IMSc Webinar

Characterization Of Periodic Orbits In Open Nonlinear Dynamical Systems

Sandip Saha

SNBNCBS Kolkata

A periodic orbit in an open nonlinear dynamical system described by two-dimensional
ordinary differential equation stemming from various phenomenological sources is one of the most
important motivations to study nonlinear dynamics. In the multi-scale perturbative treatment
(e.g., Renormalisation Group and Krylov–Bogolyubov) of dynamical systems, the amplitude
and phase of the oscillation get renormalized and suitable for several nonlinear feedback systems
including self-sustained chemical and biological oscillations, namely, a cyclic steady state in cell
division, Circadian oscillation, and Calcium oscillations. Such periodic orbits can be isochronous
or limit cycle, and in some special cases, they become center. However, a general prescription of
shape, size, and the number of stable limit cycles in a given system are not yet well established.
From the physical point of view, the response properties of a limit cycle due to parametrically
excited by an external field is also ill-understood. As the common underlying thread in this
study, we explore a class of open natural dynamical systems for the characterization of various
periodic orbits. As the multiple limit cycles in a given system is an important knotty issue, we
have investigated on the counting of limit cycles and its application in systematic construction
of multi-rhythmic oscillators.