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‘.

Investigations on Nonlinear and Radiative Properties of Certain Phatonic Materials

Light in its physical and philosophical sense has captured the imagination of human mind right from the dawn of civilization. The invention of lasers in the 60’s caused a renaissance in the field of optics. This intense, monochromatic, highly directional radiation created new frontiers in science and technology. The strong oscillating electric field of laser radiation creates a. polarisation response that is nonlinear in character in the medium through which it passes and the medium acts as a new source of optical field with alternate properties. It was in this context, that the field of optoelectronics which encompasses the generation, modulation, transmission etc. of optical radiation has gained tremendous importance. Organic molecules and polymeric systems have emerged as a class of promising materials of optoelectronics because they offer the flexibility, both at the molecular and bulk levels, to optimize the nonlinearity and other suitable properties for device applications. Organic nonlinear optical media, which yield large third-order nonlinearities, have been widely studied to develop optical devices like high speed switches, optical limiters etc. Transparent polymeric materials have found one of their most promising applicationsin lasers, in which they can be used as active elements with suitable laser dyes doped in it. The solid-matrix dye lasers make possible combination of the advantages of solid state lasers with the possibility of tuning the radiation over a broad spectral range. The polymeric matrices impregnated with organic dyes have not yet widely used because of the low resistance of the polymeric matrices to laser damage, their low dye photostability, and low dye stability over longer time of operation and storage. In this thesis we investigate the nonlinear and radiative properties of certain organic materials and doped polymeric matrix and their possible role in device development

Preparation and Study of Optical Properties of CdS, CdS-TiO2 and CdS-Au Nanocomposites for Photonic Applications

