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.

A Comparative Study Of The Organisation And Management Of Notihed Markets In Bangladesh With The Organisation And Management Of Regulated Markets In West Bengal India

For improving agricultural marketing, which has been discussed in the previous chapter, the Government has intervened in different ways. The direct regulatory role through the regulation of markets and market practices is one of the ways in which governmental intervention can improve agricultural marketing. This study is an enquiry of the direct regulatory role of the government through regulation of markets and market practices. By restructuring the operational methods and redesigning the existing physical markets, this system gives direct benefit to the cultivating class and protects them from the market manipulations of organised and powerful private traders. If traders do not continue their trade for the time being they will not be affected financially, because they are resourceful or financially solvent. On the other hand, Cultivators must sell their produce immediately after harvesting for the lack of additional facilities or to satisfy other needs for which finance is required. Another important reason is that Cultivators/farmers are not organised and because of lack of their organisation, they sell their produces individually. In this situation, a farmer is helpless when astute traders indulge in manipulations at the time of purchase of the produces. So it is the government's obligation to protect the interest of the farmers. Protection of the farmer/cultivator is necessary not only from the point of social justice but also from that of economic growth. If the farmers are assured of a remunerative or incentive price for their produce, they will get the inspiration to produce more and through more production, economy will be developed and the nation as a whole will be benefitted. This study will examine the management system of the markets through the direct regulatory role played by the governments to control markets and market practices in West Bengal and Bangladesh.

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.