trajectory {phaseR}

Phase plane trajectory plotting

Description

Performs numerical integration of the chosen ODE system, for a user specified set of initial conditions. Plots the resulting solution(s) in the phase plane.

Usage

trajectory(
  deriv,
  y0 = NULL,
  n = NULL,
  tlim,
  tstep = 0.01,
  parameters = NULL,
  system = "two.dim",
  col = "black",
  add = TRUE,
  state.names = if (system == "two.dim") c("x", "y") else "y",
  method = "ode45",
  ...
)

Arguments

deriv

A function computing the derivative at a point for the ODE system to be analysed. Discussion of the required structure of these functions can be found in the package vignette, or in the help file for the function ode.

y0

The initial condition(s). In the case of a one-dimensional system, this can either be a numeric vector of length one, indicating the location of the dependent variable initially, or a numeric vector indicating multiple initial locations of the independent variable. In the case of a two-dimensional system, this can either be a numeric vector of length two, reflecting the location of the two dependent variables initially, or it can be numeric matrix where each row reflects a different initial condition. Alternatively this can be specified as NULL, and then locator can be used to specify initial condition(s) on a plot. In this case, for one-dimensional systems, all initial conditions are taken at tlim[1], even if not selected so on the graph. Defaults to NULL.

n

If y0 is left NULL, such initial conditions can be specified using locator, n sets the number of initial conditions to be chosen. Defaults to NULL.

tlim

Sets the limits of the independent variable for which the solution should be plotted. Should be a numeric vector of length two. If tlim[2] > tlim[1], then tstep should be negative to indicate a backwards trajectory.

tstep

The step length of the independent variable, used in numerical integration. Decreasing the absolute magnitude of tstep theoretically makes the numerical integration more accurate, but increases computation time. Defaults to 0.01.

parameters

Parameters of the ODE system, to be passed to deriv. Supplied as a numeric vector; the order of the parameters can be found from the deriv file. Defaults to NULL.

system

Set to either "one.dim" or "two.dim" to indicate the type of system being analysed. Defaults to "two.dim".

col

The colour(s) to plot the trajectories in. Should be a character vector. Will be reset accordingly if it is not of the length of the number of initial conditions. Defaults to "black".

add

Logical. If TRUE, the trajectories added to an existing plot. If FALSE, a new plot is created. Defaults to TRUE.

state.names

The state names for ode functions that do not use positional states.

method

Passed to ode. See there for further details. Defaults to "ode45".

...

Additional arguments to be passed to plot.

Value

Returns a list with the following components (the exact make up is dependent on the value of system):

add

As per input.

col

As per input, but with possible editing if a character vector of the wrong length was supplied.

deriv

As per input.

n

As per input.

method

As per input.

parameters

As per input.

system

As per input.

tlim

As per input.

tstep

As per input.

t

A numeric vector containing the values of the independent variable at each integration step.

x

In the two-dimensional system case, a numeric matrix whose columns are the numerically computed values of the first dependent variable for each initial condition.

y

In the two-dimensional system case, a numeric matrix whose columns are the numerically computed values of the second dependent variable for each initial condition. In the one-dimensional system case, a numeric matrix whose columns are the numerically computed values of the dependent variable for each initial condition.

y0

As per input, but converted to a numeric matrix if supplied as a vector initially.

Author(s)

Michael J Grayling

See Also

ode, plot

Examples

# Plot the flow field, nullclines and several trajectories for the
# one-dimensional autonomous ODE system logistic
logistic_flowField  <- flowField(logistic,
                                 xlim       = c(0, 5),
                                 ylim       = c(-1, 3),
                                 parameters = c(1, 2),
                                 points     = 21,
                                 system     = "one.dim",
                                 add        = FALSE)
logistic_nullclines <- nullclines(logistic,
                                  xlim       = c(0, 5),
                                  ylim       = c(-1, 3),
                                  parameters = c(1, 2),
                                  system     = "one.dim")
logistic_trajectory <- trajectory(logistic,
                                  y0         = c(-0.5, 0.5, 1.5, 2.5),
                                  tlim       = c(0, 5),
                                  parameters = c(1, 2),
                                  system     = "one.dim")

# Plot the velocity field, nullclines and several trajectories for the
# two-dimensional autonomous ODE system simplePendulum
simplePendulum_flowField  <- flowField(simplePendulum,
                                       xlim       = c(-7, 7),
                                       ylim       = c(-7, 7),
                                       parameters = 5,
                                       points     = 19,
                                       add        = FALSE)
y0                        <- matrix(c(0, 1, 0, 4, -6, 1, 5, 0.5, 0, -3),
                                    5, 2, byrow = TRUE)

simplePendulum_nullclines <- nullclines(simplePendulum,
                                        xlim       = c(-7, 7),
                                        ylim       = c(-7, 7),
                                        parameters = 5,
                                        points     = 500)

simplePendulum_trajectory <- trajectory(simplePendulum,
                                        y0         = y0,
                                        tlim       = c(0, 10),
                                        parameters = 5)