Abstract:
Our aim is to avoid the need for calculating explicitly solutions to ODE's; we will be focusing
on determining qualitative features of these solutions. The approach is graphical and geometric and
provides a description of the solutions behavior, which allow us to understand the phenomena captured
in the modeling in a pictorial form.
To introduce the phase plane, consider the system
𝑚𝑥" + 𝑘𝑥 = 0 (1)
Governing the free oscillation of the simple harmonic mechanical oscillator shown in the figure 1. Of
course we can readily solve (1) and obtain the general solution
𝑥 𝑡 = 𝐶1 cos 𝜔𝑡 + 𝐶2 sin 𝜔𝑡 𝑤𝑒𝑟𝑒 𝜔 = 𝑘
𝑚
is the natural frequency or equivalently,
𝑥 𝑡 = 𝐴 sin(𝜔𝑡 + β
) , (2)
Where A and Γ are the amplitude and phase angle, respectively. To present this result graphically, one
can plot x verses t and obtain any number of sine waves of different amplitude and phase, but let us
proceed differently.
Figure 1. Simple harmonic Mechanical oscillator
We begin by re-expressing (1), equivalently, as the system of first βorder equations
𝑑𝑥
𝑑𝑡 = 𝑦, (3a)
𝑑𝑦
𝑑𝑡 = β
𝑘 𝑚
𝑥 (3b)
