dynamics: Dynamical systems calculations and graphics

In mosaicCalc: R-Language Based Calculus Operations for Teaching

Dynamical systems calculations and graphics

Description

• streamlines() draws raindrop-shaped paths at a randomized grid of points that follow trajectories

• flow_field() draws arrows showing the flow at a grid of points

• trajectory_euler() compute an Euler solution. ()

Usage

streamlines(..., npts = 8, dt = 0.01, nsteps = 10, color = "black", alpha = 1)

flow_field(..., npts = 8, scale = 0.8, color = "black", alpha = 1)

trajectory_euler(..., dt = 0.01, nsteps = 4, full = TRUE, every = 1)

Arguments

...	The first arguments should describe the dynamics. See details.
npts	The number of points on an edge of the grid
dt	time step (e.g. 0.01)
nsteps	how many Euler steps to take
color	What color to use
alpha	What alpha to use
scale	Number indicating how long to draw arrows. By default, the longest arrows take up a full box in the grid
full	report the derivative and the step size for each variable
every	n, report will contain every nth step

Details

The dynamical functions themselves will be formulas like dx ~ a*x*y and dy ~ y/x. Initial conditions will be arguments of the form x=3 and y=4. If there are parameters in the dynamical functions, you should also add the parameter values, for instance a=2.

For flow_field() and streamlines() you do not need to specify initial conditions. The grid will be set by the domain argument.

The graphics functions are all arranged to accept, if given, a ggplot object piped in. The new graphics layer will be drawn on top of that. If there is no ggplot object piped in, then the graphics will be made as a first layer, which can optionally piped into other ggformula functions or + into ggplot layers.

Examples

streamlines(dx ~ x+y, dy~ x-y, domain(x=0:6, y=0:3))
flow_field(dx ~ x+y, dy~ x-y, domain(x=0:6, y=0:3))
Dyn <- makeODE(dx ~ 0.06*x, dy ~-x - y, domain(t=0:20), dt=0.1,x=3, y=4)
streamlines(Dyn, domain(x=-5:5, y=-5:5), npts=15)
flow_field(Dyn, domain(x=-6:6, y=-10:10))
trajectory_euler(dx ~ -y, dy ~ .5*x, x=3, y=3)
rabbits <- drabbit ~ 0.2*rabbit - 0.01*rabbit*fox
foxes <- dfox ~ -.2*fox + 0.0005*rabbit*fox 
trajectory_euler(rabbits, foxes, rabbit=100, fox=3, dt=0.1, nsteps=500, every=10)

mosaicCalc documentation built on Sept. 11, 2024, 9:10 p.m.