›› Introduction

›› Granger Causality from Fourier and Wavelet Transforms

›› Granger Causality for Point Processes

›› Mitigating the Effects of Noise on Granger Causality

›› Synchronized Extinction of
Species

›› Movement Prediction in Neuroscience

›› Comparison of Linear Causality Measures

›› Nonlinear Causality Measures

›› Neural Networks

›› First
Passage Time Problems

›› Synchronization and Pattern Formation
in Coupled Dynamical Systems

›› Processes with Long Range Correlations

›› A New Method for Computing Lyapunov
Exponents

›› Symplectic Integration of Nonlinear
Hamiltonian Systems

›› Representations of Sp(6,R) and
SU(3)

›› Fractal Analysis in Geophysics

›› Generalized Replicator Dynamics

›› Evolutionary Games with Two
Time Scales

›› Moment Invariants

›› Time Series Analysis in Geophysics

›› Free-electron Laser (FEL) Theory**
**

**Introduction**

I work in the general areas of nonlinear dynamics,
time series analysis and applied stochastic dynamics. I apply my work
in areas like neuroscience, geophysics and accelerator physics. A more
detailed description of my research work is given below.

**Granger Causality from Fourier and Wavelet Transforms**

Extracting information flow in networks of coupled
dynamical systems from the time series measurements of their
activity is of great interest in physical, biological and social
sciences. Such knowledge holds the key to the understanding of
phenomena ranging from turbulent fluids to interacting genes and
proteins to networks of neural ensembles. Granger causality
has emerged in recent years as a leading
statistical technique for accomplishing this goal. The definition of
Granger causality is based on the theory of
linear prediction and its original estimation
framework requires autoregressive (AR) modeling of time series data.
This is a parametric approach to estimating Granger causality.

The above parametric approach has certain weaknesses. We
overcame this by proposing a nonparametric approach to
estimate Granger causality (in the spectral domain) directly from Fourier and wavelet
transforms of data, eliminating the need of explicit AR modeling.
This approach makes use of spectral factorization.
Time-domain Granger causality can be obtained by integrating the
corresponding spectral representation over frequency. This method
was successfully applied to both simulated data and data from a neuroscience
experiment.

**Granger Causality for Point Processes**

Simultaneous recordings of spike trains from
multiple single neurons are becoming commonplace. Mathematically,
spike trains can be modeled as point processes.
Understanding the interaction patterns among these spike
trains (point processes) remains a key research area. A question of interest is
the evaluation of information flow between neurons
through the analysis of whether one spike train exerts
causal influence on another. For continuous-valued time
series data, Granger causality has proven an effective
method for this purpose. However, the basis for Granger
causality estimation is autoregressive data modeling, which
is not directly applicable to spike trains. Various filtering
options distort the properties of spike trains as point
processes. We proposed a new nonparametric approach
to estimate Granger causality directly from the Fourier
transforms of spike train data. We validated the method on
synthetic spike trains generated by model networks of
neurons with known connectivity patterns and then applied it
to neurons simultaneously recorded from the thalamus and
the primary somatosensory cortex of a squirrel monkey
undergoing tactile stimulation.

**Mitigating the Effects of Noise on Granger Causality**

Computing Granger causal relations among bivariate experimentally
observed time series has received increasing attention over the past
few years. Such causal relations, if correctly estimated, can yield
significant insights into the dynamical organization of the system
being investigated. Since experimental measurements are inevitably
contaminated by noise, it is thus important to understand the
effects of such noise on Granger causality estimation. Using the theory of
stochastic processes, we showed mathematically that two effects can arise:
(1) spurious causality between two measured variables
can arise and (2) true causality can be suppressed. We also demonstrated these
effects numerically. We then provided a denoising strategy to mitigate this
problem. Specifically, we proposed a denoising algorithm based on the
combined use of the Kalman filter theory and the
Expectation-Maximization (EM) algorithm. Numerical examples were used
to demonstrate the effectiveness of the denoising approach.