Nanophotonics can be regarded as a fusion of nanotechnology and photonics and it is an emerging field providing researchers opportunities in fundamental science and new technologies. In recent times many new methodsand techniques have been developed to prepare materials at nanoscale dimensions. Most of these materials exhibit unique and interesting optical properties and behavior. Many of these have been found to be very useful to develop new devices and systems such as tracers in biological systems, optical limiters, light emitters and energy harvesters. This thesis presents a summary of the work done by the author in the field by choosing a few semiconductor systems to prepare nanomaterials and nanocomposites. Results of the study of linear and nonlinear optical properties of materials thus synthesized are also presented in the various chapters of this thesis. CdS is the material chosen here and the methods and the studies of the detailed investigation are presented in this thesis related to the optical properties of CdS nanoparticles and its composites. Preparation and characterization methods and experimental techniques adopted for the investigations were illustrated in chapter 2 of this thesis. Chapter 3 discusses the preparation of CdS, TiO2 and Au nanoparticles. We observed that the fluorescence behaviour of the CdS nanoparticles, prepared by precipitation technique, depends on excitation wavelength. It was found that the peak emission wavelength can be shifted by as much as 147nm by varyingthe excitation wavelengths and the reason for this phenomenon is the selective excitation of the surface states in the nanoparticles. This provided certain amount of tunability for the emission which results from surface states.TiO2 nanoparticle colloids were prepared by hydrothermal method. The optical absorption study showed a blue shift of absorption edge, indicating quantum confinement effect. The large spectral range investigated allows observing simultaneously direct and indirect band gap optical recombination. The emission studies carried out show four peaks, which are found to be generated from excitonic as well as surface state transitions. It was found that the emission wavelengths of these colloidal nanoparticles and annealed nanoparticles showed two category of surface state emission in addition to the excitonic emission. Au nanoparticles prepared by Turkevich method showed nanoparticles of size below 5nm using plasmonic absorption calculation. It was also found that there was almost no variation in size as the concentration of precursor was changed from 0.2mM to 0.4mM.We have observed SHG from CdS nanostructured thin film prepared onglass substrate by chemical bath deposition technique. The results point out that studied sample has in-plane isotropy. The relative values of tensor components of the second-order susceptibility were determined to be 1, zzz 0.14, xxz and 0.07. zxx These values suggest that the nanocrystals are oriented along the normal direction. However, the origin of such orientation remains unknown at present. Thus CdS is a promising nonlinear optical material for photonic applications, particularly for integrated photonic devices. CdS Au nanocomposite particles were prepared by mixing CdS nanoparticles with Au colloidal nanoparticles. Optical absorption study of these nanoparticles in PVA solution suggests that absorption tail was red shifted compared to CdS nanoparticles. TEM and EDS analysis suggested that the amount of Au nanoparticles present on CdS nanoparticles is very small. Fluorescence emission is unaffected indicating the presence of low level of Au nanoparticles. CdS:Au PVA and CdS PVA nanocomposite films were fabricated and optically characterized. The results showed a red-shift for CdS:Au PVA film for absorption tail compared to CdS PVA film. Nonlinear optical analysis showed a huge nonlinear optical absorption for CdS:Au PVA nanocomposite and CdS:PVA films. Also an enhancement in nonlinear optical absorption is found for CdS:Au PVA thin film compared to the CdS PVA thin film. This enhancement is due to the combined effect of plasmonic as well as excitonic contribution at high input intensity. Samples of CdS doped with TiO2 were also prepared and the linear optical absorption spectra of these nanocompositeparticles clearly indicated the influence of TiO2 nanoparticles. TEM and EDS studies have confirmed the presence of TiO2 on CdS nanoparticles. Fluorescence studies showed that there is an increase in emission peak around 532nm for CdS nanoparticles. Nonlinear optical analysis of CdS:TiO2 PVA nanocomposite films indicated a large nonlinear optical absorption compared to that of CdS:PVA nanocomposite film. The values of nonlinear optical absorption suggests that these nanocomposite particles can be employed for optical limiting applications. CdSe-CdS and CdSe-ZnS core-shell QDs with varying shell size were characterized using UV–VIS spectroscopy. Optical absorption and TEM analysis of these QDs suggested a particle size around 5 nm. It is clearly shown that the surface coating influences the optical properties of QDs in terms of their size. Fluorescence studies reveal the presence of trap states in CdSe-CdS and CdSe- ZnS QDs. Trap states showed an increase as a shell for CdS is introduced and increasing the shell size of CdS beyond a certain value leads to a decrease in the trap state emission. There is no sizeable nonlinear optical absorption observed. In the case of CdSe- ZnS QDs, the trap state emission gets enhanced with the increase in ZnS shell thickness. The enhancement of emission from trap states transition due to the increase in thickness of ZnS shell gives a clear indication of distortion occurring in the spherical symmetry of CdSe quantum dots. Consequently the nonlinear optical absorption of CdSe-ZnS QDs gets increased and the optical limiting threshold is decreased as the shell thickness is increased in respect of CdSe QDs. In comparison with CdSe-CdS QDs, CdSe-ZnS QDs possess much better optical properties and thereby CdSe-ZnS is a strong candidate for nonlinear as well as linear optical applications.

Interchannel interactions and relaxation in the 2p auger spectra of Mg-like ions

Relativistic Auger rates for the 2p spectra of Mg-like ions have been calculated in the atomic range 13 < Z < 36. We used the multiconfiguration Dirac-Fock method but beyond a simple frozen-orbital approach we include also relaxation for the bound electrons and the interchannel interaction between the continuum states. Both effects may alter the individual transition rates remarkably. This is analysed for a few selected states within the isoelectronic sequence. Weak transitions within the 2p spectra can be changed by an order of magnitude because of the continuum coupling. The influence of both effects for higher-Z ions is reduced but still remain visible.

Analytical Study of Light Propagation in Highly Nonlinear Media

The present dissertation is devoted to the construction of exact and approximate analytical solutions of the problem of light propagation in highly nonlinear media. It is demonstrated that for many experimental conditions, the problem can be studied under the geometrical optics approximation with a sufficient accuracy. Based on the renormalization group symmetry analysis, exact analytical solutions of the eikonal equations with a higher order refractive index are constructed. A new analytical approach to the construction of approximate solutions is suggested. Based on it, approximate solutions for various boundary conditions, nonlinear refractive indices and dimensions are constructed. Exact analytical expressions for the nonlinear self-focusing positions are deduced. On the basis of the obtained solutions a general rule for the single filament intensity is derived; it is demonstrated that the scaling law (the functional dependence of the self-focusing position on the peak beam intensity) is defined by a form of the nonlinear refractive index but not the beam shape at the boundary. Comparisons of the obtained solutions with results of experiments and numerical simulations are discussed.

