3D plots of ODE solutions
This functionality does not run in MATLAB.
plot::Ode3d(f
,[t_{0}, t_{1}, …]
,Y_{0}
, <[G_{1}, <Style = style_{1}>, <Color = c_{1}>], [G_{2}, <Style = style_{2}>, <Color = c_{2}>], …
>, <method
>, <RelativeError = rtol
>, <AbsoluteError = atol
>, <Stepsize = h
>, <a = a_{min} .. a_{max}
>,options
) plot::Ode3d(f
,[Automatic, t_{start}, t_{end}, t_{step}]
,Y_{0}
, <[G_{1}, <Style = style_{1}>, <Color = c_{1}>], [G_{2}, <Style = style_{2}>, <Color = c_{2}>], …
>, <method
>, <RelativeError = rtol
>, <AbsoluteError = atol
>, <Stepsize = h
>, <a = a_{min} .. a_{max}
>,options
) plot::Ode3d([t_{0}, t_{1}, …]
,f
,Y_{0}
, <[G_{1}, <Style = style_{1}>, <Color = c_{1}>], [G_{2}, <Style = style_{2}>, <Color = c_{2}>], …
>, <method
>, <RelativeError = rtol
>, <AbsoluteError = atol
>, <Stepsize = h
>, <a = a_{min} .. a_{max}
>,options
) plot::Ode3d([Automatic, t_{start}, t_{end}, t_{step}]
,f
,Y_{0}
, <[G_{1}, <Style = style_{1}>, <Color = c_{1}>], [G_{2}, <Style = style_{2}>, <Color = c_{2}>], …
>, <method
>, <RelativeError = rtol
>, <AbsoluteError = atol
>, <Stepsize = h
>, <a = a_{min} .. a_{max}
>,options
)
plot::Ode3d(f, [t_{0}, t_{1},...],
Y_{0})
renders threedimensional projections
of the solutions of the initial value problem given by f
, t_{0}
and Y_{0}
.
plot::Ode3d(f, [t_{0}, t_{1},...],
Y_{0}, [G])
computes a mesh of numerical
sample points Y(t_{0}), Y(t_{1}),
… representing the solution Y(t) of
the first order differential equation (dynamical system)
.
The procedure
maps these solution points (t_{i}, Y(t_{i})) in ℝ×ℂ^{n} to a mesh of 3D plot points [x_{i}, y_{i}, z_{i}]. These points can be connected by straight lines or interpolating splines.
Internally, a sequence of numerical sample points
Y_1 := numeric::odesolve(f, t_0..t_1, Y_0, Options)
,
Y_2 := numeric::odesolve(f, t_1..t_2, Y_1, Options)
,
and so on
is computed, where Options
is some combination
of method
, RelativeError = rtol
, AbsoluteError
= atol
, and Stepsize = h
. See numeric::odesolve
for
details on the vector field procedure f
, the initial
condition Y_{0}
, and the options.
The utility function numeric::ode2vectorfield
may
be used to produce the input parameters f, t_{0},
Y_{0}
from a set of differential expressions
representing the ODE.
Each of the "generators of plot data" G_{1}
, G_{2}
etc.
creates a graphical solution curve from the numerical sample points Y_{0}
, Y_{1}
etc.
Each generator G
, say, is internally called in
the form G(t_{0}, Y_{0}),
G(t_{1}, Y_{1}), …
to
produce a sequence of plot points in 3D.
The solver numeric::odesolve
returns
the solution points Y_{0}
, Y_{1}
,
and so on, as lists or onedimensional arrays (the actual type is
determined by the initial value Y_{0}
).
Consequently, each generator G
must accept two
arguments (t, Y)
: t
is a real
parameter, Y
is a "vector" (either
a list or a 1dimensional array).
Each generator must return a list with 3 elements representing
the (x, y, z) coordinates
of the graphical point associated with a solution point (t,
Y)
of the ODE. All generators must produce graphical data
of the same dimension, that is, for plot::Ode3d
,
3D data as lists with 3 elements. For example, G := (t, Y)
> [Y_1, Y_2, Y_3]
creates a 3D phase plot of the first
three components of the solution curve.
If no generators are given, plot::Ode3d
by
default plots each group of two components as functions of time with
the same style.
Note that arbitrary values associated with the solution curve
may be displayed graphically by an appropriate generator G
.
Several generators G_{1}, G_{2}
,
and so on, can be specified to generate several curves associated
with the same numerical mesh Y_{0}, Y_{1},
…
.
The graphical data produced by each of the generators G_{1},
G_{2},...
consists of a sequence of
mesh points in 3D.
With Style = Points
, the graphical
data are displayed as a discrete set of points.
With Style = Lines
, the graphical
data points are displayed as a curve consisting of straight line segments
between the sample points. The points themselves are not displayed.
With Style = Splines
, the graphical
data points are displayed as a smooth spline curve connecting the
sample points. The points themselves are not displayed.
With Style = [Splines, Points]
and Style
= [Lines, Points]
, the effects of the styles used are combined,
that is, both the evaluation points and the straight lines or splines,
respectively, are displayed.
The plot attributes accepted by plot::Ode3d
include Submesh
= n
, where n is
some positive integer. This attribute only has an effect on the curves
which are returned for the graphical generators with Style
= Splines
and Style = [Splines, Points]
,
respectively. It serves for smoothening the graphical spline curve
using a sufficiently high number of plot points.
n
is the number of plot points between two
consecutive numerical points corresponding to the time mesh. The default
value is n = 4,
that is, the splines are plotted as five straight line segments connecting
the numerical sample points.
Attribute  Purpose  Default Value 

