- Computational Neuroscience, Dynamical Systems, Chaos Theory, Turbulence, Stability and Chaos, Fluid Dynamics, and 17 moreBifurcation, Turbulence modeling, Low Dimensional Systems, Chaos/Complexity Theory, Nonlinear dynamics, Fractal Dimension, Robustness, Fractals, Laminar to Turbulence Transitions, Logistic Map, Stability, Turbulent Flows, Turbulence Modelling, Bifurcation Analysis, Perturbation Analysis, Chaotic Dynamics, and Dynamical systems and Chaosedit
- I am currently a Staff Data Scientist at Migo. Previously, I was a Courant Instructor and a Swartz Postdoctoral Fellow at New York University. I did my PhD in Applied Mathematics (Dynamical Systems and Chaos), after 5 wonderful years of ... moreI am currently a Staff Data Scientist at Migo. Previously, I was a Courant Instructor and a Swartz Postdoctoral Fellow at New York University. I did my PhD in Applied Mathematics (Dynamical Systems and Chaos), after 5 wonderful years of research at the University of Maryland College Park. My PhD thesis adviser was Prof. James A. Yorke and co-advisers were Prof. Ed. Ott and Prof. Ulrike Feudel. Before that, I finished my Dual Degree (Bachelors and Masters of Technology) in Aerospace Engineering (specializing in Dynamics and Control) at the Indian Institute of Technology Bombay.edit
Reliable signal transmission represents a fundamental challenge for cortical systems, which display a wide range of weights of feedforward and feedback connections among heterogeneous areas. We re-examine the question of signal... more
Reliable signal transmission represents a fundamental challenge for cortical systems, which display a wide range of weights of feedforward and feedback connections among heterogeneous areas. We re-examine the question of signal transmission across the cortex in network models based on recently available mesoscopic, directed- and weighted- inter-areal connectivity data of the macaque cortex. Our findings reveal that, in contrast to feed-forward propagation models, the presence of long-range excitatory feedback projections could compromise stable signal propagation. Using population rate models as well as a spiking network model, we find that effective signal propagation can be accomplished by balanced amplification across cortical areas while ensuring dynamical stability. Moreover, the activation of prefrontal cortex in our model requires the input strength to exceed a threshold, in support of the ignition model of conscious processing, demonstrating our model as an anatomically-real...
Research Interests:
Period-doubling cascades are among the most prominent features of many smooth one-parameter families of maps, F : R×M → M, where M is a locally compact manifold without boundary, typically RN. In particular, we investigate F(μ, ·) for μ ∈... more
Period-doubling cascades are among the most prominent features
of many smooth one-parameter families of maps, F : R×M → M,
where M is a locally compact manifold without boundary, typically RN.
In particular, we investigate F(μ, ·) for μ ∈ J = [μ1, μ2], when F(μ1, ·)
has only finitely many periodic orbits while F(μ2, ·) has exponential
growth of the number of periodic orbits as a function of the period. For
generic F, under additional hypotheses, we use a fixed point index argument
to show that there are infinitely many “regular” periodic orbits at
μ2. Furthermore, all but finitely many of these regular orbits at μ2 are
tethered to their own period-doubling cascade. Specifically, each orbit
ρ at μ2 lies in a connected component C(ρ) of regular orbits in J ×M;
different regular orbits typically are contained in different components,
and each component contains a period-doubling cascade. These components
are one-manifolds of orbits, meaning that we can reasonably say
that an orbit ρ is “tethered” or “tied” to a unique cascade. When F(μ2)
has horseshoe dynamics, we show how to count the number of regular
orbits of each period, and hence the number of cascades in J ×M.
As corollaries of our main results, we give several examples, we
prove that the map in each example has infinitely many cascades, and
we count the cascades.
of many smooth one-parameter families of maps, F : R×M → M,
where M is a locally compact manifold without boundary, typically RN.
In particular, we investigate F(μ, ·) for μ ∈ J = [μ1, μ2], when F(μ1, ·)
has only finitely many periodic orbits while F(μ2, ·) has exponential
growth of the number of periodic orbits as a function of the period. For
generic F, under additional hypotheses, we use a fixed point index argument
to show that there are infinitely many “regular” periodic orbits at
μ2. Furthermore, all but finitely many of these regular orbits at μ2 are
tethered to their own period-doubling cascade. Specifically, each orbit
ρ at μ2 lies in a connected component C(ρ) of regular orbits in J ×M;
different regular orbits typically are contained in different components,
and each component contains a period-doubling cascade. These components
are one-manifolds of orbits, meaning that we can reasonably say
that an orbit ρ is “tethered” or “tied” to a unique cascade. When F(μ2)
has horseshoe dynamics, we show how to count the number of regular
orbits of each period, and hence the number of cascades in J ×M.
As corollaries of our main results, we give several examples, we
prove that the map in each example has infinitely many cascades, and
we count the cascades.
