TITLE:
On Existence of Periodic Solutions of Certain Second Order Nonlinear Ordinary Differential Equations via Phase Portrait Analysis
AUTHORS:
Olaniyi S. Maliki, Ologun Sesan
KEYWORDS:
ODE, Stability, Periodic Solutions, Limit Cycles, MathCAD Solution
JOURNAL NAME:
Applied Mathematics,
Vol.9 No.11,
November
20,
2018
ABSTRACT: The global phase portrait describes the qualitative behaviour of the solution
set of a nonlinear ordinary differential equation, for all time. In general, this
is as close as we can come to solving nonlinear systems. In this research work
we study the dynamics of a bead sliding on a wire with a given specified
shape. A long wire is bent into the shape of a curve with equation z = f (x)
in a fixed vertical plane. We consider two cases, namely without friction and
with friction, specifically for the cubic shape f (x) = x3−x . We derive the
corresponding differential equation of motion representing the dynamics of
the bead. We then study the resulting second order nonlinear ordinary differential
equations, by performing simulations using MathCAD 14. Our main
interest is to investigate the existence of periodic solutions for this dynamics
in the neighbourhood of the critical points. Our results show clearly that periodic
solutions do indeed exist for the frictionless case, as the phase portraits
exhibit isolated limit cycles in the phase plane. For the case with friction, the
phase portrait depicts a spiral sink, spiraling into the critical point.