with (plots):
with (DEtools):
courbes := proc (sys, t0, x0, y0)
local p, fcns;
fcns := {x(t), y(t)};
p := dsolve ({op(sys), x(t0)=x0, y(t0)=y0}, fcns, type = numeric):
odeplot (p, [x(t), y(t)], 0..10, numpoints=500, scaling=constrained, color=black):
end:
portrait := proc (sys, tmax,x0,y0)
local vars, ini, domaine;
vars := [x(t), y(t)];
domaine := 0..tmax;
ini := [[x(0)=x0,y(0)=y0]];
phaseportrait (sys, vars, t=domaine,ini,stepsize=.05,scene=[x(t),y(t)], linecolour = sin(2*t), method=classical[foreuler], scaling=constrained);
end:
sys :=[diff(x(t),t) = y(t), diff(y(t),t) = -x(t)+x(t)^2];
courbes (sys, 0, .9, 0);
portrait (sys, 10, .9, 0);
sys := [diff(x(t),t) = x(t) * (1-y(t)) , diff(y(t),t) = -y(t) * (1-x(t))];
courbes (sys, 0, .9, .5);
portrait (sys, 10, 1.3, .5);
fieldplot([x*(1-y), -y*(1-x)], x=0..5, y=0..5, fieldstrength=fixed);
sys := [D(x)(t) = y(t), D(y)(t) = -y(t)-sin(x(t))^2];
portrait (sys, 15, 1, 1);
sys := [D(x)(t) = y(t), D(y)(t) = -x(t) + (1-x(t)^2-y(t)^2)*y(t)];
portrait (sys, 100, 0., .01);
sys := [diff(x(t),t)=y(t), diff(y(t),t)=-x(t)+x(t)^2];
portrait (sys, 41, 0, .3);