# example12: Example ODE system 12 In mjg211/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` ```example12(t, y, parameters) ```

## Arguments

 `t` The value of t, the independent variable, to evaluate the derivative at. Should be a `numeric` `vector` of `length` one. `y` The values of x and y, the dependent variables, to evaluate the derivative at. Should be a `numeric` `vector` of `length` two. `parameters` The values of the parameters of the system. Not used here.

## Details

`example12` evaluates the derivatives of the following coupled ODE system at the point (t, x, y):

dx/dt = x - y, dy/dt = x2 + y2 - 2.

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``` ```# Plot the velocity field, nullclines and several trajectories example12_flowField <- flowField(example12, xlim = c(-4, 4), ylim = c(-4, 4), points = 17, add = FALSE) y0 <- matrix(c(2, 2, -3, 0, 0, 2, 0, -3), 4, 2, byrow = TRUE) example12_nullclines <- nullclines(example12, xlim = c(-4, 4), ylim = c(-4, 4), points = 200) example12_trajectory <- trajectory(example12, y0 = y0, tlim = c(0, 10)) # Determine the stability of the equilibrium points example12_stability_1 <- stability(example12, ystar = c(1, 1)) example12_stability_2 <- stability(example12, ystar = c(-1, -1)) ```