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.

Impacts of Climate variability on Agriculture in Kerala

The study revealed that southwest monsoon rainfall in Kerala has been declining while increasing in post monsoon season. The annual rainfall exhibits a cyclic trend of 40-60 years, with a significant decline in recent decades. The intensity of climatological droughts was increasing across the State of Kerala through it falls under heavy rainfall zone due to unimodal rainfall pattern. The moisture index across the State of Kerala was moving from B4 to B3 humid, indicating that the State was moving from wetness to dryness within the humid climate.The study confirms that a warming Kerala is real as maximum, minimum and mean temperatures and temperature ranges are increasing. The rate of increase in maximum temperature was high (1.46°C) across the high ranges, followed by the coastal belt (1.09°C) of Kerala while the rate of increase was relatively marginal (0.25°C) across the midlands. The rate of increase in temperature across the high ranges is probably high because of deforestation. It indicates that the highranges and coastal belts in Kerala are vulnerable to global warming and climate change when compared to midlands.Interestingly, the trend in annual rainfall is increasing at Pampadumpara (Idukki), while declining at Ambalavayal across the highranges. In the case of maximum temperature, it was showing increasing trend at Pampadumpara while declining trend at Ambalavayal. In the case of minimum temperature it is declining at Pampadumpara while increasing in Ambalavalal.The paddy productivity in Kerala during kharif / virippu is unlikely to decline due to increasing temperature on the basis of long term climate change, but likely to decline to a considerable extent due to prolonged monsoon season, followed by unusual summer rains as noticed in 2007-08 and 2010-11.All the plantation crops under study are vulnerable to climate variability such as floods and droughts rather than long term changes in temperature and rainfall.

IA Critical Study of the Impact of Institutional Finance on Agriculture in Selected Villages of Ernakulam Revenue District

Vast changes have taken place in the field of institutional rural credit in India since the nationalisation of nineteen commercial banks in 1969. The supply of institutional finance to cultivators amounted to 63.2 percent of the total credit in 1981 compared to 31.2 percent in 1971. Insti tutionalisation of agricultural credit envisaged two objectives in general. One was to emancipate cultivators and farmers from the clutches of indigenous financiers and money lenders. The second was to make farmers financially capable of adopting the new technology or improved practices in agriculture so as to increase their agricultural production and thereby contributing to the development of agriculture in India. Though vast literature on Institutional Credi t and agriculture is available, no indepth and serious work examining thoroughly the cause of credit diversion has been undertaken so far. The present study is an attempt to fill up this gap. The study will be helpful to lending insti tutions, viz. Co-ope:r-atives, Commercial banks and various other insti tutional agencies in connection with their lending activity_ Also, the study will help government in .formulating proper policies that will insure a preferential treatment in favour of the most needy category of farmers and cultivators with respect to agricultural credit disbursement

Refinance operations of the national bank for agriculture and rural development-the kerala experience

NABARD has completed 14 years of operation.ln the light of its experiences and achievements, the performance ev.ilu.ilion of the National Bank need to be looked into.This could provide certain criteria for its strength and weakness which may help in consolidating the institution for better utilisation of its potentialities. It is also noteworthy that no evaluative study on the National Bank has been conducted in Kerala. The Major objective of this study is to evaluate the role of NABARD in catering to the long-term agricultural requirements of Kerala for 1982 to 1992.This is done by analysing the quantum and quality of NABARD's schematic refinance. The qualitative indices like (1) the efficiency of loan recovery, (2) the impact or financial viability of NABARD refinanced schemes, (3) the credit gap, (4) the commitment-disbursement gap, and (5) the imbalances in the NABARD refinance form the core of the study.Hypotheses were formulated inorder to study and analyse these qualitative indices. The study is presented in eight chapters

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.