The Lorenz system

Chapter 4 The Lorenz system

The Lorenz equations were first derived and studied by American mathemati- cian and meteorologist Edward Lorenz in the 1960s. The system models the Rayleigh-Bénard convection, a natural atmospheric heat convection. In the model,

a fluid cell is located between two parallel plates with different temperatures[ 5 ], which is described by a system of nonlinear partial differential equations. The sys- tem is then subjected to a series of approximations[ 19 ] with some terms neglected. The resulting system is a third-order autonomous system of nonlinear equations, usually written as:

where σ, ρ and β are positive constants corresponding to physical constants and parameters[ 20 ]. According to [ 5 ] , “the variable x is proportional to the intensity of convective motion, y is proportional to the temperature difference between ascending and descending currents, and z is proportional to the distortion from linearity of the vertical temperature profile.”

During his research, Lorenz encountered and identified a new kind of dynamic behavior, now called the strange attractor. His groundbreaking discovery became the starting point of a new branch of applied sciences, the chaos theory, which became one of the most important scientific theorem of the 20th century. He described the so-called butterfly effect, which refers the extreme sensitivity to differences in initial conditions, which he used to explain why weather forecasts are inaccurate. The strange attractor of the system, called Lorenz attractor was

a subject of extensive research in mathematics.

4.1 Fixed points

We shall start the examination of the system as usual, by finding the fixed points:

From the first equation, x = y follows, which after substituting into the second one yields:

(4.3) Therefore the origin is a trivial solution, which corresponds to no convection. Here

0 = x(ρ − 1 − z).

the Jacobian matrix is

0 0 −β From the structure of the matrix the eigenvalue λ = −b is obvious. The other

y  x −β

(x,y,z)=(0,0,0)

two roots are the λ 1,2 solutions of

For ρ < 1 both eigenvalues are negative, so the origin is a stable node. For ρ > 1 J 0 will have one positive and one negative eigenvalue, thus the origin becomes a saddle point. Another solution of 4.3 is z =ρ−1, in which case the third equation of 4.2 yields

(4.7) for ρ > 1.

2 0=x 2 − β(ρ − 1) ⇒ x = ± β(ρ − 1)

In this case there are two new equilibrium points

± q β(ρ − 1), ± β(ρ − 1), ρ − 1 , which correspond to steady convection. The Ja- cobian matrix at these points is

The eigenvalues at these points are difficult to calculate analytically. However, the system has the symmetry (x, y, z) → (−x, −y, z), thus we know, that these two equilibrium points have the same eigenvalues[ 5 ]. It can be shown, that for

they are stable[ 20 ], which for the parameter values –as set by σ−β−1

470 Lorenz– β ≡ 8/3 and σ ≡ 10 yields the stability condition ρ <

≈ 24.74. At ρ = 1

19 the system suffers a so called pitchfork bifurcation[ 5 ]. This type of bifurcation

occurs when a stable fix point (in our case the origin) loses stability and “gives

birth” to a pair of new, stable fix points (see Figure C.12 ).

4.2 Strange attractor

The Lorenz attractor was the first chaotic attractor found and it since that it has been the subject of enormous literature. The question of the geometry of the attractor was one of the 14 unsolved problems for the 21st century, proposed by

Steve Smale in 1998[ 3 ].

The attractor starts to show up somewhat before ρ = 24.74. In fact, for a short interval of ρ, the chaotic attractor coexists with the stable nodes[ 21 ]. These nodes, on the other hand, are very sensitive to perturbation and can be found only by starting trajectories from the nodes’ basins of attraction.

max(z) 200

Figure 4.1. Bifurcation diagram of the Lorenz system, with the parameter ρ

While increasing ρ further, we can find periodic solutions. To find them, it is

useful to plot the bifurcation diagram of the system (see Figure 4.1 ). It can be shown by analytical methods[ 5 , 21 ], that for ρ → ∞ the solutions are periodic.

Indeed, the system has a stable periodic solution for ρ > 313, which is also sym- metrical. At ρ = 313 the limit cycle loses stability through a pitchfork bifurcation and two stable asymmetric limit cycles appear. At ρ = 230 they both suffer period doubling, which can be seen on the bifurcation diagram too. The phase portraits

of the limit cycles for these values can be seen on Figure C.13 . More periodic solutions appear in “windows” of the bifurcation diagram: at ρ = 162 the system again has a stable symmetric limit cycle, while at ρ = 100 two asymmetric limit

cycles exist. These can be seen on Figure C.14 . When the system is in chaotic state, a single trajectory “draws” the whole attractor, which is possible because of the attractor’s fractal geometry. The image of the attractor –which became a symbol of chaos– can be seen on Figure 4.2 .

Figure 4.2. 3D picture of the chaotic Lorenz attractor at ρ = 40

(Projections on Figure C.15 )