We applied the above method to denoise two
datasets of local field potentials recorded from monkeys performing a
visuomotor task. For the first dataset, it was found that the analysis
of the high gamma band (60-90 Hz) neural activity in the prefrontal cortex
is highly susceptible to the effect of noise, and denoising leads to
markedly improved results that were physiologically interpretable.
For the second dataset, Granger causality between primary motor and
primary somatosensory cortices was not consistent across two monkeys
and the effect of noise was suspected. After denoising, the discrepancy
between the two subjects was significantly reduced.

**Synchronized Extinction
of Species**

More than 99% of the species that ever existed on the
surface of the earth are now extinct and their extinction on a global
scale has been a puzzle. One may think that a species under an external
threat may survive in some isolated locations leading to the revival of
the species. Using a general model we show that, under a common external
forcing, the species with a quadratic saturation term first undergoes
spatial synchronization and then extinction. The effect can be observed
even when the external forcing acts only on some locations provided the
dynamics contains a synchronizing term. Absence of the quadratic
saturation term can help the species to avoid extinction.

**Movement Prediction in Neuroscience**

Among the awesome repertoire of tasks that the human
brain can accomplish, one of the more fascinating ones is how the electrical
activity of millions of brain cells (neurons) is translated into precise
sequences of movements.

One of the greatest challenges in applied neuroscience
is to build prosthetic limbs controlled by neural signals from the brain.
The ultimate goal is to provide paralytic patients and amputees with
the means to move and communicate by controlling the prosthetic device
using brain activity. Scientists and engineers are slowly getting closer
to building such devices thanks to studies revealing a strong connection
between the activity of neurons in the brain's cerebral cortex and the
movements of limbs.

With the above goal in mind, we have performed some
preliminary analysis of experimental data from a macaque monkey. We considered an experiment where the monkey moves
its hand in one of the eight directions based on visual cues. The goal
was to predict the direction of movement correctly before the hand movement
actually starts. Signals were recorded from an array of microelectrodes
implanted in the monkey's motor cortex. As a first step, we computed
the summed power in the gamma frequency band for each channel and trial. Using the Mahalanobis distance, we then predicted the
direction of movement from a single trial multichannel data. We
obtained successful predictions significantly above the random chance
level.

Presently we are conducting a major brain-machine interface
experiment and results are expected soon.

**Comparison of Linear Causality Measures**

Given the deluge of multi-channel data generated by
experiments in neuroscience, the role of multivariate time series processing,
especially the nonlinear time series processing, has become crucial.
In particular, obtaining causal relations among signals is important
in identifying functional relations between different regions of the
brain. Over the years Gersch causality (which uses partial
coherence analysis) has been employed explicitly or implicitly by many
researchers as a way of identifying connectivity, sources of driving
or causal influence.

For real world neurobiological data, a complicating
factor is that the time series picked up by a sensor is inevitably the
mixture of the signal of interest (e.g. theta oscillation in the hippocampus)
and other unrelated processes, collectively referred to as the measurement
noise. We studied the effectiveness of the two different causality measures,
the Gersch causality and the Granger causality, in the above noisy situation.

We studied experimental recordings from the limbic
system of the rat during theta oscillations and showed that the application
of partial coherence and Gersch causality can lead to contradictory
conclusions. We hypothesized that the observed phenomena can be explained
by the differential levels of signal to noise ratio. We performed extensive
simulations involving multiple time series to confirm this. The main
point of our work was that, despite its wide use, partial coherence
based Gersch causality is extremely sensitive to measurement noise and
often could lead to erroneous results. We then applied Granger causality
to the data and simulation results and showed that it is robust against
moderate levels of noise.

**Nonlinear Causality Measures**

We addressed the problem of obtaining nonlinear causality
measures. We extended the Granger causality theory to nonlinear time
series by incorporating the embedding reconstruction technique for multivariate
time series. A three-step algorithm was presented and used to analyze
various nonlinear time series. We demonstrated that our method always
gives reliable results.

