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

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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

  • y : (state) variable values at which the Jacobian is estimated.

  • parms: parameter vector - need not be used.

  • time: time at which the Jacobian is estimated - in general, time will not be used.

  • ...: (optional) any other arguments.

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 <karline.soetaert@nioz.nl>

See Also

jacobian.band, estimates the Jacobian matrix assuming a banded structure.

hessian, estimates the Hessian matrix.

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

uniroot.all, to solve for all roots of one (nonlinear) equation

multiroot, to solve n roots of n (nonlinear) equations

Examples

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## =======================================================================
## 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 (2009). 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 (2009). 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