jacobian.full {rootSolve} | R Documentation |
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
jacobian.full(y, func, dy =NULL, time=0, parms=NULL, pert=1e-8, ...)
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 |
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.
The square jacobian matrix; the elements on i-th row and j-th column are given by: d(f(x)_i)/d(x_j)
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.
Karline Soetaert <k.soetaert@nioo.knaw.nl>
jacobian.band
, for a banded jacobian matrix
gradient
, for a full (not necessarily square) gradient matrix and where the function call is simpler
# 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