When three time series have to be analyzed, the conditional
nonlinear causality index proposed by us can distinguish between direct
and indirect causal relationships between any two of the time series.
This is not possible using simple pairwise analysis.

**Neural Networks**

For oscillatory neural networks, the stability of the
equilibrium point is an important property for many neural computational
applications. Aided by results from nonnegative matrices we demonstrated,
using two simple models of coupled neural populations of arbitrary size,
a general methodology that can yield explicit bounds on the individual
coupling strengths to ensure the stability of the equilibrium point.

**First Passage Time Problems**
First passage time problem for stochastic processes
is a well known problem with applications in physics, chemistry, biology
and engineering. The exact first passage time distribution is known
only for a few specific stochastic processes (the most notable being
the Brownian motion). Finding the exact first passage time distribution
for a broader class of stochastic processes is therefore an important
problem. We obtained the exact first passage density function for Levy
type stochastic processes with zero and nonzero drifts. This work involved
application of results from probability theory, fractional calculus
and Fox or H-functions. Further, asymptotic results for another class
of stochastic processes (fractional Brownian motion and its generalizations)
were also obtained.

**Synchronization and Pattern Formation in Coupled
Dynamical Systems **
Coupled dynamical systems are used to model spatially
extended nonlinear systems (such as the brain) by utilizing well understood
dynamical systems as building blocks. In coupled dynamical systems,
the individual constituent systems (called nodes) are coupled to one
another using specified coupling strengths. A very interesting form
of behaviour which occurs in coupled dynamical systems is the synchronization
of the nodes.

This has found applications in a variety of fields
including communications, optics, neuroscience, neural networks and
geophysics. An essential prerequisite for these applications is to know
the bounds on the coupling strengths so that the stability of the synchronous
state is ensured. Previous attempts aimed at obtaining such conditions
have typically looked either at systems of very small size or at very
specific coupling schemes (diffusive coupling, global all to all coupling
etc. with a single coupling strength).

We obtained stability bounds for the synchronized state
in coupled systems with arbitrary coupling. These bounds extend previously
available results. Based on the Gershgorin disc theory and the stability
region/master stability function, we developed general approaches that
yield constraints directly on the coupling strengths which ensure the
stability of synchronized dynamics.

In a reaction-diffusion system, at the Turing bifurcation
point, the global equilibrium state becomes unstable and an inhomogeneous
spatial pattern known as a Turing pattern emerges. Following Turing's
classic work, pattern formation in reaction-diffusion systems has become
a major topic of investigation both theoretically and experimentally
with applications in diverse fields of science and engineering.

We derived explicit analytical expressions defining
the stability region in the parameter space spanned by the coupling
strengths. We demonstrated that, by following different paths
in the parameter space, different spatiotemporal patterns can be selectively
realized. Conversely, given a desired spatiotemporal pattern, we were
able to design coupling schemes which give rise to that pattern as the
coupled system evolves. Although the spatial patterns are the same for a given
coupling matrix, we showed that, depending on whether the original synchronized
state is a fixed point or a chaotic attractor, the temporal evolution
of the patterns is either constant in time or modulated in an on-off
intermittent fashion.

**Processes with Long Range Correlations**
To assess whether a given time series can be modeled
by a stochastic process possessing long memory one usually applies one
of two types of analysis methods: the spectral method and the random
walk analysis. We showed that each one of these methods used alone can
be susceptible to producing false results. We thus advocated an integrated
approach which requires the use of both methods in a consistent fashion.
We provided the theoretical foundation of this approach and illustrated
the main ideas using examples. We also analysed the observation of long
range anticorrelation (Hurst exponent $H<1/2$) in real world time series
data. The very peculiar nature of such processes was emphasized in light
of the stringent condition under which such processes can occur. Using
examples we discussed the possible factors that could contribute to
the false claim of long range anticorrelations and demonstrated the
particular importance of the integrated approach in this case.

