VecDecomAll: Vector decomposition and remainder fields

In bmarkslash7/QPot: Quasi-Potential Analysis for Stochastic Differential Equations

Description

This function calculates the vector, gradient, and remainder fields.

Usage

VecDecomAll(surface, x.rhs, y.rhs, x.bound, y.bound)

Arguments

surface	matrix output from QPGlobal or QPotential.
x.rhs	a string containing the right hand side of the equation for x.
y.rhs	a string containing the right hand side of the equation for y.
x.bound	the x boundaries denoted at c(minimum, maximum).
y.bound	the y boundaries denoted at c(minimum, maximum).

Value

returns an array of all three vector fields: the deterministic skeleton, the negative gradient of the quasi-potential, and the remainder. The array has three dimensions with the respective lengths of xstepnumber, ystepnumber, and 6. The six are the x- and y-values for each of the three vector fields, as x-deterministic skeleton, y-deterministic skeleton, x-negative gradient of the quasi-potential, y-negative gradient of the quasi-potential, x-remainder, and y-remainder.

Examples

# First, the system of equations
	equationx <- "1.54*x*(1.0-(x/10.14)) - (y*x*x)/(1.0+x*x)"
	equationy <- "((0.476*x*x*y)/(1+x*x)) - 0.112590*y*y"

# Second, shared parameters for each quasi-potential run
	xbounds <- c(-0.5, 10.0)
	ybounds <- c(-0.5, 10.0)
	xstepnumber <- 100
	ystepnumber <- 100

# Third, first local quasi-potential run
	xinit1 <- 1.40491
	yinit1 <- 2.80808
	storage.eq1 <- QPotential(x.rhs = equationx, x.start = xinit1, 
	x.bound = xbounds, x.num.steps = xstepnumber, y.rhs = equationy, 
	y.start = yinit1, y.bound = ybounds, y.num.steps = ystepnumber)

# Fourth, second local quasi-potential run
	xinit2 <- 4.9040
	yinit2 <- 4.06187
	storage.eq2 <- QPotential(x.rhs = equationx, x.start = xinit2, 
	x.bound = xbounds, x.num.steps = xstepnumber, y.rhs = equationy, 
	y.start = yinit2, y.bound = ybounds, y.num.steps = ystepnumber)

# Fifth, determine global quasi-potential 
	unst.x <- c(0, 4.2008)
	unst.y <- c(0, 4.0039)
	ex1.global <- QPGlobal(local.surfaces = list(storage.eq1, storage.eq2), 
	unstable.eq.x = unst.x, unstable.eq.y = unst.y, x.bound = xbounds, 
	y.bound = ybounds)

# Sixth, decompose the global quasi-potential into the 
# deterministic skeleton, gradient, and remainder vector fields
	VDAll <- VecDecomAll(surface = ex1.global, x.rhs = equationx, y.rhs = equationy, 
	x.bound = xbounds, y.bound = ybounds)

bmarkslash7/QPot documentation built on Jan. 11, 2020, 11:11 a.m.