Continuer example: a curve object

In this section a simple example is presented to illustrate the basic use of the continuer. This example generates a curve $g$ in the $(x,y)$-plane such that $x^2+y^2=1$. So if the user specifies a point reasonably close to this curve we get the unit circle. The defining function is

$$F(x,y) = x^2+y^2-1$$ (97)

In the following listing this curve is implemented. Do note that while there are no options needed, the curve file must return an option structure (see section 3.3).

curve.m
1	function out = curve
2	%
3	% Curve file of circle
4	%
5
6	out{1} = @curve_func;
7	out{2} = @defaultprocessor;
8	out{3} = @options;
9	out{4} = []; %@jacobian;
10	out{5} = []; %@hessians;
11	out{6} = []; %@testf;
12	out{7} = []; %@userf;
13	out{8} = []; %@process;
14	out{9} = []; %@singmat;
15	out{10} = []; %@locate;
16	out{11} = []; %@init;
17	out{12} = []; %@done;
18	out{13} = @adapt;
19	function f = curve_func(arg)
20	x = arg;
21	f = x(1)^2+x(2)^2-1;
22
23	function varargout = defaultprocessor(varargin)
24	if nargin > 2
25	s = varargin{3};
26	varargout{3} = s;
27	end
28	% no special data
29	varargout{2} = [];
30	% all done succesfully
31	varargout{1} = 0;
32
33	function option = options
34	option = contset;
35
36	function [res,x,v] = adapt(x,v)
37	res=[];
38
curve.m

The file curve.m is stored in the directory Testruns/TestSystems. Starting computations at $(x,y)=(1,0)$, the output in MATLAB looks like:

>> init;
>> [x,v,s]=cont(@curve,[1;0]);
first point found
tangent vector to first point found
Closed curve detected at step 70

elapsed time  = 0.1 secs
npoints curve = 70

The generated curve is plotted in Figure 32 with the command:

>> cpl(x,v,s)

In this case x has dimension 2, so a 2D-plot is drawn with the first component of x (value of the state variable $x$) on the x-axis and the second component of x (value of the state variable $y$) on the y-axis. The above commands are also executed by running testdrawcurve; the file testdrawcurve.m is in the directory Testruns.

Figure 32: Computed curve of curve.m