**A New Method for Computing Lyapunov Exponents**
Chaotic dynamics is an important field of research
with applications in many areas of science and engineering. Specific
examples of such areas include astrophysical, biological and chemical
systems, mechanical devices, models of the weather, lasers, plasmas
and fluids, to mention a few. It is important to be able to identify
chaotic systems since one can predict the future behaviour of such systems
only for a limited period of time. This is due to the fact that these
systems exhibit sensitive dependence on initial conditions, i.e. two
trajectories which start out close to each other diverge exponentially.
The Lyapunov exponents precisely quantify this exponential divergence
and a positive Lyapunov exponent implies the presence of chaotic behaviour
in the system. Moreover, practical algorithms to compute Lyapunov exponents
can be devised. For these reasons, Lyapunov exponents have become the
primary tool to detect chaos in a given system. Any good method for
computing Lyapunov exponents has, therefore, wide applicability in several
areas of science and engineering.

We obtained a new, accurate method for the computation
of Lyapunov exponents for arbitrary continuous-time dynamical systems.
Our method makes full use of the group-theoretical representations of
orthogonal matrices applied to decompositions of M where M is the tangent
map of the system. Our method analytically obviates the need for rescaling
and reorthogonalization. Our method also does away with the other shortcomings
listed above: A partial Lyapunov spectrum can be computed using a fewer
number of equations as compared to the computation of the full spectrum,
there is no difficulty in evaluating degenerate Lyapunov spectra, the
equations are straightforward to generalize to higher dimensions, and
the method uses the minimal set of dynamical variables. Since our method
is based on exact differential equations for the Lyapunov exponents,
global invariances of the Lyapunov spectrum can be preserved. Preservation
of such invariants is important in stationary, thermostatted nonequilibrium
systems. The above method has now been extended to computation of Lyapunov
exponents for arbitrary discrete maps. This method also has all the
nice properties mentioned above.

**Symplectic Integration of Nonlinear Hamiltonian
Systems**
Hamiltonian systems form an important class of dynamical
systems and can be used to describe many physical systems. An important
question concerning any such system is its long-term nonlinear stability.
Specifically, we are interested in the long-term stability of particles
being transported through this system. Generically these systems are
non-integrable and it is therefore very difficult to give stability
criteria in an analytic form. A possible solution is to numerically
follow the trajectories of particles through the system for a large
number of periods (a process that goes by the name of tracking). One
could then attempt to infer the stability of motion in the system by
analyzing these tracking results.

The most straightforward method that can be used to
perform this long-term tracking is numerical integration. However, this
method is too slow for analyzing the stability of very complicated systems.
Further, any method that we use should preserve the symplectic structure
that is inherent in any Hamiltonian system. Conventional numerical integration
methods, like the Runge-Kutta method, do not preserve this structure.
This can lead to either spurious chaotic behaviour or damping. Consequently,
there is a possibility of getting wrong stability results when using
such non-symplectic integrators. Therefore, we need a symplectic integrator
that is both fast and accurate.

Several symplectic integration methods have been discussed
in the literature using generating functions. These are typically implicit
methods and employing these methods requires one to use Newton's method
with its attendent questions of convergence etc. We have obtained a
new method for symplectic integration of nonlinear Hamiltonian systems.
In this method, the Hamiltonian system is first represented as a symplectic
map. This map (in general) has an infinite Taylor series expansion.
In practice, one can compute only a finite number of terms in this series.
This gives rise to a truncated map approximation of the original map.
This truncated map is, however, not symplectic and can lead to wrong
stability results when iterated (as noted earlier). To overcome this
problem, the map is refactorized as a product of special maps called
"jolt maps" in such a manner that symplecticity is maintained. These
jolt maps are nilpotent operators of rank 2 and hence have only two
nonzero terms in their Taylor series expansion when acting on phase
space variables. The method makes use of properties of the Lie group
SU(3) and its associated discrete subgroups. Moreover, a scheme to optimize
the number of jolt maps required in this method have been outlined.
This involves splitting the integration over SU(3) into an integration
over the group SO(3) followed by an integration over the coset space
SU(3)/SO(3).

