# Numerical Integration of models in ODE of polynomial form Function for the numerical integration of Ordinary Differential Equations of polynomial form. ```r numicano( nVar, dMax, dMin = 0, Istep = 1000, onestep = 1/125, KL = NULL, PolyTerms = NULL, v0 = NULL, method = "rk4" ) ``` ## Arguments - `nVar`: Number of variables considered in the polynomial formulation. - `dMax`: Maximum degree of the polynomial formulation. - `dMin`: The minimum negative degree of the polynomial formulation (0 by default). - `Istep`: The number of integration time steps - `onestep`: Time step length - `KL`: Matrix formulation of the model to integrate numerically - `PolyTerms`: Vectorial formulation of the model (only for models of canonical form) - `v0`: The initial conditions (a vector which length should correspond to the model dimension `nVar`) - `method`: The integration method (See package `deSolve`), by default `method = 'rk4'`. ## Returns A list of two variables: `$KL` The model in its matrix formulation `$reconstr` The integrated trajectory (first column is the time, next columns are the model variables) ## Examples ```r ############# # Example 1 # ############# # For a model of general form (here the rossler model) # model dimension: nVar = 3 # maximal polynomial degree dMax = 2 # Number of parameter number (by default) pMax <- d2pMax(nVar, dMax) # convention used for the model formulation poLabs(nVar, dMax) # Definition of the Model Function a = 0.520 b = 2 c = 4 Eq1 <- c(0,-1, 0,-1, 0, 0, 0, 0, 0, 0) Eq2 <- c(0, 0, 0, a, 0, 0, 1, 0, 0, 0) Eq3 <- c(b,-c, 0, 0, 0, 0, 0, 1, 0, 0) K <- cbind(Eq1, Eq2, Eq3) # Edition of the equations visuEq(K, nVar, dMax) # initial conditions v0 <- c(-0.6, 0.6, 0.4) # model integration reconstr <- numicano(nVar, dMax, Istep=1000, onestep=1/50, KL=K, v0=v0, method="ode45") # Plot of the simulated time series obtained dev.new() plot(reconstr$reconstr[,2], reconstr$reconstr[,3], type='l', main='phase portrait', xlab='x(t)', ylab = 'y(t)') ############# # Example 2 # ############# # For a model of canonical form # model dimension: nVar = 4 # maximal polynomial degree dMax = 3 # Number of parameter number (by default) pMax <- d2pMax(nVar, dMax) # Definition of the Model Function PolyTerms <- c(281000, 0, 0, 0, -2275, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 861, 0, 0, 0, -878300, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) # terms used in the model poLabs(nVar, dMax, findIt=(PolyTerms!=0)) # initial conditions v0 <- c(0.54, 3.76, -90, -5200) # model integration reconstr <- numicano(nVar, dMax, Istep=500, onestep=1/250, PolyTerms=PolyTerms, v0=v0, method="ode45") # Plot of the simulated time series obtained plot(reconstr$reconstr[,2], reconstr$reconstr[,3], type='l', main='phase portrait', xlab='x', ylab = 'dx/dt') # Edition of the equations visuEq(reconstr$KL, nVar, dMax) ############# # Example 3 # ############# # For a model of general form (here the rossler model) # model dimension: nVar = 3 # maximal polynomial degree dMax = 2 dMin = -1 # Number of parameter number (by default) pMax <- regOrd(nVar, dMax, dMin)[2] # convention used for the model formulation poLabs(nVar, dMax, dMin) # Definition of the Model Function a = 0.520 b = 2 c = 4 Eq1 <- c(0,-1, 0,-1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0) Eq2 <- c(0, 0, 0, a, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0) Eq3 <- c(b,-c, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0) K <- cbind(Eq1, Eq2, Eq3) # Edition of the equations #visuEq(K, nVar, dMax) # initial conditions v0 <- c(-0.6, 0.6, 0.4) # model integration reconstr <- numicano(nVar, dMax, dMin, Istep=1000, onestep=1/50, KL=K, v0=v0, method="ode45") # Plot of the simulated time series obtained dev.new() plot(reconstr$reconstr[,2], reconstr$reconstr[,3], type='l', main='phase portrait', xlab='x(t)', ylab = 'y(t)') ``` ## See Also `derivODE2`, `numinoisy` ## Author(s) Sylvain Mangiarotti

numicano function