailed analysis of the fluctuations of the rheological parameters viz. shear and normal stresses, simulated by means of the Stokesian Dynamics method, of a macroscopically homogeneous sheared suspension of neutrally buoyant non-Brownian suspension of identical spheres in the Couette gap between two parallel walls in the limit of vanishingly small Reynolds numbers using the tools of non-linear dynamics and chaos theory for a range of particle concentration and Couette gaps. The thesis used the tools of nonlinear dynamics and chaos theory viz. average mutual information, space-time separation plots, visual recurrence analysis, principal component analysis, false nearest-neighbor technique, correlation integrals, computation of Lyapunov exponents for a range of area fraction of particles and for different Couette gaps. The thesis observed that one stress component can be predicted using another stress component at the same area fraction. This implies a type of synchronization of one stress component with another stress component. This finding suggests us to further analysis of the synchronization of stress components with another stress component at the same or different area fraction of particles. The different model equations of stress components for different area fraction of particles hints at the possible existence a general formula for stress fluctuations with area fraction of particle as a parameter

Nonlinear Dynamics of Nd:YAG lasers: Hopf bifurcation, Multistability and Chaotic Synchronization

In this thesis we have presented some aspects of the nonlinear dynamics of Nd:YAG lasers including synchronization, Hopf bifurcation, chaos control and delay induced multistability.We have chosen diode pumped Nd:YAG laser with intracavity KTP crystal operating with two mode and three mode output as our model system.Different types of orientation for the laser cavity modes were considered to carry out the studies. For laser operating with two mode output we have chosen the modes as having parallel polarization and perpendicular polarization. For laser having three mode output, we have chosen them as two modes polarized parallel to each other while the third mode polarized orthogonal to them.

Nonlinear Dynamics of Multiple Quantum Well Lasers: Chaos and Multistability

This thesis presents analytical and numerical results from studies based on the multiple quantum well laser rate equation model. We address the problem of controlling chaos produced by direct modulation of laser diodes. We consider the delay feedback control methods for this purpose and study their performance using numerical simulation. Besides the control of chaos, control of other nonlinear effects such as quasiperiodicity and bistability using delay feedback methods are also investigated.A number of secure communication schemes based on synchronization of chaos semiconductor lasers have been successfully demonstrated theoretically and experimentally. The current investigations in these field include the study of practical issues on the implementations of such encryption schemes. We theoretically study the issues such as channel delay, phase mismatch and frequency detuning on the synchronization of chaos in directly modulated laser diodes. It would be helpful for designing and implementing chaotic encryption schemes using synchronization of chaos in modulated semiconductor lasers.

Studies on Some Aspects of Light Beam Propagation Through Certain Nonlinear Media

This thesis deals with the study of light beam propagation through different nonlinear media. Analytical and numerical methods are used to show the formation of solitonS in these media. Basic experiments have also been performed to show the formation of a self-written waveguide in a photopolymer. The variational method is used for the analytical analysis throughout the thesis. Numerical method based on the finite-difference forms of the original partial differential equation is used for the numerical analysis.In Chapter 2, we have studied two kinds of solitons, the (2 + 1) D spatial solitons and the (3 + l)D spatio-temporal solitons in a cubic-quintic medium in the presence of multiphoton ionization.In Chapter 3, we have studied the evolution of light beam through a different kind of nonlinear media, the photorcfractive polymer. We study modulational instability and beam propagation through a photorefractive polymer in the presence of absorption losses. The one dimensional beam propagation through the nonlinear medium is studied using variational and numerical methods. Stable soliton propagation is observed both analytically and numerically.Chapter 4 deals with the study of modulational instability in a photorefractive crystal in the presence of wave mixing effects. Modulational instability in a photorefractive medium is studied in the presence of two wave mixing. We then propose and derive a model for forward four wave mixing in the photorefractive medium and investigate the modulational instability induced by four wave mixing effects. By using the standard linear stability analysis the instability gain is obtained.Chapter 5 deals with the study of self-written waveguides. Besides the usual analytical analysis, basic experiments were done showing the formation of self-written waveguide in a photopolymer system. The formation of a directional coupler in a photopolymer system is studied theoretically in Chapter 6. We propose and study, using the variational approximation as well as numerical simulation, the evolution of a probe beam through a directional coupler formed in a photopolymer system.