We have applied the above symplectic integration method
using jolt factorization to the symplectic map describing the nonlinear
pendulum Hamiltonian. This example was chosen since the pendulum is
the prototypical nonlinear Hamiltonian system which has the further
advantage that analytical solutions are available facilitating easy
comparisons. The symplectic map corresponding to the pendulum Hamiltonian
was obtained and truncated at the eighth order. This map was then refactorized
using jolt maps. Using this for integration, results which agreed well
with the exact results were obtained. In contrast, a non-symplectic
eighth order integration method was shown to give rise to spurious chaotic
behaviour. This demonstrated the advantages of using the symplectic
integration method outlined above.

Using solvable and polynomial symplectic maps, symplectic
integration algorithms which can be easily generalized to higher dimensions
were also obtained.

**Representations of
Sp(6,R) and SU(3)**** **

In the study of nonlinear Hamiltonian dynamics, the
real symplectic group Sp(2n,R) and its compact subgroups play an important
role. Quite often, one studies the single particle dynamics of nonlinear
Hamiltonian systems. Since this has three degrees of freedom, the relevant
group is Sp(6,R). Moreover, the equations of motion are formulated in
terms of phase space variables (generalized coordinates and momenta).
In particular, in Lie perturbation theory of Hamiltonian dynamics, homogeneous
polynomials of phase space variables play a central role. Therefore,
it is important to study the representations of Sp(6,R) carried by these
polynomials. Further, in deriving symplectic integration algorithms
for Hamiltonian systems (which was discussed in the previous section),
representations of compact subgroups of Sp(6,R) (especially SU(3)) carried
by homogeneous polynomials are required. They may also be useful in
deriving metric invariants for symplectic maps.

For the above reasons, we studied representations
of Sp(6,R) and SU(3) carried by homogeneous polynomials of phase space
variables. It was proved that homogeneous polynomials of degree m carry
the irreducible representation (m,0,0) of both the Lie algebra sp(6,R)
and the corresponding Lie group Sp(6,R). Relations between representations
of sp(6,R) and Sp(6,R) were established. Under the action of SU(3),
the representations obtained above are reducible and can be written
as a direct sum of irreducible representations of SU(3). Finally, explicit
expressions for SU(3) states within these representations were given
in terms of phase space variables. Such expressions are useful in symplectic
integration theory.

We also obtained a rather unconventional real basis
for the symplectic algebra sp(2n,R). We demonstrated the utility of
this basis for practical computations by giving a simple derivation
of the second and fourth order indices of irreducible representations
of sp(2n,R).

**Fractal Analysis in
Geophysics**

Understanding climatic dynamics and variability is
an important problem in geophysics. One can attempt to solve this problem
by developing elaborate climate models and studying them numerically.
This requires tremendous amount of computer time and effort. Therefore,
it would be good to have some simple measures of climatic variablity
which can be used to understand the underlying dynamics in a gross sense
and to exhibit linkages between different climatic dynamics.

We proposed a novel climate predictability index.
It quantifies, in a simple manner, the predictability of three major
components comprising the climate - temperature, pressure and precipitation.
The quantification was done using a fractal dimensional analysis of
the corresponding time series as described below. It is well known that
a geophysical time series can be modelled as a fractional Brownian motion.
Using this fact, it is possible to apply rescaled range (R/S) analysis
to the time series and calculate its so-called Hurst exponent. This
Hurst exponent is related in a simple manner to the fractal dimension
of the time series curve. Fractal dimensions close to 1.5 and 1.0 correspond
to low and high predictability respectively. This correspondence was
exploited to define a predictability index. The climate predictability
index was calculated for stations in India using the Global Historical
Climatology Network dataset. Change in the index with the seasons suggested
a strong influence of more than one climatic dynamics. In such cases,
calculations done using mean yearly data were shown to be suspect. The
index was shown to be useful in studying the interplay between various
climatic components (viz. temperature, pressure and precipitation).
Changes in predictability indices for temperature and pressure were
seen to affect the index for precipitation. The index can be used as
a discriminant for determining which stations are selected for use in
developing regional climatic models.

