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

Scaling relations in the Lyapunov exponents of one dimensional maps

We establish numerically the validity of Huberman-Rudnick scaling relation for Lyapunov exponents during the period doubling route to chaos in one dimensional maps. We extend our studies to the context of a combination map. where the scaling index is found to be different.

Post Environmental Evaluation of the Rajjaprabha Dam in Thailand

This thesis Entitled Post-Environmental Evaluation of The Rajjaprabha Dam In Thailand. This post evaluation of environmental consequences of Rajjaprabha dam IS conducted ten years after its commencement. The Rajjaprabha dam project was planned and implemented as a multipurpose project, mainly for hydropower production, flood protection, fisheries, recreation and irrigation. The project includes the dam and reservoir with a 240 MW hydropower plant located about 90 km upstream from Surat Thani province, and irrigation systems covering the coastal plain in Surat Thani. The upstream storage reservoir (with about 5,639 mcm storage) and the hydropower plant had already been implemented. The first phase of irrigation system covers an area of 23,100 hectares. The second phase is envisaged to cover about 50,000 hectares. This study was conducted with the following objectives: (I) to assess all existing environmental resources and their values with the help of input-output analysis (2) to findout the beneficial impacts of the project (3) to evaluate the actual positive effects vis-a-vis the estimated effects before the project was implemented and (4) to identify all significant changes in relatives to the impacts previously assessed. The study area includes the Phum Duang river basin of about 4,668 km2 (placed on the areas that are upstream and downstream to the damsite), The duration of study is limited to 10 years after the dam has become operational i.e. from 1987-1997. The results of the study reveal that there is no significant changes in climatic and ground water resources, with respect to the study area inspte of the fact that the physical and chemical properties of the soil have slightly changed. Sedimentation in the reservoir does not have much effect on the function of the dam.

Analytical Studies on Diffusion and lntermittency in Chaotic Maps

The study of simple chaotic maps for non-equilibrium processes in statistical physics has been one of the central themes in the theory of chaotic dynamical systems. Recently, many works have been carried out on deterministic diffusion in spatially extended one-dimensional maps This can be related to real physical systems such as Josephson junctions in the presence of microwave radiation and parametrically driven oscillators. Transport due to chaos is an important problem in Hamiltonian dynamics also. A recent approach is to evaluate the exact diffusion coefficient in terms of the periodic orbits of the system in the form of cycle expansions. But the fact is that the chaotic motion in such spatially extended maps has two complementary aspects- - diffusion and interrnittency. These are related to the time evolution of the probability density function which is approximately Gaussian by central limit theorem. It is noticed that the characteristic function method introduced by Fujisaka and his co-workers is a very powerful tool for analysing both these aspects of chaotic motion. The theory based on characteristic function actually provides a thermodynamic formalism for chaotic systems It can be applied to other types of chaos-induced diffusion also, such as the one arising in statistics of trajectory separation. It was noted that there is a close connection between cycle expansion technique and characteristic function method. It was found that this connection can be exploited to enhance the applicability of the cycle expansion technique. In this way, we found that cycle expansion can be used to analyse the probability density function in chaotic maps. In our research studies we have successfully applied the characteristic function method and cycle expansion technique for analysing some chaotic maps. We introduced in this connection, two classes of chaotic maps with variable shape by generalizing two types of maps well known in literature.