# 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` The value of t, the independent variable, to evaluate the derivative at. Should be a single number. `y` The values of x and y, the dependent variables, to evaluate the derivative at. Should be a vector of length two. `parameters` The values of the parameters of the system. Should be a single number prescribing the value of μ.

## Details

`vanDerPol` evaluates the derivative of the following ODE at the point (t, x, y):

dx/dt = y, dy/dt = μ(1 - x2)y - x.

Its format is designed to be compatible with `ode` from the `deSolve` package.

## Value

Returns a list 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 15 16 17 18 19 20 21 22 23 24 25 26 27``` ```# Plot the velocity field, nullclines and several trajectories. vanDerPol.flowField <- flowField(vanDerPol, xlim = c(-5, 5), ylim = c(-5, 5), parameters = 3, points = 15, add = FALSE) y0 <- matrix(c(2, 0, 0, 2, 0.5, 0.5), 3, 2, byrow = TRUE) vanDerPol.nullclines <- nullclines(vanDerPol, xlim = c(-5, 5), ylim = c(-5, 5), parameters = 3, points = 500) vanDerPol.trajectory <- trajectory(vanDerPol, y0 = y0, tlim = c(0, 10), parameters = 3) # Plot x and y against t vanDerPol.numericalSolution <- numericalSolution(vanDerPol, y0 = c(4, 2), tlim = c(0, 100), parameters = 3) # Determine the stability of the equilibrium point vanDerPol.stability <- stability(vanDerPol, ystar = c(0, 0), parameters = 3) ```