**Generalized Replicator Dynamics**
One of the striking phenomena exhibited by a wide
variety of complex adaptive systems is that individual agents or components
of the system evolve to perform highly specialized tasks, and at the
same time the system as a whole evolves towards a greater diversity
in terms of the kinds of individual agents or components it contains
or the tasks that are performed in it. Some examples of this include
living systems which have evolved increasingly specialized and diverse
kinds of interacting protein molecules, ecologies which develop diverse
species with specialized traits, early human societies which evolve
from a state where everyone shares in a small number of chores to a
state with many more activities performed largely by specialists, and
firms in an economic web that explore and occupy increasingly specialized
and diverse niches. Since the above phenomenon is so widespread, it
would be useful to have a mathematical model which captures this.

We have proposed a mathematical model of economic
communities that exhibits these twin evolutionary phenomena of specialization
and diversity. The community consists of N interacting agents. Each
agent's activity consists of an individual mix out of a fixed set of
strategies. Agents receive a payoff depending on activities of the entire
community, and independently modify their own mix of strategies to increase
their payoff. The dynamics obtained is a generalized replicator dynamics
and consists of a set of coupled nonlinear ordinary differential equations
describing the time evolution of the activities of individual agents.
We proved certain theorems and obtained numerical results for the attractors
of the system. We focused our attention on attractors that have the
property of individual specialization and global diversity. Under certain
conditions, the desired attractors were shown to exist and have basins
of attraction that cover the entire configuration space. Thus the evolutionary
phenomena mentioned above occur generically in the model. We have described
properties a new strategy should have (in the context of already existing
activities) in order for it to be "accepted" by the community. Thus
the model suggests a natural mechanism for the emergence of context
dependent innovations in the community. It was also shown that under
certain generic conditions on the payoff matrix parameters, the agents
exhibit a collective behaviour, and for sufficiently large N, the community
exhibits diversity and self-organization.

**Evolutionary Games with Two Time Scales**
One of the classical problems in game theory and economics
is that of equilibrium selection. In a typical situation, one has many
candidate equilibria and the problem of selecting the right one has
to be solved. One approach towards this problem is to construct dynamic
models of disequilibrium wherein the desired equilibrium arises as an
attractor of the proposed dynamics. The dynamics are intended to model
economic agents in an interactive environment where they `learn' from
experience and adapt their strategy in real times. Exisitng models fall
into two distinct classes representing two different modes of learning.
In the first class of models, one retains the identity of individual
agents in a finite collection and postulates adaptation rules for individual
learning which lead to the aggregate behaviour being studied. In the
second class of models, the population itself adapts by incrementally
reinforcing or discouraging (implicitly, through birth-death mechanisms)
the strategies that give respectively higher and lower payoffs than
the current average. There are, however, no models which combine both
these features even though this is perfectly acceptable in economic
situations.

We proposed a new two tier model of learning. In this
model, active or `ontogenetic' learning by individuals takes place on
a fast time scale and passive or `phylogenetic' learning by society
as a whole on a slow time scale, each affecting the evolution of the
other. The former is modelled by the Monte Carlo dynamics of statistical
physics where the individual updates his strategy to pick moves that
yield higher payoffs with higher probability, retaining at the same
time `expensive' moves, albeit with a low probability. The latter may
be attributed to random exploration or errors. We superimpose on this,
on a slower time scale, the replicator dynamics, describing the evolution
of a conglomeration of economic agents under selection pressure from
the environment (which may include other such conglomerations). The
two dynamics interact through the payoff matrices that enter their velocity
fields. This leads to a two time scale evolution, which can then be
cast in the framework of singularly perturbed ordinary differential
equations. The Monte Carlo dynamics for ontogenetic learning was analyzed
in some detail, particularly for the n=2 symmetric case. We carried
out numerical simulations which showed a rich variety of qualitative
behaviour. In some cases, the two dynamics counter act each other so
as to have the system poised in between two distinct kinds of equilibria,
reminiscent of (but distinct from) self-organized criticality. In other
cases, under similar situations, one obtains sustained oscillations
between two equilibria, a phenomenon reminiscent of (but distinct from)
noise-induced transitions on one hand and heteroclinic cycles on the
other.

