An analogue of the logistic map in two dimensions

This is a sequel to our earlier work on the modulated logistic map. Here, we first show that the map comes under the universality class of Feigenbaum. We then give evidence for the fact that our model can generate strange attractors in the unit square for an uncountable number of parameter values in the range μ∞<μ<1. Numerical plots of the attractor for several values of μ are given and the self-similar structure is explicity shown in one case. The fractal and information dimensions of the attractors for many values of μ are shown to be greater than one and the variation in their structure is analysed using the two Lyapunov exponents of the system. Our results suggest that the map can be considered as an analogue of the logistic map in two dimensions and may be useful in describing certain higher dimensional chaotic phenomena.

Significance of Time Scales in Nonlinear Dynamical Analysis of Electroencephalogram Signals

We propose to show in this paper, that the time series obtained from biological systems such as human brain are invariably nonstationary because of different time scales involved in the dynamical process. This makes the invariant parameters time dependent. We made a global analysis of the EEG data obtained from the eight locations on the skull space and studied simultaneously the dynamical characteristics from various parts of the brain. We have proved that the dynamical parameters are sensitive to the time scales and hence in the study of brain one must identify all relevant time scales involved in the process to get an insight in the working of brain.

Speech Analysis using Modern Techniques of Nonlinear Dynamics

Medical fields requires fast, simple and noninvasive methods of diagnostic techniques. Several methods are available and possible because of the growth of technology that provides the necessary means of collecting and processing signals. The present thesis details the work done in the field of voice signals. New methods of analysis have been developed to understand the complexity of voice signals, such as nonlinear dynamics aiming at the exploration of voice signals dynamic nature. The purpose of this thesis is to characterize complexities of pathological voice from healthy signals and to differentiate stuttering signals from healthy signals. Efficiency of various acoustic as well as non linear time series methods are analysed. Three groups of samples are used, one from healthy individuals, subjects with vocal pathologies and stuttering subjects. Individual vowels/ and a continuous speech data for the utterance of the sentence "iruvarum changatimaranu" the meaning in English is "Both are good friends" from Malayalam language are recorded using a microphone . The recorded audio are converted to digital signals and are subjected to analysis.Acoustic perturbation methods like fundamental frequency (FO), jitter, shimmer, Zero Crossing Rate(ZCR) were carried out and non linear measures like maximum lyapunov exponent(Lamda max), correlation dimension (D2), Kolmogorov exponent(K2), and a new measure of entropy viz., Permutation entropy (PE) are evaluated for all three groups of the subjects. Permutation Entropy is a nonlinear complexity measure which can efficiently distinguish regular and complex nature of any signal and extract information about the change in dynamics of the process by indicating sudden change in its value. The results shows that nonlinear dynamical methods seem to be a suitable technique for voice signal analysis, due to the chaotic component of the human voice. Permutation entropy is well suited due to its sensitivity to uncertainties, since the pathologies are characterized by an increase in the signal complexity and unpredictability. Pathological groups have higher entropy values compared to the normal group. The stuttering signals have lower entropy values compared to the normal signals.PE is effective in charaterising the level of improvement after two weeks of speech therapy in the case of stuttering subjects. PE is also effective in characterizing the dynamical difference between healthy and pathological subjects. This suggests that PE can improve and complement the recent voice analysis methods available for clinicians. The work establishes the application of the simple, inexpensive and fast algorithm of PE for diagnosis in vocal disorders and stuttering subjects.

