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## Project, EGME 511 (Advanced Mechanical Vibration)

Analysis of Van Der Pol diﬀerential equation

Nasser M. Abbasi

June 28, 2015

### Contents

This report in PDF

### 1 Introduction

Van der Pol diﬀerential equation is given by

In this analysis, we will consider the case only for positive . We will analyze the stability of this equation
and generate a phase diagram.

### 2 Stability

The ﬁrst step in examining stability of a non-linear diﬀerential equation is to convert it to state space by
introducing 2 state variables.

Therefore

Equilibrium points are found by solving , hence from the above, we see that and from
we conclude that as well. Hence

The system matrix is now found. First we note that , and
, hence

Hence at becomes

Now we ﬁnd the characteristic equation

Hence , therefore

If then both roots are on the RHS, hence system is unstable (equilibrium point is a repelling
point).

If then we have , and we have spiral out equilibrium point, unstable.

### 3 Phase diagram

We need to obtain a relation between and . From the diﬀerential equation

rewrite in state space variables, we obtain

Hence the above is in the form , therefore the isoclines lines can be found by
setting

Where is a constant. Hence we obtain the parameterize equation to use to plot the gradient lines
as

### 4 Phase diagram

To generate the phase diagram ,
a program was written which allows one to adjust the initial conditions and the parameters and and
observe the eﬀect on the shape of the limit cycle. We see that starting from diﬀerent initial conditions, the
solution trajectory always ends up in a limit cycle.

The following is a screen shot of the program written for this project.