Constraint satisfaction techniques under uncertain conditions for fault diagnosis in nonlinear dynamic systems

The speed of fault isolation is crucial for the design and reconfiguration of fault tolerant control (FTC). In this paper the fault isolation problem is stated as a constraint satisfaction problem (CSP) and solved using constraint propagation techniques. The proposed method is based on constraint satisfaction techniques and uncertainty space refining of interval parameters. In comparison with other approaches based on adaptive observers, the major advantage of the presented method is that the isolation speed is fast even taking into account uncertainty in parameters, measurements and model errors and without the monotonicity assumption. In order to illustrate the proposed approach, a case study of a nonlinear dynamic system is presented

Nuclear relaxation contribution to static and dynamic (infinite frequency approximation) nonlinear optical properties by means of electrical property expansions: Application to HF, CH4, CF4, and SF6

Electrical property derivative expressions are presented for the nuclear relaxation contribution to static and dynamic (infinite frequency approximation) nonlinear optical properties. For CF4 and SF6, as opposed to HF and CH4, a term that is quadratic in the vibrational anharmonicity (and not previously evaluated for any molecule) makes an important contribution to the static second vibrational hyperpolarizability of CF4 and SF6. A comparison between calculated and experimental values for the difference between the (anisotropic) Kerr effect and electric field induced second-harmonic generation shows that, at the Hartree-Fock level, the nuclear relaxation/infinite frequency approximation gives the correct trend (in the series CH4, CF4, SF6) but is of the order of 50% too small

Initial convergence of the perturbation series expansion for vibrational nonlinear optical properties

Initial convergence of the perturbation series expansion for vibrational nonlinear optical (NLO) properties was analyzed. The zero-point vibrational average (ZPVA) was obtained through first-order in mechanical plus electrical anharmonicity. Results indicated that higher-order terms in electrical and mechanical anharmonicity can make substantial contributions to the pure vibrational polarizibility of typical NLO molecules

Approximate Gauss-Newton methods for nonlinear least squares problems

The Gauss–Newton algorithm is an iterative method regularly used for solving nonlinear least squares problems. It is particularly well suited to the treatment of very large scale variational data assimilation problems that arise in atmosphere and ocean forecasting. The procedure consists of a sequence of linear least squares approximations to the nonlinear problem, each of which is solved by an “inner” direct or iterative process. In comparison with Newton’s method and its variants, the algorithm is attractive because it does not require the evaluation of second-order derivatives in the Hessian of the objective function. In practice the exact Gauss–Newton method is too expensive to apply operationally in meteorological forecasting, and various approximations are made in order to reduce computational costs and to solve the problems in real time. Here we investigate the effects on the convergence of the Gauss–Newton method of two types of approximation used commonly in data assimilation. First, we examine “truncated” Gauss–Newton methods where the inner linear least squares problem is not solved exactly, and second, we examine “perturbed” Gauss–Newton methods where the true linearized inner problem is approximated by a simplified, or perturbed, linear least squares problem. We give conditions ensuring that the truncated and perturbed Gauss–Newton methods converge and also derive rates of convergence for the iterations. The results are illustrated by a simple numerical example. A practical application to the problem of data assimilation in a typical meteorological system is presented.

Experimental and theoretical study of the infrared spectra of BrHI- and BrDI-

Gas phase vibrational spectra of BrHI- and BrDI- have been measured from 6 to 17 mum (590-1666 cm-1) using tunable infrared radiation from the free electron laser for infrared experiments in order to characterize the strong hydrogen bond in these species. BrHI-.Ar and BrDI-.Ar complexes were produced and mass selected, and the depletion of their signal due to vibrational predissociation was monitored as a function of photon energy. Additionally, BrHI- and BrDI- were dissociated into HBr (DBr) and I- via resonant infrared multiphoton dissociation. The spectra show numerous transitions, which had not been observed by previous matrix studies. New ab initio calculations of the potential-energy surface and the dipole moment are presented and are used in variational ro-vibrational calculations to assign the spectral features. These calculations highlight the importance of basis set in the simulation of heavy atoms such as iodine. Further, they demonstrate extensive mode mixing between the bend and the H-atom stretch modes in BrHI- and BrDI- due to Fermi resonances. These interactions result in major deviations from simple harmonic estimates of the vibrational energies. As a result of this new analysis, previous matrix-isolation spectra assignments are reevaluated. (C) 2004 American Institute of Physics.

