Doctoral Research

Aspects of Temporal Perturbation in Dynamical Systems

An inverted Kapitza pendulum, stabilized by a vibrating pivot — the thesis’s opening model

Overview

My PhD, completed in January 2019 at the Department of Physical Sciences, IISER Kolkata, under Prof. Soumitro Banerjee, explored how time-dependent perturbations reshape the behavior of nonlinear dynamical systems — from a single mechanical pendulum to networks of coupled oscillators, superconducting junctions, and the evolution of the universe itself.

The unifying question across the thesis: when a system is periodically or randomly perturbed in time, does it settle into a new, stable state that wouldn’t exist otherwise — and what governs that transition?

What the Thesis Covered

Pendulum with a vibrating suspension point (Kapitza pendulum)

A vertically vibrating pivot can stabilize a pendulum in its upside-down position — normally the least stable configuration possible. I derived the effective potential governing this “dynamic stabilization,” mapped how damping and coupling shift the basin of stability, and extended the analysis to interacting pairs of pendula.

Networks of coupled inverted pendula

Extending the single-pendulum result to many-body systems — 1D chains, 2D lattices, and fully-connected (all-to-all) networks — to see how network topology itself affects whether the stabilized state survives.

Hamiltonian Mean-Field model with time-varying coupling

In a system with long-range interactions, I showed that periodically modulating the coupling strength shifts the critical point of the system’s phase transition — the shift depends predictably on the frequency and amplitude of the modulation.

Noise-induced transitions in a Josephson junction

Introduced a temporal (noisy) modulation into a Josephson junction with both fundamental and second-harmonic current-phase terms, showing that the normally-unstable ϕ = π state can become stabilized — with direct relevance to real superconducting circuit behavior.

Dynamical systems analysis in cosmology

Applied the same dynamical-systems toolkit (phase space, stability analysis) to phantom dark energy models under five different scalar-field potentials — results consistent with current cosmological observations.

Mathematical Background

Nonlinear Dynamical Systems Bifurcation Theory Linear Stability Analysis Perturbation Theory Statistical Mechanics of Long-Range Systems Stochastic Processes Coupled Oscillator / Network Theory General Relativity & Cosmology (FRW Metric) Condensed Matter Physics

Technical Skills Earned

Numerical ODE Simulation Molecular Dynamics Simulation Stochastic Simulation Bifurcation & Stability Mapping Scientific Visualization LaTeX Independent Research & Publication

Publications from the Thesis

  1. Noise induced transition in Josephson junction with second harmonic. Eur. Phys. J. B (2018) 91: 13
  2. Hamiltonian mean field model: effect of temporal perturbation in coupling matrix. (with Soumen Patra) Mod. Phys. Lett. B 32, 1850147 (2018)
  3. Dynamical systems analysis of phantom dark energy models. (with Nandan Roy) J. Cosmol. Astropart. Phys. 2018, 06 (2018)
  4. Dynamics of a system of coupled inverted pendula with vertical forcing. arXiv:1803.01643

This foundation in nonlinear dynamics, stochastic methods, and large-scale numerical simulation carries directly into my current work in statistical genetics and computational modeling — the mathematics of stability, perturbation, and noise turn out to matter just as much in a genome as in a pendulum.