# Results

The `xmaple`

worksheet , or the postscript version.

The graphics page.

### Steady States

The only steady state solution is for *x* = 0, *y* = 0,
*z* = 0, *w* = 0. (else refered to as (0,0,0,0).)

### Steady State Stability

The Jacobian matrix for the model has the following form after being
evaluated at (0,0,0,0).
Jacobian matrix
entries
0 | 1 | 0 | 0 |

-1 | -c_{1} | 0 | 3 |

0 | 0 | | -3/10 |

0 | 0 | 3/10 | |

The eigenvalues are:

at (0,0,0,0), the following stability can be found:

- If > 0 and c
_{1} >
0, then (0,0,0,0) is a saddle.
- If > 0 and c
_{1} <
0, then (0,0,0,0) is a source.
- If < 0 and c
_{1} >
0, then (0,0,0,0) is a sink.
- If < 0 and c
_{1} <
0, then (0,0,0,0) is a source.
- If = 0 and c
_{1} = 0, we
can say nothing about the stability of (0,0,0,0). Bifurcation theory is
needed to analyse the system in this case.

### Phase Portraits and Bifurcation diagram

`xrk`

was used, along with `xmaple`

, to discover the
model's behavior. A bifurcation diagram is necessary to find the behavior
of the point (0,0,0,0) for = 0 and
c_{1} = 0. Please see the `xmaple`

worksheet or the postscript version of the main
work for this results section. Additional work is on the graphics page. *Warning! Lots
of large graphics! May take a while to load!*

### Relation to the van der Pol model

The van der Pol oscillator is similar to the linear pendulum equation
```
x" = -ax' -bx
```

(where the right hand side of the equation is the sum of two forces: the
restoring force *bx* and the linear frictional force *ax'*),
except that the frictional term is nonlinear in the van der Pol model.
The van der Pol equation is

```
x" + ( + x
```^{2})*x' + x = 0

where is constant.

The behavior of the x-y system (when plotted), with positive c_{1}
and , acts much like the van der Pol
forced oscillation model with < 0.
(see case on graphics page)

`xmaple`

worksheet on the van der
Pol oscillator, and the postscript version

Michelle Vadeboncoeur
<mrvahs@wpi.edu>

Forest Lee-Elkin
<yusuf@wpi.edu>