Error estimates for a fully discrete spectral scheme for a class of nonlinear, nonlocal dispersive wave equations

We analyze a fully discrete spectral method for the numerical solution of the initial- and periodic boundary-value problem for two nonlinear, nonlocal, dispersive wave equations, the Benjamin–Ono and the Intermediate Long Wave equations. The equations are discretized in space by the standard Fourier–Galerkin spectral method and in time by the explicit leap-frog scheme. For the resulting fully discrete, conditionally stable scheme we prove an L2-error bound of spectral accuracy in space and of second-order accuracy in time.

On the spectra of certain integro-differential-delay problems with applications in neurodynamics

We investigate the spectrum of certain integro-differential-delay equations (IDDEs) which arise naturally within spatially distributed, nonlocal, pattern formation problems. Our approach is based on the reformulation of the relevant dispersion relations with the use of the Lambert function. As a particular application of this approach, we consider the case of the Amari delay neural field equation which describes the local activity of a population of neurons taking into consideration the finite propagation speed of the electric signal. We show that if the kernel appearing in this equation is symmetric around some point a= 0 or consists of a sum of such terms, then the relevant dispersion relation yields spectra with an infinite number of branches, as opposed to finite sets of eigenvalues considered in previous works. Also, in earlier works the focus has been on the most rightward part of the spectrum and the possibility of an instability driven pattern formation. Here, we numerically survey the structure of the entire spectra and argue that a detailed knowledge of this structure is important within neurodynamical applications. Indeed, the Amari IDDE acts as a filter with the ability to recognise and respond whenever it is excited in such a way so as to resonate with one of its rightward modes, thereby amplifying such inputs and dampening others. Finally, we discuss how these results can be generalised to the case of systems of IDDEs.

A new interpretation of the bonding properties and UV–vis spectra of [M3(CO)12] clusters (M = Ru, Os): a TD-DFT study

DFT and TD-DFT calculations (ADF program) were performed in order to analyze the electronic structure of the [M-3(CO)(12)] clusters (M = Ru, Os) and interpret their electronic spectra. The highest occupied molecular orbitals are M-M bonding (sigma) involving different M-M bonds, both for Ru and Os. They participate in low-energy excitation processes and their depopulation should weaken M-M bonds in general. While the LUMO is M-NI and M-CO anti-bonding (sigma*), the next, higher-lying empty orbitals have a main contribution from CO (pi*) and either a small (Ru) or an almost negligible one (Os) from the metal atoms. The main difference between the two clusters comes from the different nature of these low-energy unoccupied orbitals that have a larger metal contribution in the case of ruthenium. The photochemical reactivity of the two clusters is reexamined and compared to earlier interpretations.

The relative order and inverses of recurrent networks

Differential geometry is used to investigate the structure of neural-network-based control systems. The key aspect is relative order—an invariant property of dynamic systems. Finite relative order allows the specification of a minimal architecture for a recurrent network. Any system with finite relative order has a left inverse. It is shown that a recurrent network with finite relative order has a local inverse that is also a recurrent network with the same weights. The results have implications for the use of recurrent networks in the inverse-model-based control of nonlinear systems.

Adaptive general predictive controller for nonlinear systems

A nonlinear general predictive controller (NLGPC) is described which is based on the use of a Hammerstein model within a recursive control algorithm. A key contribution of the paper is the use of a novel, one-step simple root solving procedure for the Hammerstein model, this being a fundamental part of the overall tuning algorithm. A comparison is made between NLGPC and nonlinear deadbeat control (NLDBC) using the same one-step nonlinear components, in order to investigate NLGPC advantages and disadvantages.