σ = 10;
b = 8/3;
r = 28;

fixedpoints =
Graphics3D[
{Red,PointSize[Medium],
Point[{0,0,0}],
Point[{Sqrt[b*(r-1)],Sqrt[b*(r-1)],r-1}],
Point[{-Sqrt[b*(r-1)],-Sqrt[b*(r-1)],r-1}]
}
];


tmax=40;
sol1 =
NDSolve[
{x'[t]==σ(y[t]-x[t]),
y'[t]==r*x[t]-y[t]-x[t]*z[t],
z'[t]==x[t]*y[t]-b*z[t],
x[0]==0,
y[0]==2,
z[0]==0
},
{x,y,z},
{t,tmax}
];

plot1 =
ParametricPlot3D[
Evaluate[
{x[t],y[t],z[t]}/.sol1,
{t,0,tmax}
],
PlotRange->All
];

Show[
fixedpoints,
plot1
]

sol2 =
NDSolve[
{x'[t]==σ(y[t]-x[t]),
y'[t]==r*x[t]-y[t]-x[t]*z[t],
z'[t]==x[t]*y[t]-b*z[t],
x[0]==0,
y[0]==-2,
z[0]==0
},
{x,y,z},
{t,tmax}
];

plot2 =
ParametricPlot3D[
Evaluate[
{x[t],y[t],z[t]}/.sol2,
{t,0,tmax}
],
PlotRange->All
];

Show[
fixedpoints,
plot2
]

sol3 =
NDSolve[
{x'[t]==σ(y[t]-x[t]),
y'[t]==r*x[t]-y[t]-x[t]*z[t],
z'[t]==x[t]*y[t]-b*z[t],
x[0]==0,
y[0]==2.01,
z[0]==0
},
{x,y,z},
{t,tmax}
];

plot3 = 
ParametricPlot3D[
Evaluate[
{x[t],y[t],z[t]}/.sol3,
{t,0,tmax}
],
PlotRange->All
];

Show[
fixedpoints,
plot3
]


Plot[
Evaluate[{x[t]/.sol1,x[t]/.sol3}],
{t,0,25},
PlotRange->All
]