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11.84 Sample ‘ode’ ¶

Example of phase plain created by ode solving, contour lines (cont) and flow threads.

MGL code:

subplot 2 2 0 '<_':title 'Cont':box
axis:xlabel 'x':ylabel '\dot{x}'
new f 100 100 'y^2+2*x^3-x^2-0.5':cont f

subplot 2 2 1 '<_':title 'Flow':box
axis:xlabel 'x':ylabel '\dot{x}'
new fx 100 100 'x-3*x^2'
new fy 100 100 'y'
flow fy fx 'v';value 7

subplot 2 2 2 '<_':title 'ODE':box
axis:xlabel 'x':ylabel '\dot{x}'
for $x -1 1 0.1
  ode r 'y;x-3*x^2' 'xy' [$x,0]
  plot r(0) r(1)
  ode r '-y;-x+3*x^2' 'xy' [$x,0]
  plot r(0) r(1)
next

C++ code:

void smgl_ode(mglGraph *gr)
{
	gr->SubPlot(2,2,0,"<_");	gr->Title("Cont");	gr->Box();
	gr->Axis();	gr->Label('x',"x");	gr->Label('y',"\\dot{x}");
	mglData f(100,100);	gr->Fill(f,"y^2+2*x^3-x^2-0.5");
	gr->Cont(f);
	gr->SubPlot(2,2,1,"<_");	gr->Title("Flow");	gr->Box();
	gr->Axis();	gr->Label('x',"x");	gr->Label('y',"\\dot{x}");
	mglData fx(100,100), fy(100,100);	gr->Fill(fx,"x-3*x^2");	gr->Fill(fy,"y");
	gr->Flow(fy,fx,"v","value 7");
	gr->SubPlot(2,2,2,"<_");	gr->Title("ODE");	gr->Box();
	gr->Axis();	gr->Label('x',"x");	gr->Label('y',"\\dot{x}");
	for(double x=-1;x<1;x+=0.1)
	{
		mglData in(2), r;	in.a[0]=x;
		r = mglODE("y;x-3*x^2","xy",in);
		gr->Plot(r.SubData(0), r.SubData(1));
		r = mglODE("-y;-x+3*x^2","xy",in);
		gr->Plot(r.SubData(0), r.SubData(1));
	}
}