Phase space

1
Phase space
Whats a phase space diagram for this?
Qualitatively how does this change the phase
space diagram? Lets be quantiative now.
How can we relate velocity to potential and an
initial condition?
2
Van der Pols equation
What system does this look similar to? (assume m
is small and positive)
What happens then?
What does this imply about the phase diagram?
3
Van der Pols equation Phase diagram
It has what we call a limit cycle with stationary
amplitude, an attractor
How might we look at this quantiatively?
4
Van der Pols equation Phase diagram
It has what we call a limit cycle with stationary
amplitude, an attractor
How might we look at this quantiatively?
5
Simplest Transition from linear to Nonlinear
A pendulum
l
q
6
Pendulum
l
q
Does this look familiar? Are there any
approximations that might make this look
familiar?
For small amplitudes, q is small and so.
7
Nonlinear Pendulum
l
q
For non-small amplitudes, then the solution is
best done using energy methods and elliptical
integrals
Energy is a constant of motion and so,
At the highest point in the motion what is T?
8
Nonlinear Pendulum
l
q
9
Nonlinear Pendulum
We can now do a phase diagram! (other info would
require integration) Where is the initial
condition of the system?
10
Nonlinear Pendulum Small Oscillations
For small oscillations, we need to do a power
series
Can anyone tell we what this looks like?
Rewrite, is this surprising?
11
Nonlinear Pendulum Damped and/or driven and the
emergence of chaos
Undamped, non-chaotic
We can have chaotic motion, best examined via
Poincare sections
12
Poincare Sections
13
Poincare Sections
Started sampling at long time (why?)
Poincare section sampled at integral multiples of
the driving frequences (why?)