# stability: Stability Analysis In phaseR: Phase Plane Analysis of One and Two Dimensional Autonomous ODE Systems

## 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

 ```1 2``` ```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.

## Author(s)

Michael J. Grayling

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```# 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)) ```

phaseR documentation built on Aug. 20, 2018, 5:03 p.m.