# example5: Example ODE System 5 In phaseR: Phase Plane Analysis of One and Two Dimensional Autonomous ODE Systems

## Description

The derivative function of an example two-dimensional autonomous ODE system.

## Usage

 `1` ```example5(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. Not required here.

## Details

`example5` evaluates the derivatives of the following coupled ODE system at the point (t, x, y):

dx/dt = 2x + y, dy/dt = 2x - y.

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, manifolds and several trajectories example5.flowField <- flowField(example5, xlim = c(-3, 3), ylim = c(-3, 3), points = 19, add = FALSE) y0 <- matrix(c(1, 0, -1, 0, 2, 2, -2, 2, 0, 3, 0, -3), 6, 2, byrow = TRUE) example5.nullclines <- nullclines(example5, xlim = c(-3, 3), ylim = c(-3, 3)) example5.trajectory <- trajectory(example5, y0 = y0, tlim = c(0,10)) example5.manifolds <- drawManifolds(example5, y0 = c(0, 0), tend = 100, col = c("green", "red"), add.legend = TRUE) # Plot x and y against t example5.numericalSolution <- numericalSolution(example5, y0 = c(0, 3), tlim = c(0, 3)) # Determine the stability of the equilibrium point example5.stability <- stability(example5, ystar = c(0, 0)) ```