jacobian.full {rootSolve}R Documentation

Full square jacobian matrix for a system of ODEs (ordinary differential equations)

Description

Given a vector of (state) variables, and a function that estimates one function value for each (state) variable (e.g. the rate of change), estimates the Jacobian matrix (d(f(x))/d(x))
Assumes a full and square Jacobian matrix

Usage

jacobian.full(y, func, dy =NULL, time=0, parms=NULL, pert=1e-8, ...)

Arguments

y (state) variables, a vector; if y has a name attribute, the names will be used to label the Jacobian matrix columns
func function that calculates one function value for each element of y; if an ODE system, func calculates the rate of change (see details)
dy reference function value; if not specified, it will be estimated by calling func
time time, passed to function func
parms parameter values, passed to function func
pert numerical perturbation factor; increase depending on precision of model solution
... other arguments passed to function func

Details

The function func that estimates the rate of change of the state variables has to be consistent with functions called from R-package deSolve, which contains integration routines.

This function call is as: function(time,y,parms,...) where

The Jacobian is estimated numerically, by perturbing the x-values.

Value

The square jacobian matrix; the elements on i-th row and j-th column are given by: d(f(x)_i)/d(x_j)

Note

This function is useful for stability analysis of ODEs, which start by estimating the Jacobian at equilibrium points. The type of equilibrium then depends on the eigenvalue of the Jacobian.

Author(s)

Karline Soetaert <k.soetaert@nioo.knaw.nl>

See Also

jacobian.band, for a banded jacobian matrix
gradient, for a full (not necessarily square) gradient matrix and where the function call is simpler

Examples

# 1. Structure of the Jacobian
#------------------------------
mod <- function (t=0,y, parms=NULL,...)
{
 dy1<-  y[1] + 2*y[2]
 dy2<-3*y[1] + 4*y[2] + 5*y[3]
 dy3<-         6*y[2] + 7*y[3] + 8*y[4]
 dy4<-                  9*y[3] +10*y[4]
 return(as.list(c(dy1,dy2,dy3,dy4)))
}
jacobian.full(y=c(1,2,3,4),func=mod)

# 2. Stability properties of a physical model
#------------------------------
coriolis <- function (t,velocity,pars,f)
 {
  dvelx <- f*velocity[2]
  dvely <- -f*velocity[1]
  list(c(dvelx,dvely))
 }

# neutral stability; f is coriolis parameter
Jac <- jacobian.full(y=c(velx=0,vely=0),func=coriolis,
                     parms=NULL,f=1e-4)
print(Jac)
eigen(Jac)$values

# 3. Type of equilibrium
#----------------------------------------
# From Soetaert and Herman (2008). A practical guide to ecological 
# modelling. Using R as a simulation platform. Springer

eqn <- function (t,state,pars)
 {
  with (as.list(c(state,pars)),
  {
  dx<-a*x + cc*y
  dy<-b*y + dd*x
  list(c(dx,dy))
  })
 }

# stable equilibrium
A<-eigen(jacobian.full(y=c(x=0,y=0),func=eqn,
                       parms=c(a=-0.1,b=-0.3,cc=0,dd=0)))$values
# unstable equilibrium
B<-eigen(jacobian.full(y=c(x=0,y=0),func=eqn,
                       parms=c(a=0.2,b=0.2,cc=0.0,dd=0.2)))$values
# saddle point
C<-eigen(jacobian.full(y=c(x=0,y=0),func=eqn,
                       parms=c(a=-0.1,b=0.1,cc=0,dd=0)))$values
# neutral stability
D<-eigen(jacobian.full(y=c(x=0,y=0),func=eqn,
                       parms=c(a=0,b=0,cc=-0.1,dd=0.1)))$values
# stable focal point
E<-eigen(jacobian.full(y=c(x=0,y=0),func=eqn,
                       parms=c(a=0,b=-0.1,cc=-0.1,dd=0.1)))$values
# unstable focal point
F<-eigen(jacobian.full(y=c(x=0,y=0),func=eqn,
                       parms=c(a=0.,b=0.1,cc=0.1,dd=-0.1)))$values

data.frame(type=c("stable","unstable","saddle","neutral",
           "stable focus","unstable focus"),
           eigenvalue_1=c(A[1],B[1],C[1],D[1],E[1],F[1]),
           eigenvalue_2=c(A[2],B[2],C[2],D[2],E[2],F[2]))

# 4. Limit cycles
#----------------------------------------
# From Soetaert and Herman (2008). A practical guide to ecological 
# modelling. Using R as a simulation platform. Springer

eqn2 <- function (t,state,pars)
 {
  with (as.list(c(state,pars)),
  {
  dx<-  a*y   +e*x*(x^2+y^2-1)
  dy<-  b*x   +f*y*(x^2+y^2-1)
  list(c(dx,dy))
  })
 }

# stable limit cycle with unstable focus
eigen(jacobian.full(c(x=0,y=0),eqn2,parms=c(a=-1,b=1,e=-1,f=-1)))$values
# unstable limit cycle with stable focus
eigen(jacobian.full(c(x=0,y=0),eqn2,parms=c(a=-1,b=1,e=1,f=1)))$values

[Package rootSolve version 1.3 Index]