Recurrence Quantification Analysis of System Signals for Detecting Tool and Chatter in Turning

In this thesis, the applications of the recurrence quantification analysis in metal cutting operation in a lathe, with specific objective to detect tool wear and chatter, are presented.This study is based on the discovery that process dynamics in a lathe is low dimensional chaotic. It implies that the machine dynamics is controllable using principles of chaos theory. This understanding is to revolutionize the feature extraction methodologies used in condition monitoring systems as conventional linear methods or models are incapable of capturing the critical and strange behaviors associated with the metal cutting process.As sensor based approaches provide an automated and cost effective way to monitor and control, an efficient feature extraction methodology based on nonlinear time series analysis is much more demanding. The task here is more complex when the information has to be deduced solely from sensor signals since traditional methods do not address the issue of how to treat noise present in real-world processes and its non-stationarity. In an effort to get over these two issues to the maximum possible, this thesis adopts the recurrence quantification analysis methodology in the study since this feature extraction technique is found to be robust against noise and stationarity in the signals.The work consists of two different sets of experiments in a lathe; set-I and set-2. The experiment, set-I, study the influence of tool wear on the RQA variables whereas the set-2 is carried out to identify the sensitive RQA variables to machine tool chatter followed by its validation in actual cutting. To obtain the bounds of the spectrum of the significant RQA variable values, in set-i, a fresh tool and a worn tool are used for cutting. The first part of the set-2 experiments uses a stepped shaft in order to create chatter at a known location. And the second part uses a conical section having a uniform taper along the axis for creating chatter to onset at some distance from the smaller end by gradually increasing the depth of cut while keeping the spindle speed and feed rate constant.The study concludes by revealing the dependence of certain RQA variables; percent determinism, percent recurrence and entropy, to tool wear and chatter unambiguously. The performances of the results establish this methodology to be viable for detection of tool wear and chatter in metal cutting operation in a lathe. The key reason is that the dynamics of the system under study have been nonlinear and the recurrence quantification analysis can characterize them adequately.This work establishes that principles and practice of machining can be considerably benefited and advanced from using nonlinear dynamics and chaos theory.

Studies on Chaos and Synchronization in ac-driven Josephson junctions

The main goal of this thesis is to study the dynamics of Josephson junction system in the presence of an external rf-biasing.A system of two chaotically synchronized Josephson junction is studied.The change in the dynamics of the system in the presence of at phase difference between the applied fields is considered. Control of chaos is very important from an application point of view. The role Of phase difference in controlling chaos is discussed.An array of three Josephson junctions iS studied for the effect of phase difference on chaos and synchronization and the argument is extended for a system of N Josephson junctions. In the presence of a phase difference between the external fields, the system exhibits periodic behavior with a definite phase relationship between all the three junctions.Itdeals with an array of three Josephson junctions with a time delay in the coupling term. It is observed that only the outer systems synchronize while the middle system remain uncorrelated with t-he other two. The effect of phase difference between the applied fields and time-delay on system dynamics and synchronization is also studied. We study the influence of an applied ac biasing on a serniannular Josephson junction. It is found the magnetic field along with the biasing induces creation and annihilation of fluxons in the junction. The I-V characteristics of the junction is studied by considering the surface loss term also in the model equation. The system is found to exhibit chaotic behavior in the presence of ac biasing.

Dynamics of coupled Josephson junctions under the influence of applied fields

We investigate the effect of the phase difference of appliedfields on the dynamics of mutually coupledJosephsonjunctions. A phase difference between the appliedfields desynchronizes the system. It is found that though the amplitudes of the output voltage values are uncorrelated, a phase correlation is found to exist for small values of applied phase difference. The dynamics of the system is found to change from chaotic to periodic for certain values of phase difference. We report that by keeping the value of phase difference as π, the system continues to be in periodic motion for a wide range of values of system parameters. This result may find applications in devices like voltage standards, detectors, SQUIDS, etc., where chaos is least desired.

Phase synchronization in an array of driven Josephson junctions

We consider an array of N Josephson junctions connected in parallel and explore the condition for chaotic synchronization. It is found that the outer junctions can be synchronized while they remain uncorrelated to the inner ones when an external biasing is applied. The stability of the solution is found out for the outer junctions in the synchronization manifold. Symmetry considerations lead to a situation wherein the inner junctions can synchronize for certain values of the parameter. In the presence of a phase difference between the applied fields, all the junctions exhibit phase synchronization. It is also found that chaotic motion changes to periodic in the presence of phase differences.

