and internal radii, respectively. If the ratio w
R
=w
r
is irrational, it is
said that the frequencies are incommensurate. Then the trajectories
(7.3.11) never close on themselves and eventually cover the entire
torus surface. Nevertheless, such a motion is predictable, and thus it
is not chaotic. Another type of attractor that appears in three-dimen-
sional systems is the strange attractor. It will be introduced using the
famous Lorenz model in the next section.
7.4 LORENZ MODEL
The Lorenz model describes the convective dynamics of a fluid
layer with three dimensionless variables:
dX
dt
¼ p(Y X)
dY
dt
¼XZ þ rX Y
dZ
dt
¼ XY bZ (7:4:1)
12
10
8
6
4
2
0
2
4
6
8
10
12
12 10 8 6 4 2
024681012
Figure 7.6 Toroidal trajectories (7.3.11) in the X-Y plane for R ¼ 10, r ¼ 1,
w
R
¼ 100, w
r
¼ 3.
Nonlinear Dynamical Systems 79
In (7.4.1), the variable X characterizes the fluid velocity distribution,
and the variables Y and Z describe the fluid temperature distribution.
The dimensionless parameters p, r, and b characterize the thermo-
hydrodynamic and geometric properties of the fluid layer. The Lorenz
model, being independent of the space coordinates, is a result of signifi-
cant simplifications of the physical process under consideration [5, 7].
Yet, this model exhibits very complex behavior. As it is often done in
the literature, we shall discuss the solutions to the Lorenz model for
the fixed parameters p ¼ 10 and b ¼ 8=3. The parameter r (which is the
vertical temperature difference) will be treated as the control parameter.
At small r 1, any trajectory with arbitrary initial conditions ends
at the state space origin. In other words, the non-convective state at
X ¼ Y ¼ Z ¼ 0 is a fixed point attractor and its basin is the entire
phase space. At r > 1, the system acquires three fixed points. Hence,
the point r ¼ 1 is a bifurcation. The phase space origin is now repel-
lent. Two other fixed points are attractors that correspond to the
steady convection with clockwise and counterclockwise rotation, re-
spectively (see Figure 7.7). Note that the initial conditions define
8
6
4
2
0
2
4
6
8
10
8 6 4 202468
A : X-Y, Y(0) = 1
B : X-Z, Y(0) = 1
C : X-Y, Y(0) = 1
D : X-Z, Y(0) = 1
X
YZ
A
B
C
D
Figure 7.7 Trajectories of the Lorenz model for p ¼ 10, b ¼ 8/3, r ¼ 6, X(0)
¼ Z(0) ¼ 0, and different Y(0).
80 Nonlinear Dynamical Systems
which of the two attractors is the trajectory’s final destination. The
locations of the fixed points are determined by the stationary solution
dX
dt
¼
dY
dt
¼
dZ
dt
¼ 0(7:4:2)
Namely,
Y ¼ X, Z ¼ 0:5X
2
,X¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b(r 1)
p
(7:4:3)
When the parameter r increases to about 13.93, the repelling
regions develop around attractors. With further growth of r, the
trajectories acquire the famous ‘‘butterfly’’ look (see Figure 7.8). In
this region, the system becomes extremely sensitive to initial condi-
tions. An example with r ¼ 28 in Figure 7.9 shows that the change of
Y(0) in 1% leads to completely different trajectories Y(t). The system
is then unpredictable, and it is said that its attractors are ‘‘strange.’’
With further growth of the parameter r, the Lorenz model reveals
new surprises. Namely, it has ‘‘windows of periodicity’’ where the
trajectories may be chaotic at first but then become periodic. One of
the largest among such windows is in the range 144 < r < 165. In this
parameter region, the oscillation period decreases when r grows. Note
30
20
10
0
10
20
30
40
50
60
20 15 1010 55 0 5 10152025
X-Y
X-Z
X
YZ
Figure 7.8 Trajectories of the Lorenz model for p ¼ 10, b ¼ 8/3 and r ¼ 28.
Nonlinear Dynamical Systems 81

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