G.6.a) Undamped oscillators

The differential equation of the undamped oscillator is
           with  .
Here is one such:

To make this second order diffeq into a system of two first order diffeq's, you put
             
and then replace with and replace with :

Clean this up:

Now look at:

Throw in a trajectory:

Look at the corresponding solution :

Another:

And the corresponding solution :

Compare the trajectories on the same axes:

Compare the solution curves :

Play with other starting points and then describe how solutions of the undamped oscillator
          
look.
Why do you think folks call these things undamped oscillators?

G.6.b) Damped oscillators

Here is the diffeq of an undamped oscillator as studied above:

To get a damped version of this oscillator, you throw in a shock absorber term
            
with positive but not too large:

To make this second order differential equation into a system of two first order differential equations, you put
             
and then replace with and replace with :

Clean this up:

Now look at:

Throw in a trajectory:

Look at the corresponding solution :

Another:

Look at the corresponding solution :

Compare the trajectories on the same axes:

Look at the two solution curves:

Play with other starting points and then describe how solutions of the damped oscillator
          
with small and positive, and positive look.
Why do you think folks call these things damped oscillators?

G.6.d.i) The nonlinear van der Pol heartbeat oscillator

You can get an undamped oscillator this way:

You can get a damped oscillator by making positive and small:

You can get a negatively damped (self-excited) oscillator by making negative and small:

Here's how you can get a van der Pol oscillator:

When  , do you expect the van der Pol oscillator to behave like an undamped, a damped or a negatively damped oscillator?

When or , do you expect the van der Pol oscillator to behave like an undamped, a damped or a negatively damped oscillator?

G.6.d.ii)

Another look at a van der Pol oscillator:

To make this second order differential equation into a system of two first order differential equations, you put
             
and then replace by and replace by :

Clean this up:

Now look at:

Cowabunga.
Throw in a trajectory:

Look at the corresponding solution :

Just like heartbeats in E.R.
Another:

And the corresponding solution :

Compare the two trajectories on the same axes:

Folks like to call this a "limit cycle."

See the two solution curves:

Heartbeats.
Look carefully at the flow plot and try to see how the flow forces the limit cycle to happen.