# flowField: Flow field In mjg211/phaseR: Phase Plane Analysis of One- And Two-Dimensional Autonomous ODE Systems

## Description

Plots the flow or velocity field for a one- or two-dimensional autonomous ODE system.

## Usage

 ```1 2 3 4 5 6``` ```flowField(deriv, xlim, ylim, parameters = NULL, system = "two.dim", points = 21, col = "gray", arrow.type = "equal", arrow.head = 0.05, frac = 1, add = TRUE, state.names = if (system == "two.dim") c("x", "y") else "y", xlab = if (system == "two.dim") state.names else "t", ylab = if (system == "two.dim") state.names else state.names, ...) ```

## Arguments

 `deriv` A function computing the derivative at a point for the ODE system to be analysed. Discussion of the required format of these functions can be found in the package vignette, or in the help file for the function `ode`. `xlim` In the case of a two-dimensional system, this sets the limits of the first dependent variable in which gradient reflecting line segments should be plotted. In the case of a one-dimensional system, this sets the limits of the independent variable in which these line segments should be plotted. Should be a `numeric` `vector` of `length` two. `ylim` In the case of a two-dimensional system this sets the limits of the second dependent variable in which gradient reflecting line segments should be plotted. In the case of a one-dimensional system, this sets the limits of the dependent variable in which these line segments should be plotted. Should be a `numeric` `vector` of `length` two. `parameters` Parameters of the ODE system, to be passed to `deriv`. Supplied as a `numeric` `vector`; the order of the parameters can be found from the `deriv` file. Defaults to `NULL`. `system` Set to either `"one.dim"` or `"two.dim"` to indicate the type of system being analysed. Defaults to `"two.dim"`. `points` Sets the density of the line segments to be plotted; `points` segments will be plotted in the x and y directions. Fine tuning here, by shifting `points` up and down, allows for the creation of more aesthetically pleasing plots. Defaults to `11`. `col` Sets the colour of the plotted line segments. Should be a `character` `vector` of `length` one. Will be reset accordingly if it is of the wrong `length`. Defaults to `"gray"`. `arrow.type` Sets the type of line segments plotted. If set to `"proportional"` the `length` of the line segments reflects the magnitude of the derivative. If set to `"equal"` the line segments take equal lengths, simply reflecting the gradient of the derivative(s). Defaults to `"equal"`. `arrow.head` Sets the length of the arrow heads. Passed to `arrows`. Defaults to `0.05`. `frac` Sets the fraction of the theoretical maximum length line segments can take without overlapping, that they can actually attain. In practice, `frac` can be set to greater than 1 without line segments overlapping. Fine tuning here assists the creation of aesthetically pleasing plots. Defaults to `1`. `add` Logical. If `TRUE`, the flow field is added to an existing plot. If `FALSE`, a new plot is created. Defaults to `TRUE`. `state.names` The state names for `ode` functions that do not use positional states. `xlab` Label for the x-axis of the resulting plot. `ylab` Label for the y-axis of the resulting plot. `...` Additional arguments to be passed to either `plot` or `arrows`.

## Value

Returns a `list` with the following components (the exact make up is dependent on the value of `system`):

 `add` As per input. `arrow.head` As per input. `arrow.type` As per input. `col` As per input, but with possible editing if a `character` `vector` of the wrong `length` was supplied. `deriv` As per input. `dx` A `numeric` `matrix`. In the case of a two-dimensional system, the values of the derivative of the first dependent derivative at all evaluated points. `dy` A `numeric` `matrix`. In the case of a two-dimensional system, the values of the derivative of the second dependent variable at all evaluated points. In the case of a one-dimensional system, the values of the derivative of the dependent variable at all evaluated points. `frac` As per input. `parameters` As per input. `points` As per input. `system` As per input. `x` A `numeric` `vector`. In the case of a two-dimensional system, the values of the first dependent variable at which the derivatives were computed. In the case of a one-dimensional system, the values of the independent variable at which the derivatives were computed. `xlab` As per input. `xlim` As per input. `y` A `numeric` `vector`. In the case of a two-dimensional system, the values of the second dependent variable at which the derivatives were computed. In the case of a one-dimensional system, the values of the dependent variable at which the derivatives were computed. `ylab` As per input. `ylim` As per input.

## Author(s)

Michael J Grayling

`arrows`, `plot`
 ``` 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 29 30 31 32 33 34 35 36 37 38 39``` ```# Plot the flow field, nullclines and several trajectories for the # one-dimensional autonomous ODE system logistic logistic_flowField <- flowField(logistic, xlim = c(0, 5), ylim = c(-1, 3), parameters = c(1, 2), points = 21, system = "one.dim", add = FALSE) logistic_nullclines <- nullclines(logistic, xlim = c(0, 5), ylim = c(-1, 3), parameters = c(1, 2), system = "one.dim") logistic_trajectory <- trajectory(logistic, y0 = c(-0.5, 0.5, 1.5, 2.5), tlim = c(0, 5), parameters = c(1, 2), system = "one.dim") # Plot the velocity field, nullclines and several trajectories for the # two-dimensional autonomous ODE system simplePendulum 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) ```