AbsoluteError  maximal absolute discretization error  
AffectViewingBox  influence of objects on the ViewingBox of
a scene  TRUE 
Colors  list of colors to use  [RGB::Blue , RGB::Red , RGB::Green , RGB::MuPADGold , RGB::Orange , RGB::Cyan , RGB::Magenta , RGB::LimeGreen , RGB::CadmiumYellowLight , RGB::AlizarinCrimson , RGB::Aqua , RGB::Lavender , RGB::SeaGreen , RGB::AureolineYellow , RGB::Banana , RGB::Beige , RGB::YellowGreen , RGB::Wheat , RGB::IndianRed , RGB::Black ] 
Frames  the number of frames in an animation  50 
Function  function expression or procedure  
InitialConditions  initial conditions of the ODE  
Legend  makes a legend entry  
LegendText  short explanatory text for legend  
LegendEntry  add this object to the legend?  FALSE 
LineWidth  width of lines  0.35 
LineStyle  solid, dashed or dotted lines?  Solid 
LinesVisible  visibility of lines  TRUE 
Name  the name of a plot object (for browser and legend)  
ODEMethod  the numerical scheme used for solving the ODE  DOPRI78 
ParameterEnd  end value of the animation parameter  
ParameterName  name of the animation parameter  
ParameterBegin  initial value of the animation parameter  
ParameterRange  range of the animation parameter  
PointSize  the size of points  1.5 
PointStyle  the presentation style of points  FilledCircles 
PointsVisible  visibility of mesh points  TRUE 
Projectors  project an ODE solution to graphical points  
RelativeError  maximal relative discretization error  
Stepsize  set a constant step size  
Submesh  density of submesh (additional sample points)  4 
TimeEnd  end time of the animation  10.0 
TimeMesh  the numerical time mesh  
TimeBegin  start time of the animation  0.0 
TimeRange  the real time span of an animation  0.0 .. 10.0 
Title  object title  
TitleFont  font of object titles  [" sansserif " , 11 ] 
TitlePosition  position of object titles  
TitleAlignment  horizontal alignment of titles w.r.t. their coordinates  Center 
TitlePositionX  position of object titles, x component  
TitlePositionY  position of object titles, y component  
TitlePositionZ  position of object titles, z component  
USubmesh  density of additional sample points for parameter "u"  4 
Visible  visibility  TRUE 
VisibleAfter  object visible after this time value  
VisibleBefore  object visible until this time value  
VisibleFromTo  object visible during this time range  
VisibleAfterEnd  object visible after its animation time ended?  TRUE 
VisibleBeforeBegin  object visible before its animation time starts?  TRUE 
Consider the nonlinear oscillator , . As a dynamical system for , solve the following initial value problem , Y(0) = Y_{0}:
f := (t, Y) > [Y[2], sin(t)  Y[1]^3]: Y0 := [0, 0.5]:
The following generator produces a phase plot in the (x, y) plane, embedded in a 3D plot:
G1 := (t, Y) > [Y[1], Y[2], 0]:
Further, use the z coordinate of the 3D plot to display the value of the "energy" function over the phase curve:
G2 := (t, Y) > [Y[1], Y[2], (Y[1]^2 + Y[2]^2)/2]:
The phase curve in the (x, y) plane is combined with the graph of the energy function:
p := plot::Ode3d(f, [i/5 $ i = 0..100], Y0, [G1, Style = Splines, Color = RGB::Red], [G2, Style = Points, Color = RGB::Black], [G2, Style = Lines, Color = RGB::Blue]):
Set an explicit size of the points used in the representation of the energy:
p::PointSize := 2*unit::mm:
The renderer is called:
plot(p, AxesTitles = ["y", "y'", "E"], CameraDirection = [10, 15, 5]):
The Lorenz ODE is the system
with fixed parameters p, r, b. As a dynamical system for Y = [x, y, z], solve the ODE with the following vector field:
f := proc(t, Y) local x, y, z; begin [x, y, z] := Y: [p*(y  x), x*z + r*x  y, x*y  b*z] end_proc:
Consider the following parameters and the following initial
condition Y0
:
p := 10: r := 28: b := 1: Y0 := [1, 1, 1]:
The following generator Gxyz
produces a 3D
phase plot of the solution. The generator Gyz
projects
the solution curve to the (y, z) plane
with x = 20;
the generator Gxz
projects the solution curve to
the (x, z) plane
with y =  15;
the generator Gxy
projects the solution curve to
the (x, y) plane
with z = 0:
Gxyz := (t, Y) > Y: Gyz := (t, Y) > [ 20, Y[2], Y[3]]: Gxz := (t, Y) > [Y[1], 15, Y[3]]: Gxy := (t, Y) > [Y[1], Y[2], 0 ]:
With these generators, create a 3D plot object consisting of the phase curve and its projections.
object := plot::Ode3d(f, [i/10 $ i=1..100], Y0, [Gxyz, Style = Splines, Color = RGB::Red], [Gyz, Style = Splines, Color = RGB::Grey50], [Gxz, Style = Splines, Color = RGB::Grey50], [Gxy, Style = Splines, Color = RGB::Grey50], Submesh = 7):
Finally, the plot is rendered. This call is somewhat time consuming
because it calls the numerical solver numeric::odesolve
to produce the graphical
data:
plot(object, CameraDirection = [220, 110, 150])

The vector field of the ODE: a procedure.
See


The time mesh: real numerical values. If data are displayed
with


The time mesh: real numerical values.


The initial condition of the ODE: a list or
a 1dimensional array. See


"generators of plot data": procedures mapping
a solution point


Use a specific numerical scheme (see 

Animation parameter, specified as 

Option, specified as Sets the style in which the plot data are displayed. The following
styles are available: 

Option, specified as Sets the RGB
color 

Option, specified as Sets a numerical discretization tolerance (see 

Option, specified as Sets a numerical discretization tolerance (see 

Option, specified as Sets a constant stepsize (see 