1. Lorenz system Stationary states: pitchfork bifurcation: r= 1
Jacobi matrix:
Characteristic equation: Det[lI – (Jacobi)] = 0

2. Hopf bifurcation characteristic equation
characteristic equation at Hopf bifurcation ( l= – a, l = ±iw) :
Condition for Hopf bifurcation:
Substitute stationary solution:
for =8/3

3. Stable and unstable manifolds
The trivial equilibrium at r>1 is a saddle: there is one unstable and two stable directions. The stable manifold is two-dimensional:. there is a plane, spanned by the first and second eigenvectors, where all trajectories move towards the stationary point. Any deviation from this plane would grow, so that the trajectory will be deflected to either side along the unstable direction, along the vector
The nontrivial equilibria beyond the Hopf bifurcation line are saddle-nodes: they have one stable and two unstable directions. There is a line (one--dimensional stable manifold) along which a trajectory approaches the equilibrium, The trajectories spiral out in the unstable manifold.

4. Homoclinic bifurcation: r=13.926
s=10, b=8/3, r=13.92: trajectories going out of the trivial equilibrium along the unstable direction approach the closest nontrivial equilibrium
s=10, b=8/3, r=13.93: trajectories trajectories turn around and converge to the nontrivial equilibrium of the opposite sign

5. Preturbulence x–y plane z–y plane
s=10, b=8/3, r=22: trajectories wander around both nontrivial equilibria before settling at one of them
x–y plane
z–y plane

6. Preturbulence – Poincare map
s=10, b=8/3, r=22: trajectories wander around both nontrivial equilibria before settling at one of them
Poincare map in plane z=21

7. Turbulence Poincare map in plane z=27 x–y plane
s=10, b=8/3, r=28:
(above the Hopf bifurcation)
x–y plane
Injected trajectory spirals out

8. Turbulence –statistics
Visiting alternative parts of the attractor
Distribution of “residence times” in one part of the attractor

9. Time-delay plots Plot in {x,y,z} space Plot of {x(t), x(t+1), x(t+2)}
s=10, b=8/3, r=28:
Plot in {x,y,z} space
Plot of {x(t), x(t+1), x(t+2)}

10. Lorenz map
positions relative to the nearest equilibrium point at two consecutive iterations approximated by
b = 4.726, n = 4
Iterations of the Lorenz map

11. Statistics of Lorenz map
Iterations of the Lorenz map
b = 4.726, n = 4
Distribution of “residence times” in one part of the attractor
Table of “residence times”