findEquilibrium: Equilibrium point identification
In phaseR: Phase Plane Analysis of One- And Two-Dimensional Autonomous ODE Systems

findEquilibrium

R Documentation

Equilibrium point identification

Description

Searches for an equilibium point of a system, taking the starting point of the search as a user specified location. On identifying such a point, a classification is performed, and an informatively shaped point can be added to the plot.

Usage

findEquilibrium(
  deriv,
  y0 = NULL,
  parameters = NULL,
  system = "two.dim",
  tol = 1e-16,
  max.iter = 50,
  h = 1e-06,
  plot.it = FALSE,
  summary = TRUE,
  state.names = if (system == "two.dim") c("x", "y") else "y"
)

Arguments

`deriv`	A function computing the derivative at a point for the ODE system to be analysed. Discussion of the required structure of these functions can be found in the package vignette, or in the help file for the function `ode`.
`y0`	The starting point of the search. In the case of a one-dimensional system, this should be a `numeric` `vector` of `length` one indicating the location of the dependent variable initially. In the case of a two-dimensional system, this should be a `numeric` `vector` of `length` two reflecting the location of the two dependent variables initially. Alternatively this can be specified as `NULL`, and then `locator` can be used to specify the initial point on a plot. Defaults to `NULL`.
`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"`.
`tol`	The tolerance for the convergence of the search algorithm. Defaults to `1e-16`.
`max.iter`	The maximum allowed number of iterations of the search algorithm. Defaults to `50`.
`h`	Step length used to approximate the derivative(s). Defaults to `1e-6`.
`plot.it`	Logical. If `TRUE`, a point is plotted at the identified equilibrium point, with shape corresponding to its classification.
`summary`	Set to either `TRUE` or `FALSE` to determine whether a summary of the progress of the search procedure is returned. Defaults to `TRUE`.
`state.names`	The state names for `ode` functions that do not use positional states.

Value

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

`classification`	The classification of the identified equilibrium point.
`Delta`	In the two-dimensional system case, value of the Jacobian's determinant at the equilibrium point.
`deriv`	As per input.
`discriminant`	In the one-dimensional system case, the value of the discriminant used in perturbation analysis to assess stability. In the two-dimensional system case, the value of `tr^2 - 4*Delta`.
`eigenvalues`	In the two-dimensional system case, the value of the Jacobian's eigenvalues at the equilibrium point.
`eigenvectors`	In the two-dimensional system case, the value of the Jacobian's eigenvectors at the equilibrium point.
`jacobian`	In the two-dimensional system case, the Jacobian at the equilibrium point.
`h`	As per input.
`max.iter`	As per input.
`parameters`	As per input.
`plot.it`	As per input.
`summary`	As per input.
`system`	As per input.
`tr`	In the two-dimensional system case, the value of the Jacobian's trace at the equilibrium point.
`tol`	As per input.
`y0`	As per input.
`ystar`	The location of the identified equilibrium point.