with(plots): with(DEtools):
phase := (EQS, inicond) -> DEplot( EQS, [x(t),y(t)], t=0..40, x=-1..1, y=-1..1,
[inicond], arrows=medium):
Line\303\241ris rendszerek
Minden ir\303\241nyban ny\303\272jt\303\263:
EQS := [diff(y(t),t) = y(t), diff(x(t), t) = x(t)];
phase(EQS, [x(0)=0.5, y(0)=0.5]);
Minden ir\303\241nybn zsugor\303\255t\303\263:
EQS := [diff(y(t),t) = -y(t), diff(x(t), t) = -x(t)];
phase(EQS, [x(0)=.5, y(0)=.5]);
Felem\303\241s:
EQS := [diff(y(t),t) = y(t), diff(x(t), t) = -x(t)];
phase(EQS, [x(0)=.5, y(0)=.5]);
Nem hiperbolikus:
EQS := [diff(y(t),t) = 0, diff(x(t), t) = -x(t)];
phase(EQS, [x(0)=.5, y(0)=.5]);
Nem tengely ir\303\241ny\303\272 saj\303\241t\303\251rt\303\251k:
EQS := [diff(y(t),t) = -y(t)-3*x(t), diff(x(t), t) = -y(t)-x(t)];
phase(EQS, [x(0)=.5, y(0)=.5]);
Komplex saj\303\241t\303\251rt\303\251kek (zsugoror\303\255t\303\263):
EQS := [diff(y(t),t) = -2*y(t)-x(t), diff(x(t), t) = y(t)-x(t)];
phase(EQS, [x(0)=.5, y(0)=.5]);
Komplex saj\303\241t\303\251rt\303\251kek (ny\303\272jt\303\263):
EQS := [diff(y(t),t) = 2*y(t)-x(t), diff(x(t), t) = y(t)+x(t)];
phase(EQS, [x(0)=.5, y(0)=.5]);
Komplex saj\303\241t\303\251rt\303\251k (nem hiperbolikus):
EQS := [diff(y(t),t) = -x(t), diff(x(t), t) = y(t)];
phase(EQS, [x(0)=.5, y(0)=.5]);
3 dimenzi\303\263:
EQS := [diff(z(t),t) = 100*z(t), diff(y(t),t) = -0.05*y(t)-x(t), diff(x(t), t) = y(t)-0.05*x(t)];
DEplot3d( EQS, [x(t),y(t),z(t)], t=0..100, stepsize=.5, x=-1..1, y=-1..1, z=0..10,
[[x(0)=.5, y(0)=.5, z(0)=0]], arrows=medium);
T\303\263rusz
R := 2: r := 0.5: tore := (x, y)->[evalf((R+r*cos(y)) * cos(x)), evalf((R+r*cos(y)) * sin(x)), evalf(sin(y))]:
plot3d(tore(x,y), x=0..2*Pi, y=0..2*Pi, scaling=constrained);
spacecurve(tore(x,21/13*x), x=0..1000, numpoints=1000, scaling = constrained);
animate(spacecurve, [[frac(x),frac(sqrt(2)*x),0], x=t..t+1, numpoints = 10], t=0..50, trace=50, frames=50, scaling=constrained);
animate(spacecurve, [tore(x,sqrt(2)*x), x=t..t+1, numpoints = 10], t=0..50, trace=50, frames=50, scaling=constrained);
Bifurk\303\241ci\303\263k
EQS := [diff(x(t),t) = mu-x(t)^2, diff(y(t), t) = -y(t)];
animate(DEplot,[ EQS, [x(t),y(t)], t=0..40, x=-1..1, y=-1..1,
[[x(0)=.5, y(0)=.5]], arrows=medium], mu=-1..1);
EQS := [diff(x(t),t) = mu*x(t)-x(t)^3, diff(y(t), t) = -y(t)];
animate(DEplot,[ EQS, [x(t),y(t)], t=0..40, x=-1..1, y=-1..1,
[[x(0)=.5, y(0)=.5]], arrows=medium], mu=-1..1);
EQS := [diff(x(t),t) = -y(t)+ mu*(x(t)-y(t))+(x(t)-y(t))*(x(t)^2+y(t)^2),
diff(y(t), t) = x(t)+ mu*(x(t)+y(t))+(x(t)+y(t))*(x(t)^2+y(t)^2)];
animate(DEplot,[ EQS, [x(t),y(t)], t=0..-40, x=-1..1, y=-1..1,
[[x(0)=.2, y(0)=.2]], arrows=medium], mu=-2..1);