phaseR (version 2.0)

stability: Stability Analysis

Description

Uses stability analysis to classify equilibrium points. Uses the Taylor Series approach (also known as Perturbation Analysis) to classify equilibrium points of a one dimensional autonomous ODE system, or the Jacobian approach to classify equilibrium points of a two dimensional autonomous ODE system. In addition, it can be used to return the Jacobian at any point of a two dimensional system.

Usage

stability(deriv, ystar = NULL, parameters = NULL, system = "two.dim",
  h = 1e-07, summary = TRUE, state.names = c("x", "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 guide.

ystar

The point at which to perform stability analysis. For a one variable system this should be a single number, for a two variable system this should be a vector of length two (i.e. presently only one equilibrium points stability can be evaluated at a time). Alternatively this can be left blank and the user can use locator to choose a point to perform the analysis. However, given you are unlikely to locate exactly the equilibrium point, if possible enter y.star yourself. Defaults to NULL.

parameters

Parameters of the ODE system, to be passed to deriv. Supplied as a 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".

h

Step length used to approximate the derivative(s). Defaults to 1e-7.

summary

Set to either TRUE or FALSE to determine whether a summary of the stability analysis is returned. Defaults to TRUE.

state.names

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 upon the value of system):

classification

The classification of y.star.

Delta

In the two dimensional system case, value of the Jacobians determinant at y.star.

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 T^2 - 4*Delta.

eigenvalues

In the two dimensional system case, the value of the Jacobians eigenvalues at y.star.

eigenvectors

In the two dimensional system case, the value of the Jacobians eigenvectors at y.star.

jacobian

In the two dimensional system case, the Jacobian at y.star.

h

As per input.

parameters

As per input.

summary

As per input.

system

As per input.

tr

In the two dimensional system case, the value of the Jacobians trace at y.star.

ystar

As per input.

Examples

# NOT RUN {
# Determine the stability of the equilibrium points of the one dimensional
# autonomous ODE system example2.
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")

# Determine the stability of the equilibrium points of the two dimensional autonomous
# ODE system example11.
example11.stability.1 <- stability(example11, ystar = c(0, 0))
example11.stability.2 <- stability(example11, ystar = c(0, 2))
example11.stability.3 <- stability(example11, ystar = c(1, 1))
example11.stability.4 <- stability(example11, ystar = c(3, 0))

# }