We consider a parametrically-driven nonlinear ODE, which encompasses a simple model of an electronic circuit known as a parametric amplifier, whose linearisation has a zero eigenvalue. By adopting two different approaches, we obtain conditions for the origin to be a global attractor which is approached (a) non-monotonically and (b) monotonically. In case (b), we obtain an asymptotic expression ...

In this paper, by utilizing the fractional calculus theory and computer simulations, dynamics of the fractional order system is studied. Further, we have extended the nonlinear feedback control in ODE systems to fractional order systems, in order to eliminate the chaotic behavior. The results are proved analytically by stability condition for fractional order system. Moreover numerical simulati...

This paper addresses the modeling of nonlinear constitutive relations and hysteresis inherent to piezoceramic materials at moderate to high drive levels. Such models are necessary to realize the full potential of the materials in high performance control applications, and a necessary prerequisite is the development of techniques which permit control implementation. The approach employed here is...

A normal form is derived for Hamiltonian–Hopf bifurcations of solitary waves in nonlinear Schrödinger equations with general external potentials. This normal form is a simple second-order nonlinear ordinary differential equation (ODE) that is asymptotically accurate in describing solution dynamics near Hamiltonian–Hopf bifurcations. When the nonlinear coefficient in this normal form is complex,...

My current research interests include mathematical modelling of physical, chemical and biological phenomena using deterministic and stochastic tools, and using analytical and numerical tools to more fully understand both ODE and PDE in the stochastic setting. The two main thrusts of my research of late include stochastic perturbations of nonlinear dynamical systems, and energy transfer in compl...

In this article, we present a continuation method, which transforms spatially distributed ordinary differential equation (ODE) systems into continuous partial (PDE). We show that can be performed for both linear and nonlinear systems, including multidimensional, space-varying, time-varying systems. When applied to large-scale network, the provides PDE describing evolution of continuous-state ap...

Ordinary differential equations (ODEs) provide a classical framework to model the dynamics of biological systems, given temporal experimental data. Qualitative analysis of the ODE model can lead to further biological insight and deeper understanding compared to traditional experiments alone. Simulation of the model under various perturbations can generate novel hypotheses and motivate the desig...