R/example2.R

In phaseR: Phase Plane Analysis of One- And Two-Dimensional Autonomous ODE Systems

#' Example ODE system 2
#'
#' The derivative function of an example one-dimensional autonomous ODE system.
#'
#' \code{example2} evaluates the derivative of the following ODE at the point
#' \ifelse{html}{\out{(<i>t</i>, <i>y</i>)}}{\eqn{(t, y)}}:
#'
#' \ifelse{html}{\out{<i>dy</i>/<i>dt</i> = <i>y</i>(1 - <i>y</i>)(2 -
#' <i>y</i>).}}{\deqn{\frac{dy}{dt} = y(1 - y)(2 - y).}}
#'
#' Its format is designed to be compatible with \code{\link[deSolve]{ode}} from
#' the \code{\link[deSolve]{deSolve}} package.
#'
#' @param t The value of \ifelse{html}{\out{<i>t</i>}}{\eqn{t}}, the independent
#' variable, to evaluate the derivative at. Should be a
#' \code{\link[base]{numeric}} \code{\link[base]{vector}} of
#' \code{\link[base]{length}} one.
#' @param y The value of \ifelse{html}{\out{<i>y</i>}}{\eqn{y}}, the dependent
#' variable, to evaluate the derivative at. Should be a
#' \code{\link[base]{numeric}} \code{\link[base]{vector}} of
#' \code{\link[base]{length}} one.
#' @param parameters The values of the parameters of the system. Not used here.
#' @return Returns a \code{\link[base]{list}} containing the value of the
#' derivative at \ifelse{html}{\out{(<i>t</i>, <i>y</i>)}}{\eqn{(t, y)}}.
#' @author Michael J Grayling
#' @seealso \code{\link[deSolve]{ode}}
#' @examples
#' # Plot the flow field and several trajectories
#' example2_flowField     <- flowField(example2,
#'                                     xlim   = c(0, 4),
#'                                     ylim   = c(-1, 3),
#'                                     system = "one.dim",
#'                                     add    = FALSE,
#'                                     xlab   = "t")
#' example2_trajectory    <- trajectory(example2,
#'                                      y0     = c(-0.5, 0.5, 1.5, 2.5),
#'                                      tlim   = c(0, 4),
#'                                      system = "one.dim")
#' # Plot the phase portrait
#' example2_phasePortrait <- phasePortrait(example2,
#'                                         ylim = c(-0.5, 2.5),
#'                                         frac = 0.5)
#' # Determine the stability of the equilibrium points
#' example2_stability_1   <- stability(example2,
#'                                     ystar  = 0,
#'                                     system = "one.dim")
#' example2_stability_2   <- stability(example2,
#'                                     ystar  = 1,
#'                                     system = "one.dim")
#' example2_stability_3   <- stability(example2,
#'                                     ystar  = 2,
#'                                     system = "one.dim")
#' @export
example2 <- function(t, y, parameters) {
  list(y*(1 - y)*(2 - y))
}