**Moment Invariants**
We studied the behaviour of the moments of a particle
distribution as it is transported through a Hamiltonian system. Functions
of moments that remain invariant for an arbitrary linear Hamiltonian
system were constructed using Lie algebraic techniques. Several new
invariants were obtained. We have also constructed dynamic moment invariants
for nonlinear Hamiltonian systems.

**Time Series Analysis in Geophysics**
We analyzed multi-proxy climatic records from the
74KL marine core and compare it with the oxygen isotope record from
the Guliya ice core. These records have important implications for the
evolution of the Southwest Monsoon System. We observed three distinct
climatic events (viz. 19 ka to 13.5 ka, 13.5 ka to 10 ka and 3.4 ka
to present) from the last glacial phase to the present. Even though
the first two events are well documented in the literature, our combined
analysis of data from both the cores suggested alternative mechanisms
which could have complemented and strengthened these events. The third,
most recent event (occurring between 3.4 ka to present) does not seem
to have been well documented earlier. We proposed a possible mechanism
to explain this event. We identified representative elements which better
capture the above events as compared to more commonly used elements.

We studied paleoclimatic records from various sites
spread around the earth, focusing on the start of the last glacial-interglacial
transition. The warming, as recorded in the oxygen isotope record started
first in the tropics, then propagated to the Antarctic and then finally
to the Arctic. Our analysis of the data suggested that it took about
7.6 ka for onset of climate change to propagate globally. We proposed
that the tropical Pacific played a major role in initiating the warming
in the tropics. We discussed mechanisms that could have transported
this heat from the tropics to Antarctica and then to the Arctic during
transition to the interglacial.

**Free-electron Laser (FEL) Theory **
We proposed a novel type of free-electron laser (called
the standing-wave free-electron laser) for use as a power source in
generating high accelerating gradients in next generation accelerators.
Theoretical behaviour of this laser was studied. Stability of the system
when perturbed was investigated both analytically and numerically. These
studies established that the proposed free-electron laser was stable
and could be built as a practical power source.

We proposed a novel type of FEL as a practical device
to generate brief, intense pulses of radiation. This FEL (which we called
the multi-cavity FEL) has many optical cavities where the cavities communicate
with each other, not optically, but through the electron beam. The idea
is to simply make the FEL optical cavities sufficiently short that the
slippage length, in any one optical cavity, is less than the electron
pulse width. When electrons reach the end of one cavity they go into
the next, but the radiation remains trapped within that cavity. We also
studied a mathematical model of this FEL both analytically and numerically.
In the one-dimensional approximation, a general expression for the growth
rate in the exponential (high current) regime was obtained. In the regime
where lethargy is important, expressions were given in the two opposite
limits of small and large number of cavities and bunches. Extensive
numerical investigations of the nonlinear effects were performed. The
multi-cavity FEL was shown to be useful to avoid slippage phenomena,
and thus to make pico-second pulses of infra-red radiation. Some examples
of this application were studied.

In the wigglers of future FELs, the electron beam
will be required to travel over a length of 10m or more in pipes with
small diameters. Transverse resistive wall effects could lead to beam
breakup during this transport. To investigate this possibility, we solved
the equation of motion for a bunched beam analytically. We showed that
a steady state solution is reached for times larger than the diffusion
time. This solution can either oscillate or grow exponentially with
the length of the pipe, depending on the relative magnitudes of the
resistive wall effect and the focusing force in the wiggler. The possibility
of a significant transient was also studied in great detail.