Numerical Methods

Outline

Introduction

Linear
Equations

Interpolation

Numerical
Solutions

Computer
Measurement

      

Van der pol oscillator

A relaxation oscillator is a system that produces a periodic signal. Heart beats and the squeaking of fingernails on a blackboard are examples of relaxation oscillations. One relaxation oscillator that can be built with electronic components is called the Van der Pol oscillator. This system is described by the differential equation,

\[ \begin{equation} \frac{d^2x}{dt^2}-\mu (1-x^2)\frac{dx}{dt}+x=0. \end{equation} \]

For small values of $x$, the damping term $ \mu (1-x^2)\frac{dx}{dt}$ is negative and the signal grows until $x>1$ when the damping becomes positive and the signal decays again. The frequency of the oscillations can be controlled by changing $\mu$. In an electronic circuit this can be achieved by using a variable resistor.

$\mu=$ 1

 Numerical 2nd order differential equation solver 

$ \large \frac{dx}{dt}=$

$v_x$

$ \large a_x=\frac{F_x}{m}=\frac{dv_x}{dt}=$

Intitial conditions: 

$x(t_0)=$

 

$\Delta t=$

$v_x(t_0)=$

 

$N_{steps}$

$t_0=$

 

Plot:

vs.

 

 $t$       $x$      $v_x$