Courant Institute of Mathematical Sciences

New York University

**Email:**dass@cims.nyu.edu**Phone:****Office:**Warren Weaver Hall, Room 925

**School:**Don Bosco Park Circus, 2011**BTech, Electrical engineering:**Indian Institute of Technology, Kharagpur, India, 2011**MA in Mathematics:**Department of Mathematics, University of Maryland, College Park, USA, 2014.**PhD in Mathematics:**Department of Mathematics, University of Maryland, College Park, USA, December 2015.**Post-doctoral research associate:**Department of Mathematics, University of Maryland, College Park, USA, December 2016.**Currently :**Post-doctoral research associate, working with Dr. Dimitris Giannkis at Courant, September 2016 – present.

**Characterizing the spectrum of Koopman operators.**Associated to every discrete or continuous time dynamical system is the Koopman semi-group. Koopman operators act on observables by time-shifts. These operators provide an alternative to studying the dynamical systems as a linear operator instead of a nonlinear system on a manifold. The spectral properties of is directly related to ergodic properties, like mixing, rates of mixing, ergodicity etc. By the spectral representation of unitary maps, the Koopman operator is unitarily equivalent to a multiplication operator on L^{2}(S^{1}, &nu), where &nu is the*spectral measure*. The measure splits into an atomic, Lebesgue-absolutely continuous and singular continuous parts. The Lebesgue-absolutely continuous part has a density &rho and represents the continuous spectrum of the dynamics. So far, there has been no general approach towards obtaining this spectral density. Here are some possibilities I am exploring with Dimitris Giannakis.**The generator.**Give a time flow, the Koopman semi-group U^{t}is a strongly continuous unitary semi-group and hence has a generator V. If the flow was generated from a vector field, V coincides with this field in its action on smooth functions. V is an unbounded, skew-Hermitian operator which leads to many numerical challenges, but in certain situations, there are methods to obtain its eigenvalues. See our papers which use Galerkin methods for a smooth attractor , and one using delay coordinates in a more general setting.**Singular value decomposition (SVD).**The idea is to come up with a subspace H of L^{2}and give it an inner-product structure, so that restricted to H, V is a compact operator. The SVD of compact operators on Hilbert spaces provide a lot of information, for example, enable functional calculus with the operator.**Restricting to time-averaged sub-space.**Time averaging using the Koopman semi-group produces a smaller subspace on which V is a bounded operator. Read this for more details.**Using kernel methods.**

**Data driven approach.**A data-driven approach means inferring properties of the system from a time-series, which is typically generated from some observation map, which could be of dimension much smaller than the underlying system. Data-driven methods fall under the category of "non-parametric methods" for investigating dynamical systems, since one does not have a model to start with, or parameters to tune/fit. See for example the papers on Koopman spectra mentioned above, or a paper with Y Saiki, E Sander and J Yorke on a robust data-driven method for computing rotation numbers for invariant torii. Proving that data-driven methods work require showing that operators of various kind converge in some sense and in some functional space. These convergence properties are often liked to the spectral properties of the dynamics / Koopman operator and is an interesting avenue of research. Many of our results have a data-driven realization and sometimes the methods seem to work under conditions weaker than those stated in our results.**A stronger concept of genericity.**The underlying principle of all data-driven methods is that the data is “typical”, i.e., arises from a generic point. Given a measure &mu, a point x is said to be &mu-generic if the average of the Dirac-delta measures on the trajectory starting at x converge strongly to &mu. The Birkhoff ergodic theorem guarantees that if &mu is an invariant measure, then &mu-almost every point is generic. Additional assumptions, like physical or SRB measures, guarantee that there are points outside the support of the ergodic measure which are also generic. However, the concept of genericity needs to be generalized to take into account exponentially weighted ergodic averages. For &mu-a.e. point, these converge in the L^{2}-sense to a projection onto an eigenspace. These averages arise very frequently in spectral analysis of dynamical systems, or of signals. Here is a more detailed description of the problem.**Studying the discrete spectra**The other component of the spectrum of the Koopman operator is the discrete spectrum, the closed subspace spanned by the Koopman eigenfunctions. Koopman eigenfunctions represent repeating spatio-temporal patterns of the dynamics and has many uses, both in the theoretical understanding and numerical investigation into the dynamics. The eigenvalues are countable in number and form a subgroup of S^{1}.**Smoothness of Koopman eigenfunctions.**The Koopman operator can be defined on L^{2}space, which contains the space C^{0}. To the best of my knowledge, very little is known or studied about the smoothness of eigenfunctions of a smooth dynamical system. For example, an ergodic flow with a fixed point cannot have a continuous Koopman eigenfunction. Often, an ergodic component of the dynamics has a fractal structure and no notion of smoothness can be ascribed to it. I am also interested in the question of whether for these systems, the Koopman eigenfunction can be continuously extended to a neighborhood of the ergodic component.**Delay coordinate embeddings.**In one of our recent works, we show that if one builds a kernel on the attractor by incorporating Q delays, then in the limit of Q going to infinity, the resulting integral operator annihilates the continuous spectrum and commutes with the Koopman operator. One useful consequence is that the eigenfunctions of the integral operator are also Koopman eigenfunctions. These integral operators can be well approximated by data-driven methods and we demonstrate spectral convergence using our methods.