Development and Evaluation of Blind Identification Techniques for Nonlinear Systems

Identification and Control of Non‐linear dynamical systems are challenging problems to the control engineers.The topic is equally relevant in communication,weather prediction ,bio medical systems and even in social systems,where nonlinearity is an integral part of the system behavior.Most of the real world systems are nonlinear in nature and wide applications are there for nonlinear system identification/modeling.The basic approach in analyzing the nonlinear systems is to build a model from known behavior manifest in the form of system output.The problem of modeling boils down to computing a suitably parameterized model,representing the process.The parameters of the model are adjusted to optimize a performanace function,based on error between the given process output and identified process/model output.While the linear system identification is well established with many classical approaches,most of those methods cannot be directly applied for nonlinear system identification.The problem becomes more complex if the system is completely unknown but only the output time series is available.Blind recognition problem is the direct consequence of such a situation.The thesis concentrates on such problems.Capability of Artificial Neural Networks to approximate many nonlinear input-output maps makes it predominantly suitable for building a function for the identification of nonlinear systems,where only the time series is available.The literature is rich with a variety of algorithms to train the Neural Network model.A comprehensive study of the computation of the model parameters,using the different algorithms and the comparison among them to choose the best technique is still a demanding requirement from practical system designers,which is not available in a concise form in the literature.The thesis is thus an attempt to develop and evaluate some of the well known algorithms and propose some new techniques,in the context of Blind recognition of nonlinear systems.It also attempts to establish the relative merits and demerits of the different approaches.comprehensiveness is achieved in utilizing the benefits of well known evaluation techniques from statistics. The study concludes by providing the results of implementation of the currently available and modified versions and newly introduced techniques for nonlinear blind system modeling followed by a comparison of their performance.It is expected that,such comprehensive study and the comparison process can be of great relevance in many fields including chemical,electrical,biological,financial and weather data analysis.Further the results reported would be of immense help for practical system designers and analysts in selecting the most appropriate method based on the goodness of the model for the particular context.

Studies on nonlinear dynamics of certain reaction systems

The nonlinear dynamics of certain important reaction systems are discussed and analysed in this thesis. The interest in the theoretical and the experimental studies of chemical reactions showing oscillatory dynamics and associated properties is increasing very rapidly. An attempt is made to study some nonlinear phenomena exhibited by the well known chemical oscillator, the BelousovZhabotinskii reaction whose mathematical properties are much in common with the properties of biological oscillators. While extremely complex, this reaction is still much simpler than biological systems at least from the modelling point of view. A suitable model [19] for the system is analysed and the researcher has studied the limit cycle behaviour of the system, for different values of the stoichiometric parameter f, by keeping the value of the reaction rate (k6) fixed at k6 = l. The more complicated three-variable model is stiff in nature.

Nonlinear Analysis of Shear Dominant Prestressed Concrete Beams using ANSYS

This study reports the details of the finite element analysis of eleven shear critical partially prestressed concrete T-beams having steel fibers over partial or full depth. Prestressed T-beams having a shear span to depth ratio of 2.65 and 1.59 that failed in shear have been analyzed using the ‘ANSYS’ program. The ‘ANSYS’ model accounts for the nonlinearity, such as, bond-slip of longitudinal reinforcement, postcracking tensile stiffness of the concrete, stress transfer across the cracked blocks of the concrete and load sustenance through the bridging action of steel fibers at crack interface. The concrete is modeled using ‘SOLID65’- eight-node brick element, which is capable of simulating the cracking and crushing behavior of brittle materials. The reinforcement such as deformed bars, prestressing wires and steel fibers have been modeled discretely using ‘LINK8’ – 3D spar element. The slip between the reinforcement (rebars, fibers) and the concrete has been modeled using a ‘COMBIN39’- nonlinear spring element connecting the nodes of the ‘LINK8’ element representing the reinforcement and nodes of the ‘SOLID65’ elements representing the concrete. The ‘ANSYS’ model correctly predicted the diagonal tension failure and shear compression failure of prestressed concrete beams observed in the experiment. The capability of the model to capture the critical crack regions, loads and deflections for various types of shear failures in prestressed concrete beam has been illustrated.

New studies on the versatile roles of Polyaniline and its composites in polymer based optoelectronic and energy storage devices

