Investigations on the Dynamics of Combination Maps and the Analysis of Bifurcation in a Discontinuous Logistic Map

This thesis is a study of discrete nonlinear systems represented by one dimensional mappings.As one dimensional interative maps represent Poincarre sections of higher dimensional flows,they offer a convenient means to understand the dynamical evolution of many physical systems.It highlighting the basic ideas of deterministic chaos.Qualitative and quantitative measures for the detection and characterization of chaos in nonlinear systems are discussed.Some simple mathematical models exhibiting chaos are presented.The bifurcation scenario and the possible routes to chaos are explained.It present the results of the numerical computational of the Lyapunov exponents (λ) of one dimensional maps.This thesis focuses on the results obtained by our investigations on combinations maps,scaling behaviour of the Lyapunov characteristic exponents of one dimensional maps and the nature of bifurcations in a discontinous logistic map.It gives a review of the major routes to chaos in dissipative systems,namely, Period-doubling ,Intermittency and Crises.This study gives a theoretical understanding of the route to chaos in discontinous systems.A detailed analysis of the dynamics of a discontinous logistic map is carried out, both analytically and numerically ,to understand the route it follows to chaos.The present analysis deals only with the case of the discontinuity parameter applied to the right half of the interval of mapping.A detailed analysis for the n –furcations of various periodicities can be made and a more general theory for the map with discontinuities applied at different positions can be on a similar footing

High‐temperature phase transitions in pure and deuterated ammonium dihydrogen phosphate: Conductivity and dielectric measurements

Results of axiswise measurements of the electrical conductivity (dc and ac) and dielectric constant of NH4H2PO4 confirm the occurrence of the recently suggested high‐temperature phase transition in this crystal (at 133 °C). The corresponding transition in ND4D2PO4 observed here for the first time takes place at 141.5 °C. The mechanism involved in these transitions and those associated with the electrical conduction and dielectric anomalies are explained on the basis of the motional effects of the ammonium ions in these crystals. Conductivity values for deuterated crystals give direct evidence for the predominance of protonic conduction throughout the entire range of temperatures studied (30–260 °C).

A study on fuzzy semi inner product spaces

Mathematical models are often used to describe physical realities. However, the physical realities are imprecise while the mathematical concepts are required to be precise and perfect. Even mathematicians like H. Poincare worried about this. He observed that mathematical models are over idealizations, for instance, he said that only in Mathematics, equality is a transitive relation. A first attempt to save this situation was perhaps given by K. Menger in 1951 by introducing the concept of statistical metric space in which the distance between points is a probability distribution on the set of nonnegative real numbers rather than a mere nonnegative real number. Other attempts were made by M.J. Frank, U. Hbhle, B. Schweizer, A. Sklar and others. An aspect in common to all these approaches is that they model impreciseness in a probabilistic manner. They are not able to deal with situations in which impreciseness is not apparently of a probabilistic nature. This thesis is confined to introducing and developing a theory of fuzzy semi inner product spaces.

Studis on the climatological aspects of air pollution potential over trivandrum

The deteriorating air quality especially in urban environments is a cause of serious concern. In spite of being an effective sink, the atmosphere also has its own limitations in effectively dispersing the pollutants being dumped into it continuously by various sources, mainly industries. Many a time, it is not the higher emissions that cause alarming level of pollutants but the unfavourable atmospheric conditions under which the atmosphere is not able to disperse them effectively, leading to accumulation of pollutants near the ground. Hence, it is imperative to have an estimate of the atmospheric potential for dispersal of the substances emitted into it. This requires a knowledge of mixing height, ventilation coefficient, wind and stability of the region under study. Mere estimation of such pollution potential is not adequate, unless the probable distribution of concentration of pollutants is known. This can be obtained by means of mathematical models. The pollution potential coupled with the distribution of concentration provides a good basis for initiating steps to mitigate air pollution in any developing urban area. In this thesis, a fast developing industrial city, namely, Trivandrum is chosen for estimating the pollution potential and determining the spatial distribution of sulphur dioxide concentration. Each of the parameters required for pollution potential is discussed in detail separately. The thesis is divided into nine chapters.

A study on fuzzy semi inner product spaces

Mathematical models are often used to describe physical realities. However, the physical realities are imprecise while the mathematical concepts are required to be precise and perfect. The 1st chapter give a brief summary of the arithmetic of fuzzy real numbers and the fuzzy normed algebra M(I). Also we explain a few preliminary definitions and results required in the later chapters. Fuzzy real numbers are introduced by Hutton,B [HU] and Rodabaugh, S.E[ROD]. Our definition slightly differs from this with an additional minor restriction. The definition of Clementina Felbin [CL1] is entirely different. The notations of [HU]and [M;Y] are retained inspite of the slight difference in the concept.the 3rd chapter In this chapter using the completion M'(I) of M(I) we give a fuzzy extension of real Hahn-Banch theorem. Some consequences of this extension are obtained. The idea of real fuzzy linear functional on fuzzy normed linear space is introduced. Some of its properties are studied. In the complex case we get only a slightly weaker analogue for the Hahn-Banch theorem, than the one [B;N] in the crisp case

Studies on pulp propogation in single mode optical fibres

Studies on pulse propagation in single mode optical fibers have attracted interest from a wide area of science and technology as they have laid down the foundation for an in-depth understanding of the underlying physical principles, especially in the field of optical telecommunications. The foremost among them is discovery of the optical soliton which is considered to be one of the most significant events of the twentieth century owing to its fantastic ability to propagate undistorted over long distances and to remain unaflected after collision with each other. To exploit the important propertia of optical solitons, innovative mathematical models which take into account proper physical properties of the single mode optical fibers demand special attention. This thesis contains a theoretical analysis of the studies on soliton pulse propagation in single mode optical fibers.