文档介绍：5 Forced oscillators and limit cycles
General remarks
How may we describe a forced oscillator?
The linear equation
β¨ + ρβ˙+ γ2β= 0 (3)
is in general inadequate. Why?
Linearity if β(t) is a solution, then so is β(t), real. This is patible
∞
with bounded oscillations (., βmax < α).
We therefore introduce an equation with
a nonlinearity; and
•
an energy source pensates viscous damping.
•
Van der Pol equation
Consider a damping coefficient ρ(β) such that
ρ(β) > 0 for β large
| |
ρ(β) < 0 for β small
| |
Express this in terms of β2:
β2
ρ(β) = ρ 1
0 β2 −
0
where ρ0 > 0 and β0 is some reference amplitude.
Now, obviously,
2 2
ρ> 0 for β> β0
2 2
ρ< 0 for β< β0
28
Substituting ρ into (3), we get
2 2
d ββ dβ 2
+ ρ 1 + γβ= 0
dt2 0 β2 − dt
0
This equation is known as the van der Pol equation. It was introduced in the
1920’s as a model of nonlinear electric circuits used in the first radios.
In van der Pol’s (um tube) circuits,
high current = positive (ordinary) resistance; and
•∞
low current = negative resistance.
•∞
The basic behavior: large oscillations decay and small oscillations grow.
We shall examine this system in some detail. First, we write it in non-
dimensional form.
We define new units of time and amplitude:
unit of time = 1/γ
•
unit of amplitude = β.