Dissipation in quasistatically driven disordered systems

This work presents an analysis of hysteresis and dissipation in quasistatically driven disordered systems. The study is based on the random field Ising model with fluctuationless dynamics. It enables us to sort out the fraction of the energy input by the driving field stored in the system and the fraction dissipated in every step of the transformation. The dissipation is directly related to the occurrence of avalanches, and does not scale with the size of Barkhausen magnetization jumps. In addition, the change in magnetic field between avalanches provides a measure of the energy barriers between consecutive metastable states

Texture, composition and provinance of innershelf sediments between Narakkal and Purakkad, Kerala with special reference to the formation of Mud Banks

The continental shelf of southwest coast of India (Kerala) is broader and . flatter compared to that of the east coast. The unique characteristic feature of the study area (innershelf between Narakkal and Purakkad) is the intermittent appearance of 'mud banks' at certain locations during southwest monsoon. The strong seasonality manifests significant changes in the wind, waves, currents, rainfall, drainage etc., along this area. Peculiar geomorphological variation with high, mid and lowlands in the narrow strip of the hinterland, the geological formations mainly consisting of rocks of metamorphic origin and the humid tropical weathering conditions play significant role in regulating the shelf sedimentation. A complementary pattern of distri bution is observed for clay that shows an abundance in the nearshore. Silt, to a major extent, depicts semblance with clay distribution . Summation of the total asymmetry of grain size distribution are inferred from the variation of skewness and kurtosis.Factor I implies a low energy regime where the transportation and deposition phases are controlled mostly by pelagic suspension process as the factor loadings are dominant on finer phi sizes. The second Factor is inferred to be the result of a high energy regime which gives higher loadings on coarser size fractions. The third Factor which might be a transition phase (medium energy regime) representing the resultant flux of coastal circulation of the re-suspension/deposition and an onshoreoffshore advection by reworking and co-deposition of relict and modern sediments. The spatial variations of the energy regime based on the three end-member factor model exhibits high energy zone in the seaward portion transcending to a low energy one towards the coast.From the combined analysis of granulometry and SEM studies, it is concluded that the sandy patches beyond 20 m depth are of relict nature. They are the resultant responses of beach activity during the lower stand of sea level in the Holocene. Re-crystallisation features on the quartz grains indicate that they were exposed to subaerial weathering process subsequent to thei r deposition

Studies on Energy Exchange and Upper Ocean Thermat Structure in Arabian Sea and Heat Transport in Northern Indian Ocean

The thesis focused Studies on Energy Exchange and Upper Ocean Thermal Structure in Arabian Sea and Heat Transport in Northern Indian Ocean. The present thesis is an attempt to understand the upper ocean thermal characteristics at selected areas in the western and eastern Arabian Sea in relation to surface energy exchange and dynamics, on a climatological basis. It is also aimed to examine, the relative importance of different processes in the evolution of SST at the western and eastern Arabian Sea. Short-term variations of energy exchange and upper ocean thermal structure are also investigated. Climatological studies of upper ocean thermal structure and surface energy exchange in the western and eastern parts of Arabian Sea bring out the similarities/differences and the causative factors for the observed features. Annual variation of zonally averaged heat advection in north Indian Ocean shows that maximum export of about 100 W/m2 occurs around 15ON during southwest monsoon season. This is due to large negative heat storage caused by intense upwelling in several parts of northern Indian Ocean. By and large, northern Indian Ocean is an area of heat export

