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

¼XZ 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 −10−10 −5−5 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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