From the early stages of the twentieth century, polyaniline (PANI), a well-known and extensively studied conducting polymer has captured the attention of scientific community owing to its interesting electrical and optical properties. Starting from its structural properties, to the currently pursued optical, electrical and electrochemical properties, extensive investigations on pure PANI and its composites are still much relevant to explore its potentialities to the maximum extent. The synthesis of highly crystalline PANI films with ordered structure and high electrical conductivity has not been pursued in depth yet. Recently, nanostructured PANI and the nanocomposites of PANI have attracted a great deal of research attention owing to the possibilities of applications in optical switching devices, optoelectronics and energy storage devices. The work presented in the thesis is centered around the realization of highly conducting and structurally ordered PANI and its composites for applications mainly in the areas of nonlinear optics and electrochemical energy storage. Out of the vast variety of application fields of PANI, these two areas are specifically selected for the present studies, because of the following observations. The non-linear optical properties and the energy storing properties of PANI depend quite sensitively on the extent of conjugation of the polymer structure, the type and concentration of the dopants added and the type and size of the nano particles selected for making the nanocomposites. The first phase of the work is devoted to the synthesis of highly ordered and conducting films of PANI doped with various dopants and the structural, morphological and electrical characterization followed by the synthesis of metal nanoparticles incorporated PANI samples and the detailed optical characterization in the linear and nonlinear regimes. The second phase of the work comprises the investigations on the prospects of PANI in realizing polymer based rechargeable lithium ion cells with the inherent structural flexibility of polymer systems and environmental safety and stability. Secondary battery systems have become an inevitable part of daily life. They can be found in most of the portable electronic gadgets and recently they have started powering automobiles, although the power generated is low. The efficient storage of electrical energy generated from solar cells is achieved by using suitable secondary battery systems. The development of rechargeable battery systems having excellent charge storage capacity, cyclability, environmental friendliness and flexibility has yet to be realized in practice. Rechargeable Li-ion cells employing cathode active materials like LiCoO2, LiMn2O4, LiFePO4 have got remarkable charge storage capacity with least charge leakage when not in use. However, material toxicity, chance of cell explosion and lack of effective cell recycling mechanism pose significant risk factors which are to be addressed seriously. These cells also lack flexibility in their design due to the structural characteristics of the electrode materials. Global research is directed towards identifying new class of electrode materials with less risk factors and better structural stability and flexibility. Polymer based electrode materials with inherent flexibility, stability and eco-friendliness can be a suitable choice. One of the prime drawbacks of polymer based cathode materials is the low electronic conductivity. Hence the real task with this class of materials is to get better electronic conductivity with good electrical storage capability. Electronic conductivity can be enhanced by using proper dopants. In the designing of rechargeable Li-ion cells with polymer based cathode active materials, the key issue is to identify the optimum lithiation of the polymer cathode which can ensure the highest electronic conductivity and specific charge capacity possible The development of conducting polymer based rechargeable Li-ion cells with high specific capacity and excellent cycling characteristics is a highly competitive area among research and development groups, worldwide. Polymer based rechargeable batteries are specifically attractive due to the environmentally benign nature and the possible constructional flexibility they offer. Among polymers having electrical transport properties suitable for rechargeable battery applications, polyaniline is the most favoured one due to its tunable electrical conducting properties and the availability of cost effective precursor materials for its synthesis. The performance of a battery depends significantly on the characteristics of its integral parts, the cathode, anode and the electrolyte, which in turn depend on the materials used. Many research groups are involved in developing new electrode and electrolyte materials to enhance the overall performance efficiency of the battery. Currently explored electrolytes for Li ion battery applications are in liquid or gel form, which makes well-defined sealing essential. The use of solid electrolytes eliminates the need for containment of liquid electrolytes, which will certainly simplify the cell design and improve the safety and durability. The other advantages of polymer electrolytes include dimensional stability, safety and the ability to prevent lithium dendrite formation. One of the ultimate aims of the present work is to realize all solid state, flexible and environment friendly Li-ion cells with high specific capacity and excellent cycling stability. Part of the present work is hence focused on identifying good polymer based solid electrolytes essential for realizing all solid state polymer based Li ion cells.The present work is an attempt to study the versatile roles of polyaniline in two different fields of technological applications like nonlinear optics and energy storage. Conducting form of doped PANI films with good extent of crystallinity have been realized using a level surface assisted casting method in addition to the generally employed technique of spin coating. Metal nanoparticles embedded PANI offers a rich source for nonlinear optical studies and hence gold and silver nanoparticles have been used for making the nanocomposites in bulk and thin film forms. These PANI nanocomposites are found to exhibit quite dominant third order optical non-linearity. The highlight of these studies is the observation of the interesting phenomenon of the switching between saturable absorption (SA) and reverse saturable absorption (RSA) in the films of Ag/PANI and Au/PANI nanocomposites, which offers prospects of applications in optical switching. The investigations on the energy storage prospects of PANI were carried out on Li enriched PANI which was used as the cathode active material for assembling rechargeable Li-ion cells. For Li enrichment or Li doping of PANI, n-Butyllithium (n-BuLi) in hexanes was used. The Li doping as well as the Li-ion cell assembling were carried out in an argon filled glove box. Coin cells were assembled with Li doped PANI with different doping concentrations, as the cathode, LiPF6 as the electrolyte and Li metal as the anode. These coin cells are found to show reasonably good specific capacity around 22mAh/g and excellent cycling stability and coulombic efficiency around 99%. To improve the specific capacity, composites of Li doped PANI with inorganic cathode active materials like LiFePO4 and LiMn2O4 were synthesized and coin cells were assembled as mentioned earlier to assess the electrochemical capability. The cells assembled using the composite cathodes are found to show significant enhancement in specific capacity to around 40mAh/g. One of the other interesting observations is the complete blocking of the adverse effects of Jahn-Teller distortion, when the composite cathode, PANI-LiMn2O4 is used for assembling the Li-ion cells. This distortion is generally observed, near room temperature, when LiMn2O4 is used as the cathode, which significantly reduces the cycling stability of the cells.