Research Interests:
The character of the time-asymptotic evolution of physical systems can have complex, singular behavior with variation of a system parameter, particularly when chaos is involved. A perturbation of the parameter by a small amount ϵ can... more
The character of the time-asymptotic evolution of physical systems can have complex, singular behavior
with variation of a system parameter, particularly when chaos is involved. A perturbation of the parameter
by a small amount ϵ can convert an attractor from chaotic to nonchaotic or vice versa. We call a parameter
value where this can happen ϵ uncertain. The probability that a random choice of the parameter is ϵ
uncertain commonly scales like a power law in ϵ. Surprisingly, two seemingly similar ways of defining this
scaling, both of physical interest, yield different numerical values for the scaling exponent. We show why
this happens and present a quantitative analysis of this phenomenon.
with variation of a system parameter, particularly when chaos is involved. A perturbation of the parameter
by a small amount ϵ can convert an attractor from chaotic to nonchaotic or vice versa. We call a parameter
value where this can happen ϵ uncertain. The probability that a random choice of the parameter is ϵ
uncertain commonly scales like a power law in ϵ. Surprisingly, two seemingly similar ways of defining this
scaling, both of physical interest, yield different numerical values for the scaling exponent. We show why
this happens and present a quantitative analysis of this phenomenon.
Research Interests:
For a smooth dynamical system xn+1 = f (C,xn) (depending on a parameter C), there may be infinitely many periodic windows, that is, intervals in C having a region of stable periodic behaviour. However, the smaller of these windows are... more
For a smooth dynamical system xn+1 = f (C,xn) (depending on a parameter
C), there may be infinitely many periodic windows, that is, intervals in C having
a region of stable periodic behaviour. However, the smaller of these windows
are easily destroyed with tiny perturbations, so that only finitely many of the
windows can be detected for a given level of noise. For a fixed perturbation
size epsilon, we consider the system behaviour in the presence of noise. We look
at the ‘epsilon-robust windows’, that is, those periodic windows such that for the
superstable parameter value C in that window, the general periodic behaviour
persists despite noise of amplitude < epsilon. We focus on the quadratic map,
and numerically compute the number of periodic windows that are epsilon-robust.
We obtain a robustness exponent α = 0.51 ± 0.03, which characterizes the
robustness of periodic windows in the presence of noise.
C), there may be infinitely many periodic windows, that is, intervals in C having
a region of stable periodic behaviour. However, the smaller of these windows
are easily destroyed with tiny perturbations, so that only finitely many of the
windows can be detected for a given level of noise. For a fixed perturbation
size epsilon, we consider the system behaviour in the presence of noise. We look
at the ‘epsilon-robust windows’, that is, those periodic windows such that for the
superstable parameter value C in that window, the general periodic behaviour
persists despite noise of amplitude < epsilon. We focus on the quadratic map,
and numerically compute the number of periodic windows that are epsilon-robust.
We obtain a robustness exponent α = 0.51 ± 0.03, which characterizes the
robustness of periodic windows in the presence of noise.
Research Interests:
We investigate the geometry of the edge of chaos for a nine-dimensional sinusoidal shear flow model and show how the shape of the edge of chaos changes with increasing Reynolds number. Furthermore, we numerically compute the scaling of... more
We investigate the geometry of the edge of chaos for a nine-dimensional sinusoidal shear flow model and show
how the shape of the edge of chaos changes with increasing Reynolds number. Furthermore, we numerically
compute the scaling of the minimum perturbation required to drive the laminar attracting state into the turbulent
region. We find this minimum perturbation to scale with the Reynolds number as Re^{−2}.
how the shape of the edge of chaos changes with increasing Reynolds number. Furthermore, we numerically
compute the scaling of the minimum perturbation required to drive the laminar attracting state into the turbulent
region. We find this minimum perturbation to scale with the Reynolds number as Re^{−2}.
Research Interests:
Attracting chaotic behaviour in dynamical systems is often sensitive to small changes in parameters. If a perturbation in the parameter by a tiny amount ε can change the asymptotic behaviour of the system from being chaotic to being... more
Attracting chaotic behaviour in dynamical systems is often sensitive to small
changes in parameters. If a perturbation in the parameter by a tiny amount
ε can change the asymptotic behaviour of the system from being chaotic to
being periodic, we call it parameter value ε-uncertain. Here, using a selfsimilar
model of the intricate, intertwined parameter-space structure of the
chaotic and periodic attractors, we investigate the scaling of this uncertainty
with ε. We show that as ε approaches 0, the great majority of ε-uncertain
parameters lie in high order windows, that is, windows within windows within
windows …. The expected value of the order of the highest order window
containing this parameter approaches infinity as ε goes to zero.
changes in parameters. If a perturbation in the parameter by a tiny amount
ε can change the asymptotic behaviour of the system from being chaotic to
being periodic, we call it parameter value ε-uncertain. Here, using a selfsimilar
model of the intricate, intertwined parameter-space structure of the
chaotic and periodic attractors, we investigate the scaling of this uncertainty
with ε. We show that as ε approaches 0, the great majority of ε-uncertain
parameters lie in high order windows, that is, windows within windows within
windows …. The expected value of the order of the highest order window
containing this parameter approaches infinity as ε goes to zero.