# vanDerPol: The Van Der Pol Oscillator In phaseR: Phase Plane Analysis of One and Two Dimensional Autonomous ODE Systems

## Description

The derivative function of the Van Der Pol Oscillator, an example of a two dimensional autonomous ODE system.

## Usage

 `1` ```vanDerPol(t, y, parameters) ```

## Arguments

 `t` Value of t, the independent variable, to evaluate the derivative at. Should be a single number. `y` Values of x and y, the dependent variables, to evaluate the derivative at. Should be a vector of length 2. `parameters` Values of the parameters of the system. Should be a number for the value of mu.

## Details

Evaluates the derivative of the following coupled ODE system at the point (t, x, y):

dx/dt = y, dy/dt = mu*(1 - x^2)*y - x.

Format is designed to be compatible with ode from the deSolve package.

## Value

Returns a list dy containing the values of the two derivatives at (t, x, y).

## Author(s)

Michael J. Grayling

`ode`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```# Plot the velocity field, nullclines and several trajectories. vanDerPol.flowField <- flowField(vanDerPol, x.lim = c(-5, 5), y.lim = c(-5, 5), parameters = 3, points = 15, add = FALSE) y0 <- matrix(c(2, 0, 0, 2, 0.5, 0.5), ncol = 2, nrow = 3, byrow = TRUE) vanDerPol.nullclines <- nullclines(vanDerPol, x.lim = c(-5, 5), y.lim = c(-5, 5), parameters = 3, points = 500) vanDerPol.trajectory <- trajectory(vanDerPol, y0 = y0, t.end = 10, parameters = 3) # Plot x and y against t. vanDerPol.numericalSolution <- numericalSolution(vanDerPol, y0 = c(4, 2), t.end = 100, parameters = 3) # Determine the stability of the equilibrium point. vanDerPol.stability <- stability(vanDerPol, y.star = c(0, 0), parameters = 3) ```