2D plots of ODE solutions
This functionality does not run in MATLAB.
plot::Ode2d(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::Ode2d(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::Ode2d([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::Ode2d([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::Ode2d(f, [t_{0}, t_{1},...],
Y_{0})
renders twodimensional projections
of the solutions of the initial value problem given by f
, t_{0}
and Y_{0}
.
plot::Ode2d(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 2D plot points [x_{i}, y_{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
can
be used to produce the input parameters f, t_{0},
Y_{0}
from a set of differential expressions
representing the ODE. See Example 1.
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}
,
and so on. Each generator G
is internally called
in the form G(t_{0}, Y_{0}),
G(t_{1}, Y_{1}),...
to
produce a sequence of plot points in 2D.
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 2 elements representing
the (x, y) 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, 2D data as lists with 2 elements for plot::Ode2d
.
Here are some examples:
G := (t, Y) > [t, Y_1]
creates
a 2D plot of the first component of the solution vector along the yaxis,
plotted against the time variable t
along the xaxis
G := (t, Y) > [Y_1, Y_2]
creates
a 2D phase plot, plotting the first component of the solution along
the xaxis
and the second component along the yaxis.
The result is a solution curve in phase space (parametrized by the
time t
).
If no generators are given, plot::Ode2d
by
default plots all components of the solution as functions of time,
using [Splines, Points]
as the style.
Note that arbitrary values associated with the solution curve
may be displayed graphically by an appropriate generator G
.
See Example 2.
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},
…
. See Example 1, Example 2, and Example 3.
The graphical data produced by each of the generators G_{1},
G_{2},...
consists of a sequence of
mesh points in 2D.
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::Ode2d
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 
AntiAliased  antialiased lines and points?  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  
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 
The following procedure f
together with the
initial value Y0
represent the initial value problem
, Y(0)
= 2. In MuPAD^{®}, the onedimensional vector Y is
represented by a list with one element. The body of the function f
below
addresses the first (and only) entry of this list as Y_{1}
and
returns the onedimensional vector t Y  Y^{2} as
a list with one element. Also the initial condition Y_{0}
is
a onedimensional vector represented by a list. For details on the
format of f
, please see numeric::odesolve
:
f := (t, Y) > [t*Y[1]  Y[1]^2]: Y0 := [2]:
Alternatively, the utility function numeric::ode2vectorfield
can be used
to generate the input parameters in a more intuitive way:
[f, t0, Y0] := [numeric::ode2vectorfield( {y'(t) = t*y(t)  y(t)^2, y(0) = 2}, [y(t)])]
The numerical solution is to consist of sample points over the
time mesh t_{i} = i, i =
0, 1, …, 10. We use the default generator
of plot::Ode2d
. This generates the sample points
together with a smooth spline curve connecting these points:
p := plot::Ode2d(f, [$ 0..10], Y0, PointSize = 2*unit::mm, PointStyle = Stars):
Finally, the ode solution is rendered by a call to plot
:
plot(p, TicksDistance = 2.0, GridVisible = TRUE, SubgridVisible = TRUE):
Consider the nonlinear oscillator
,
.
As a dynamical system for
,
solve the following initial value problem
, Y(0)
= Y_{0}.
For details on the format of f
, see numeric::odesolve
:
f := (t, Y) > [Y[2],  Y[1]^7]: Y0 := [1, 0]:
The following generator produces a plot of the solution Y(t) against the time parameter t:
G1 := (t, Y) > [t, Y[1]]:
For demonstration purposes, plot the function
.
The generator G2
produces the values
along
the solution and plots these values against t:
G2 := (t, Y) > [t, Y[1]^2/2 + Y[2]^2/2]:
The energy function, or "Hamiltonian",
must
be conserved along the solution curve because the total energy of
the system is constant. Define a corresponding generator G3
to
plot
as
a function of t:
G3 := (t, Y) > [t, Y[1]^8/8 + Y[2]^2/2]:
The solution curve is combined with the graph of the function and the Hamiltonian:
p := plot::Ode2d(f, [i/2 $ i = 0..40], Y0, [G1, Style = Lines, Color = RGB::Red], [G1, Style = Points, Color = RGB::Black], [G2, Style = Lines, Color = RGB::Blue], [G2, Style = Points, Color = RGB::Black], [G3, Style = Lines, Color = RGB::Green], [G3, Style = Points, Color = RGB::Black], PointSize = 1.5*unit::mm, LineWidth = 0.2*unit::mm ):
Note that by using each generator twice, you can set different colors for the lines and points. The renderer is called:
plot(p):
The Hamiltonian G3
is constant which verifies
the accuracy of the integrator.
To visualize the dependency of the trajectory on the initial
conditions, animate plot::Ode2d
over different
values of
:
plot(plot::Ode2d(f, [i/6 $ i = 0..120], [1, a], a = 1/2..1/2, [G1, Style = Lines, Color = RGB::Red], [G2, Style = Lines, Color = RGB::Blue], [G3, Style = Lines, Color = RGB::Green], LineWidth = 0.2*unit::mm, Frames=25))
Consider the initial value problem , y(0) = 0:
f := (t, y) > t*sin(t + y^2): Y0:= [0]:
Use numeric::ode2vectorfield
to
generate the following vector field which is tangent to the solution
curves:
p1 := plot::VectorField2d([1, f(t, y)], t = 0..4, y = 1.2..1.2, Mesh = [21, 25], Color = RGB::Black):
The following object represents the plot of the solution as
a function of t
:
p2 := plot::Ode2d( (t,Y) > [f(t, Y[1])], [i/3 $ i=0..12], Y0, [(t, Y) > [t, Y[1]], Style = Points, Color = RGB::Red], [(t, Y) > [t, Y[1]], Style = Splines, Color = RGB::Blue]):
Define the point size explicitly:
p2::PointSize := 2*unit::mm:
Finally, combine the vector field and the ODE plot to a scene and call the renderer:
plot(p1, p2, XTicksDistance = 0.5, YTicksDistance = 0.2, Axes = Frame, AxesTitles = ["t", "y"], GridVisible = TRUE):
By default, numeric::odesolve
(which
is used by plot::Ode2d
internally) uses adaptive
step sizes and a method of order 8. Usually, there is no reason to
change these settings, except for demonstrative purposes. In the following
animation, use a straightforward explicit Euler method (of first order)
and show how decreasing the step size improves the quality of the
calculated solution.
The differential equation is , obviously fulfilled by the exponential function:
[f, t0, Y0] := [numeric::ode2vectorfield( {y'(t)=y(t), y(0)=1}, [y(t)])]:
To judge the quality of the numerical solution, plot the symbolic solution alongside the approximation:
plot(plot::Function2d(exp(x), x=0..3, Color = RGB::Black, LineStyle = Dashed), plot::Ode2d(f, [Automatic, 0, 3, 1/n], Y0, n = 1..50, EULER1, Stepsize = 1/n, [(t, Y) > [t, Y[1]], Style=[Lines, Points]]))

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 