Oscillators

Oscillators play a trivial role in the domain of stability analysis as they epitomize periodic nonlinear models and embody effects like Hysteresis and Phase Drift. In contrast to Flows on the Line, Periodic Solutions arise commonly, but can’t coexist with unique solutions.

Uniform Oscillators

The simplest system is defined by a constant velocity

$$\dot{\theta }=\omega$$

The general solution to the system is

$$\theta =\omega t+\theta \_0$$

The period, $T$ for $\theta$ to change by $2\pi$ is

$$T=\frac{2\pi }{\omega }=\frac{2\pi }{\omega \_1-\omega \_2}$$

The second equation models the period that ${\dot{\theta }}_1={\omega }_1$ requires to lap ${\dot{\theta }}_2={\omega }_2$ in moving around the circle.

Nonuniform Oscillators

This model equation occurs in many important principles of central disciplines

$$\dot{\theta }=\omega -a\sin \theta$$

Finding the time period involves a workflow of the following kind

$$T=\int dt=\int \_0^{2\pi }\frac{dt}{d\theta }d\theta =\int \_0^{2\pi }\frac{d\theta }{\omega -a\sin \theta }=\frac{2\pi }{\sqrt{{\omega }^2-a^2}}$$

Bottlenecks

In general, the solution $\theta$ spends most of its time in a slow bottleneck around $\frac{\pi }{2}$ and is fastest where $\theta \approx \frac{3\pi }{2}$. The time that is required to move through the bottleneck can be approximated, by thinking of the minimum of the sine wave (the bottleneck region) as quadratic function $\dot{x}=r+x^2$. The period of the bottleneck is the time required to pass through it:

$$T\_bn=\int \_{-\infty }^{\infty }\frac{dx}{r+x^2}=\frac{\pi }{\sqrt{r}}$$