**Multi-chaos.**One feature of chaotic maps or flows in high dimensions is "unstable dimensional variability", in which there are periodic points whose unstable manifolds have different dimensions. There are many mechanisms in the mathematical literature which describe how this could happen robustly, most notably, via 'blenders'. I and my advisor Jim Yorke described some very simple conditions under which multi-chaos could occur, and presented a 2-dimensional paradigm for multi-chaos which also work in higher dimensions. One of the key ingredients is the presence of a quasiperiodic orbit. We are currently interested in extending it to other manifolds and characterizing the robustness of our conditions in a measure theoretic sense.**Random circle-maps.**In many dynamical systems like in skew-product systems, there are co-existing torii or curves which get mapped into each other under the dynamics. If the dynamics is parameterized by a parameter t, then one gets a family of circle diffeomorphisms f^{t}on each of the invariant curves. Then for each n, each circle gets mapped into another though a composition of n circle diffeomorphisms. Our work in multi-chaos got us interested in characterizing the probability of any of these composition maps being irrational rotations. This requires two considerations, firstly, one needs to study how for each circle, the parameter-range splits into periodic windows, the measure of these windows, and the measure of the intersection of these periodic windows. The first has been addressed in this paper, an ongoing work aims at showing that under certain assumptions, this intersection has Lebesgue measure 0 in the parameter space. Secondly, one needs to be able to study the growth of C1 norm under these compositions. To this end, one might treat the circle diffeomorphisms as a family of random circle diffeomorphisms.**Escape from almost-invariant sets.**In a parameterized family of maps on a manifold, a forward invariant set R with non-empty interior may be created at some critical value of the parameter. Before the parameter attains this critical value, the map is topologically transitive but the average escape time of trajectories from R approaches infinity as the parameter approaches the critical value. I am interested in determining the asymptotic rate at which this quantity approaches infinity. This involves studying how the dynamics of the map determines a Rohlin-tower like structure within the set.

- S Das. Universal Bound on the Measure of Periodic Windows of Parameterized Circle Maps,
*Topology Proc.,**(2018),**52, 179-187* - S Das, J Yorke. Super convergence of ergodic averages for quasiperiodic orbits,
*Nonlinearity,**(2017)* - S Das, J Yorke. Multi-chaos from Quasiperiodicity,
*SIAM J. App. Dyn.*(2017), to appear - S Das, Y Saiki, E Sander, J Yorke. Quantitative Quasiperiodicity,
*Nonlinearity*(2017), 30.11, 4111 - S Das, et. al. Measuring quasiperiodicity,
*Europhysics Lett.*(2016), 114.4, 40005-40012 - S Das, J Yorke. Quasiperiodicity:Rotation numbers,
*The Foundations of Chaos Revisited: From Poincare to Recent Advancements*(2016), 23-37. - S Das. Dense saddles in torus maps,
*Topology Proceedings*(2015), 47, 177-190. - S Das, J Yorke. Avoiding extremes in chaotic systems,
*J. Diff. Eq. App.*(July 2015), 22.2, 217-234. - S Das, S R Chowdhury, H Saha. Accuracy enhancement in a fuzzy expert decision making system through appropriate determination of membership functions and its application in a medical diagnostic decision making system,
*Journal of medical systems*(2012), 36.3, 1607-1620.

- D Giannakis, S Das. Extraction and Prediction of Coherent Patterns in Incompressible Flows through Space-Time Koopman Analysis
- S Das, D Giannakis. Delay-coordinate maps and the spectra of Koopman operators "
- S Das, Y Saiki, E Sander, J Yorke. Solving the Babylonian Problem of quasiperiodic rotation rates
- S Das. Almost-sure quasiperiodicity in countably many co-existing circles