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Talk 01 — Suddhasattwa Das (Department of Mathematics and Statistics, Texas Tech University, USA) — [computing…]Local time: click here to view in your time zoneTitle: Approximating dynamics as zero-noise limits
Abstract
A dynamical system may be defined by a simple transition law - such as a map or a vector field. The objective of most learning techniques is to reconstruct this dynamic transition law. This is a major shortcoming, as most dynamic properties of interest are asymptotic properties such as an attractor or invariant measure. Thus approximating the dynamical law may not be sufficient to approximate these asymptotic properties. I shall present a method of representing a discrete-time deterministic dynamical system as the zero-noise limit of a Markov process. The Markov process approximation is completely data-driven. It provides a low-noise approximation of the dynamics. Simultaneously, it also approximates the invariant set, via the support of its stationary measures. Thus invariant sets of arbitrary dynamical systems, even with complicated non-smooth topology, can be approximated by this technique. Under further assumptions, it will be shown that the technique performs a convergent statistical approximation as well as approximations of true orbits.
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Talk 02 — Swapnil Malegaonkar (Visvesvaraya National Institute of Technology, India) — [computing…]Local time: click here to view in your time zoneTitle: Leveraging Digraphs in Dynamical Systems
Abstract
In this talk, I will discuss the notion of digraphs arising in one-dimensional dynamical systems and their role in understanding the orbit structure of interval maps. The focus will be on how combinatorial information encoded in digraphs can be used to study chaotic behavior and forcing relations among orbit types. In particular, digraphs associated with portions of orbits provide useful insight into the dynamical complexity and chaotic nature of the system.
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Talk 03 — Sourav Bhattacharya (Visvesvaraya National Institute of Technology, India) — [computing…]Local time: click here to view in your time zoneTitle: Mixing patterns of triods
Abstract
We derive linear orderings of the set of natural numbers that captures forcing between mixing patterns of triods.
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Talk 04 — Kaushik Kumar (Visvesvaraya National Institute of Technology, India) — [computing…]Local time: click here to view in your time zoneTitle: A survey on the geometrical structure of Mandelbrot set
Abstract
The modern theory of complex dynamical system was born by the work of mathematicians like Fatou and Julia. The subject gain new revitalisation by the enchanting computer graphics developed by Mandelbrot and the excellent work done by mathematicians like Douady, Hubbard and Sullivan. In this thesis, we will concentrate on the dynamics of complex polynomials. The parameter space of complex degree d polynomials is the space of affine conjugacy classes of these polynomials. The connectedness locus consists of classes of all degree d polynomials f whose Julia set J(f) are connected. The quadratic connectedness locus is called Mandelbrot set M.M has a complicated geometrical structure. One of the holy grail unsolved problems in complex dynamics is to prove the conjecture that Mandelbrot set M is locally connected (popularly called MLC conjecture). The goal of my presentation is to understand and do a survey of the already known geometrical structures of Mandelbrot set, so as to serve as a ready reference for the future research.
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Talk 05 — Ashish Yadav (Department of Mathematics, VNIT Nagpur, India) — [computing…]Local time: click here to view in your time zoneTitle: Triod Twist Cycles and Circle Rotations
Abstract
We study the problem of relating cycles on a \emph{triod} $Y$ to \emph{circle rotations}. We prove that the simplest cycles on a \emph{triod}~$Y$ with a given \emph{rotation number}~$\rho$, called \emph{triod--twist cycles} are conjugate, via a piece-wise monotone map of \emph{modality} at most~$m + 3$, where~$m$ is the \emph{modality} of~$P$ to the rotation on~$S^1$ by angle~$\rho$, restricted to one of its cycles.
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Talk 06 — Akash Pradhan (Department of Mathematics, VNIT Nagpur, India) — [computing…]Local time: click here to view in your time zoneTitle: Cauchy Products and Iteration-Based Functional Equations: Existence and Properties of Solutions
Abstract
In this talk, we provide a brief overview of iterative functional equations and introduce the notion of their Cauchy product. We then discuss recent results on the existence, uniqueness, and stability of solutions to equations of the form \[ \sum_{i=0}^{n} \lambda_i\, f^{i}(x)\, f^{n-i}(x)=F(x), \quad x\in[a,b], \] where $\lambda_i\in\mathbb{R}$ and $F$ is a given function. This framework generalizes the classical iterative root problem corresponding to even $n$ with $\lambda_i=0$ for all $i\neq n/2$. Numerical examples are presented to demonstrate the applicability of the existence results, and the stability of the obtained solutions is also investigated.
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Talk 07 — Richa Jain (Government College Chhapara, Seoni (M.P.), India) — [computing…]Local time: click here to view in your time zoneTitle: Some classical and fractal splines for constrained interpolation and applications in solving differential equations
Abstract
In this paper, we develop constrained classical splines in one and two variables and fractal splines in two variables. First, we obtain a classical rational spline lying above or below the prescribed piecewise line or contained in a prescribed rectangle. Using a blending technique, we construct a classical bivariate rational spline and study its constrained aspects. The proposed constrained splines are exploited for solving constraint differential equations. We construct a bivariate fractal spline as a perturbation of our classical bivariate rational spline proposed in this paper. Further, we investigate suitable conditions on the shape parameters and scaling factors so that the proposed fractal surface lies above or below the given plane, or within a prescribed cuboid. It is observed that our bivariate fractal spline converges to the original function as the mesh norm goes to zero. The proposed theoretical results are verified through numerical examples.