GPoM (version 1.4)

derivODEwMultiX: deriveODEwMultiX : A Subfonction for the numerical integration of polynomial equations in the generic form defined by function poLabs and with External Forcing F(t)

Description

This function provides the one step integration of polynomial Ordinary Differential Equations (ODE). This function requires the function ode ("deSolve" package). This function has to be run with the Runge-Kutta method (method = 'rk4')

Usage

derivODEwMultiX(t, x, K, extF, regS = NULL)

Value

xxx

Arguments

t

All the dates for which the result of the numerical integration of the model will have to be provided

x

Current state vector (input from which the next state will be estimated)

K

is the model: each column corresponds to one equation which organisation is following the convention given by function poLabs which requires the definition of the model dimension nVar (i.e. the number of variables) and the maximum polynomial degree dMax allowed. The last Equation correspond to the forcing variable that is artificially set to 0.

extF

is the external forcing. It is defined by two columns. The first colomn correspond to time t. The second column to F(t) the forcing at time t. Note that when launching the integration function ode, the forcing F(t) should be provided with a sampling time twice the sampling time used in t (because rk4 method will always use an intermediate time step).

regS

Current states of each polynomial terms used in poLabs. These states can be deduced from the current state vector x (using function regSeries). When available, it can be provided as an input to avoid unecessary computation.

Author

Sylvain Mangiarotti

Examples

# build a non autonomous model
nVar = 4
dMax = 3
omega = 0.2
gamma = 0.05
KDf=matrix(0, nrow = d2pMax(nVar = nVar, dMax = dMax), ncol = nVar)
KDf[11,1]  = 1
KDf[2,2]  = 1
KDf[5,2]  = 1
KDf[11,2]  = -gamma
KDf[35,2] = -1
KDf[2,3]  = NA
KDf[2,4]  = NA
visuEq(K = KDf, substit = c('x', 'y', 'u', 'v'))
#
# Prepare the external forcing
# number of integration time step
Istep <- 500
# time step
smpl <- 1 / 20
# output time vector
dater <- (0:Istep) * smpl
# hald step time vector (for Runge-Kutta integration)
daterdbl <- (0:(Istep*2 + 1)) * smpl / 2
# generate the forcing (here variables u and v)
extF = cbind(daterdbl, -0.1 * cos(daterdbl * omega), 0.05 * cos(daterdbl * 16/3*omega))
#
# Initial conditions to be used (external variables can be set to 0)
etatInit <- c(-0.616109362 , -0.126882584 , 0, 0)
#
# Numerical integration
reconstr2 <- ode(etatInit, dater, derivODEwMultiX,
                 KDf, extF = extF, method = 'rk4')
# Reconstruction of the output
nVarExt <- dim(extF)[2] - 1
reconstr2[,(nVar - nVarExt + 2):(nVar + 1)] <- extF[(0:Istep+1)*2, 2:(nVarExt+1)]