Studies on the universal parameters and onset of chaos in dissipative systems

It has become clear over the last few years that many deterministic dynamical systems described by simple but nonlinear equations with only a few variables can behave in an irregular or random fashion. This phenomenon, commonly called deterministic chaos, is essentially due to the fact that we cannot deal with infinitely precise numbers. In these systems trajectories emerging from nearby initial conditions diverge exponentially as time evolves)and therefore)any small error in the initial measurement spreads with time considerably, leading to unpredictable and chaotic behaviour The thesis work is mainly centered on the asymptotic behaviour of nonlinear and nonintegrable dissipative dynamical systems. It is found that completely deterministic nonlinear differential equations describing such systems can exhibit random or chaotic behaviour. Theoretical studies on this chaotic behaviour can enhance our understanding of various phenomena such as turbulence, nonlinear electronic circuits, erratic behaviour of heart and brain, fundamental molecular reactions involving DNA, meteorological phenomena, fluctuations in the cost of materials and so on. Chaos is studied mainly under two different approaches - the nature of the onset of chaos and the statistical description of the chaotic state.

Studies on the effect of randoniness on the synchronization of coupled systems and on the dynamics of intermittently driven systems

It has been shown recently that systems driven with random pulses show the signature of chaos ,even without non linear dynamics.This shows that the relation between randomness and chaos is much closer than it was understood earlier .The effect of random perturbations on synchronization can be also different. In some cases identical random perturbations acting on two different chaotic systems induce synchronizations. However most commonly ,the effect of random fluctuations on the synchronizations of chaotic system is to destroy synchronization. This thesis deals with the effect of random fluctuations with its associated characteristic timescales on chaos and synchronization. The author tries to unearth yet another manifestation of randomness on chaos and sychroniztion. This thesis is organized into six chapters.

Onset And Characterisation Of Chaos In A Two Dimensional Nonlinear Deterministic Model

Nature is full of phenomena which we call "chaotic", the weather being a prime example. What we mean by this is that we cannot predict it to any significant accuracy, either because the system is inherently complex, or because some of the governing factors are not deterministic. However, during recent years it has become clear that random behaviour can occur even in very simple systems with very few number of degrees of freedom, without any need for complexity or indeterminacy. The discovery that chaos can be generated even with the help of systems having completely deterministic rules - often models of natural phenomena - has stimulated a lo; of research interest recently. Not that this chaos has no underlying order, but it is of a subtle kind, that has taken a great deal of ingenuity to unravel. In the present thesis, the author introduce a new nonlinear model, a ‘modulated’ logistic map, and analyse it from the view point of ‘deterministic chaos‘.

Studies On Mixed Layer Depth In The Arabian Sea

This thesis is the result of an elaborate study on the mixed layer depth (MLD) and the various oceanic environmental factors controlling it in the Arabian Sea examining its predictability on annual and short term basis. To accomplish this, the study area between 100 — 250 N latitudes and 600 — 750 E longitudes in the Arabian Sea is divided into 8 subareas of 50 quadrangles. The distribution of monthly means of the surface wind field, net heat exchange mKi868€%WTmN¥tWMWF3UH9 (SST) over each subarea in the annual cycle is examined. The corresponding wind (mechanical) and convective mixing values are computed and presented along with the observed mean MLD for the subareas in the annual cycle. Effects of advection due to surface currents and surface divergence (convergence and divergence) for these subareas are examined for correlating the MLD variations. A representative time series data from typical deep water station under southwest monsoonal forcing is analysed for the spectral components to estimate the amplitude perturbations on the mean MLD variation

Study of Synchronisalion and Control of Chaos in Directly Modulated Semiconductor Lasers

Chaos is a subject oftopical interest and, studied in great detail in relation to its relevance in almost all branches of science, which include physical, chemical, and biological fields. Chaos in the literal sense signifies utter confusion, but the scientific community has differentiated chaos as deterministic chaos and white noise. Deterministic chaos implies the complex behaviour of systems, which are governed by deterministic laws. Behaviour of such systems often become unpredictable in the long run. This unpredictability arises from the sensitivity of the system to its initial conditions. The essential requirement for ‘sensitivity to initial condition’ is nonlinearity of the system. The only method for determining the future of such systems is numerically simulating its final state from a set ofinitial conditions. Synchronisation