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Â¼ Ã€XZ Ã¾ rX Ã€ Y

dt

dZ

Â¼ XY Ã€ bZ (7:4:1)

dt

80 Nonlinear Dynamical Systems

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 Â¼ 10 and b Â¼ 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 Â¼ Y Â¼ Z Â¼ 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 Â¼ 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

10

YZ

8

B

6

4

C

2

D

X

0

âˆ’8 âˆ’6 âˆ’4 âˆ’2 0 2 4 6 8

âˆ’2

= âˆ’1

A : X-Y, Y(0)

= âˆ’1

B : X-Z, Y(0)

âˆ’4 C : X-Y, Y(0) =1

A D : X-Z, Y(0) =1

âˆ’6

âˆ’8

Figure 7.7 Trajectories of the Lorenz model for p Â¼ 10, b Â¼ 8/3, r Â¼ 6, X(0)

Â¼ Z(0) Â¼ 0, and different Y(0).

81

Nonlinear Dynamical Systems

which of the two attractors is the trajectoryâ€™s final destination. The

locations of the fixed points are determined by the stationary solution

dX dY dZ

Â¼ Â¼ Â¼0 (7:4:2)

dt dt dt

Namely,

pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ

Y Â¼ X, Z Â¼ 0:5X2 , X Â¼ Ã† b(r Ã€ 1) (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 â€˜â€˜butterflyâ€™â€™ look (see Figure 7.8). In

this region, the system becomes extremely sensitive to initial condi-

tions. An example with r Â¼ 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 â€˜â€˜strange.â€™â€™

With further growth of the parameter r, the Lorenz model reveals

new surprises. Namely, it has â€˜â€˜windows of periodicityâ€™â€™ 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

60

Y Z

50

40

30

20

10

X

0

âˆ’20 âˆ’15 âˆ’10 âˆ’5 0 5 10 15 20 25

âˆ’10

âˆ’20

X-Y

X-Z

âˆ’30

Figure 7.8 Trajectories of the Lorenz model for p Â¼ 10, b Â¼ 8/3 and r Â¼ 28.

82 Nonlinear Dynamical Systems

40

Y(t)

20

0

t

0 2 4 6 8 10 12 14

âˆ’20

Y(0) = 1.00

Y(0) = 1.01

âˆ’40

Figure 7.9 Sensitivity of the Lorenz model to the initial conditions for p Â¼

10, b Â¼ 8/3 and r Â¼ 28.

that this periodicity is not described with a single frequency, and the

maximums of its peaks vary. Finally, at very high values of

r (r > 313), the system acquires a single stable limit cycle. This fascin-

ating manifold of solutions is not an exclusive feature of the Lorenz

model. Many nonlinear dissipative systems exhibit a wide spectrum of

solutions including chaotic regimes.

7.5 PATHWAYS TO CHAOS

A number of general pathways to chaos in nonlinear dissipative

systems have been described in the literature (see, e.g., [5] and refer-

ences therein). All transitions to chaos can be divided into two major

groups: local bifurcations and global bifurcations. Local bifurcations

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