# simplePendulum: The simple pendulum model In phaseR: Phase Plane Analysis of One- And Two-Dimensional Autonomous ODE Systems

## Description

The derivative function of the simple pendulum model, an example of a two-dimensional autonomous ODE system.

## Usage

 `1` ```simplePendulum(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. Should be a `numeric` `vector` prescribing the value of l.

## Details

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

dx/dt = y, dy/dt = -g sin(x)/l.

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`

## Examples

 ``` 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 28``` ```# Plot the velocity field, nullclines and several trajectories simplePendulum_flowField <- flowField(simplePendulum, xlim = c(-7, 7), ylim = c(-7, 7), parameters = 5, points = 19, add = FALSE) y0 <- matrix(c(0, 1, 0, 4, -6, 1, 5, 0.5, 0, -3), 5, 2, byrow = TRUE) simplePendulum_nullclines <- nullclines(simplePendulum, xlim = c(-7, 7), ylim = c(-7, 7), parameters = 5, points = 500) simplePendulum_trajectory <- trajectory(simplePendulum, y0 = y0, tlim = c(0, 10), parameters = 5) # Determine the stability of two equilibrium points simplePendulum_stability_1 <- stability(simplePendulum, ystar = c(0, 0), parameters = 5) simplePendulum_stability_2 <- stability(simplePendulum, ystar = c(pi, 0), parameters = 5) ```

### Example output

```Note: col has been reset as required
T: 0   Delta: 1.962   Discriminant: -7.848   Classification: CentreT: 0   Delta: -1.962   Discriminant: 7.848   Classification: Saddle
```

phaseR documentation built on Oct. 12, 2019, 9:05 a.m.