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

Interannual Variability of Upwelling Indices in the Southeastern Arabian Sea: A Satellite Based Study

Increase in sea surface temperature with global warming has an impact on coastal upwelling. Past two decades (1988 to 2007) of satellite observed sea surface temperatures and space borne scatterometer measured winds have provided an insight into the dynamics of coastal upwelling in the southeastern Arabian Sea, in the global warming scenario. These high resolution data products have shown inconsistent variability with a rapid rise in sea surface temperature between 1992 and 1998 and again from 2004 to 2007. The upwelling indices derived from both sea surface temperature and wind have shown that there is an increase in the intensity of upwelling during the period 1998 to 2004 than the previous decade. These indices have been modulated by the extreme climatic events like El–Nino and Indian Ocean Dipole that happened during 1991–92 and 1997–98. A considerable drop in the intensity of upwelling was observed concurrent with these events. Apart from the impact of global warming on the upwelling, the present study also provides an insight into spatial variability of upwelling along the coast. Noticeable fact is that the intensity of offshore Ekman transport off 8oN during the winter monsoon is as high as that during the usual upwelling season in summer monsoon. A drop in the meridional wind speed during the years 2005, 2006 and 2007 has resulted in extreme decrease in upwelling though the zonal wind and the total wind magnitude are a notch higher than the previous years. This decrease in upwelling strength has resulted in reduced productivity too.

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.

Environment and Climate Changes During the Late Quaternary: Inferences from Sedimentary Records of Southeastern Arabian Sea

Department of Marine Geology & Geophysics, Cochin University of Science and Technology

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.

Occurrence and Distribution of a Diatom-Diazotrophic Cyanobacteria Association during a Trichodesmium Bloom in the Southeastern Arabian Sea

Symbiotic diatom-diazotrophic cyanobacteria association (DDA) of Rhizosolenia hebetata and Rhizosolenia formosa with endosymbiotic cyanobacteria Richelia intracellularis was noticed and documented for the first time during a bloom of the cyanobacterium Trichodesmium erythraeum in the oligotrophic shelf waters along Kochi and Mangalore transects, southeastern Arabian Sea (SEAS), during spring intermonsoon (April 2012). Although the host is frequently seen, the symbiont is rarely reported in the Indian EEZ. The presence of nitrogen-fixing symbiotic association of Rhizosolenia-Richelia along with Trichodesmium erythraeum highlights the significance of DDAs on the nutrient and energy budgets of phytoplankton in the oligotrophic environments of the Arabian Sea during spring intermonsoon