dt=10.^(-2);
y=2.;
x=1;
z=1;
t=0;
n=0;
Do[
xp=10 (y-x);
yp=28 x-y - x z;
zp = -8/3  z+x y;

dx =xp*dt;
dy=yp*dt;
dz = zp *dt;

aux = x;
x = x+ dx;
If[ aux*x <0, n=n+1;Print["change of sign at t= ", t]];

y=y+dy;
z = z+dz;
t =t +dt,
{k,1,50 10^2}];
Print["number of changes of sign is  ",n];
Print["t = ",t,"  ",x,"  ",y,"  ",z]




change of sign at t= 36.48
change of sign at t= 38.51
change of sign at t= 39.25
change of sign at t= 39.96
change of sign at t= 42.11
change of sign at t= 45.53
change of sign at t= 48.82
change of sign at t= 49.56
number of changes of sign is  24
t = 50.  11.119  18.9299  18.2091



Now we do other things
dt=10.^(-2);
y=2.;
x=1;
z=1;
t=0;
n=0;
pts={{x,y,z}};
Do[
xp=10 (y-x);
yp=28 x-y - x z;
zp = -8/3  z+x y;

dx =xp*dt;
dy=yp*dt;
dz = zp *dt;

aux = x;
x = x+ dx;
y=y+dy;
z = z+dz;
pts=Append[pts,{x,y,z}];
t =t +dt,
{k,1,5 10^2}];



pts



Graphics3D[Line[pts]]