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PSI b)

a)

2

2

1.5

1.5

1

1

0.5

FI

0.5 0

FI âˆ’1.5 âˆ’1 âˆ’0.5 0 0.5 1 1.5

âˆ’0.5

0

âˆ’1.5 âˆ’0.5 0.5 1.5 âˆ’1

âˆ’0.5

âˆ’1.5

âˆ’1

âˆ’2

âˆ’1.5

âˆ’2.5

âˆ’2

âˆ’2.5

Figure 7.4 Trajectories of the damped oscillator with v Â¼ 2: (a) g Â¼ 2; (b)

g Â¼ 0.

77

Nonlinear Dynamical Systems

systems. Some conservative systems may have the chaotic regimes,

too (so-called non-integrable systems) [5], but this case will not be

discussed here. One can easily identify the sources of dissipation in

real physical processes, such as friction, heat radiation, and so on. In

general, flow (7.3.2) is dissipative if the condition

X @F

N

div(F) <0 (7:3:7)

@Xi

iÂ¼1

is valid on average within the phase space.

Besides the point attractor, systems with two or more dimensions

may have an attractor named the limit cycle. An example of such an

attractor is the solution of the Van der Pol equation. This equation

describes an oscillator with a variable damping coefficient

d2 u du

Ã¾ g[(u=u0 )2 Ã€ 1] Ã¾ v2 u Â¼ 0 (7:3:8)

dt2 dt

In (7.3.8), u0 is a parameter. The damping coefficient is positive at

sufficiently high amplitudes u > u0 , which leads to energy dissipation.

However, at low amplitudes (u < u0 ), the damping coefficient be-

comes negative. The negative term in (7.3.8) has a sense of an energy

source that prevents oscillations from complete decay. If one intro-

pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ

duces u0 v=g as the unit of amplitude and 1=v as the unit of time,

then equation (7.3.8) acquires the form

d2 u du

Ã¾ (u2 Ã€ e2 ) Ã¾ u Â¼ 0 (7:3:9)

dt2 dt

where e Â¼ g=v is the only dimensionless parameter that defines the

system evolution. The flow describing the Van der Pol equation has

the following form

du dw

Â¼ (e2 Ã€ u2 ) w Ã€ u

Â¼ w, (7:3:10)

dt dt

Figure 7.5 illustrates the solution to equation (7.3.1) for e Â¼ 0:4.

Namely, the trajectories approach a closed curve from the initial

conditions located both outside and inside the limit cycle. It should

be noted that the flow trajectories never intersect, even though

their graphs may deceptively indicate otherwise. This property

follows from uniqueness of solutions to equation (7.3.8). Indeed, if the

78 Nonlinear Dynamical Systems

1.5

PSI

1

0.5

FI

0

M2 M1

âˆ’1.2 âˆ’0.8 âˆ’0.4 0 0.4 0.8 1.2 1.6 2

âˆ’0.5

âˆ’1

âˆ’1.5

Figure 7.5 Trajectories of the Van der Pol oscillator with e Â¼ 0:4. Both

trajectories starting at points M1 and M2, respectively, end up on the same

limit circle.

trajectories do intersect, say at point P in the phase space, this implies

that the initial condition at point P yields two different solutions.

Since the solution to the Van der Pol equation changes qualita-

tively from the point attractor to the limit cycle at e Â¼ 0, this point is a

bifurcation. Those bifurcations that lead to the limit cycle are named

the Hopf bifurcations.

In three-dimensional dissipative systems, two new types of attractors

appear. First, there are quasi-periodic attractors. These trajectories are

associated with two different frequencies and are located on the surface

of a torus. The following equations describe the toroidal trajectories

(see Figure 7.6)

x(t) Â¼ (R Ã¾ r sin (wr t)) cos (wR t)

y(t) Â¼ (R Ã¾ r sin (wr t)) sin (wR t)

z(t) Â¼ r cos (wr t) (7:3:11)

In (7.3.11), R and r are the external and internal torus radii, respect-

ively; wR and wr are the frequencies of rotation around the external

79

Nonlinear Dynamical Systems

12

10

8

6

4

2

0

âˆ’12 âˆ’10 âˆ’8 âˆ’6 âˆ’4 âˆ’2 0 2 4 6 8 10 12

âˆ’2

âˆ’4

âˆ’6

âˆ’8

âˆ’10

âˆ’12

Figure 7.6 Toroidal trajectories (7.3.11) in the X-Y plane for R Â¼ 10, r Â¼ 1,

wR Â¼ 100, wr Â¼ 3.

and internal radii, respectively. If the ratio wR =wr 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

Â¼ p(Y Ã€ X)

dt

dY

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

occur in some parameter range, but the trajectories become chaotic

when the system control parameter reaches the critical value. Three

types of local bifurcations are discerned: period-doubling, quasi-peri-

odicity, and intermittency. Period-doubling starts with a limit cycle at

some value of the system control parameter. With further change of

83

Nonlinear Dynamical Systems

this parameter, the trajectory period doubles and doubles until it

becomes infinite. This process was proposed by Landau as the main

turbulence mechanism. Namely, laminar flow develops oscillations at

some sufficiently high velocity. As velocity increases, another (incom-

mensurate) frequency appears in the flow, and so on. Finally, the

frequency spectrum has the form of a practically continuous band. An

alternative mechanism of turbulence (quasi-periodicity) was proposed

by Ruelle and Takens. They have shown that the quasi-periodic

trajectories confined on the torus surface can become chaotic due to

high sensitivity to the input parameters. Intermittency is a broad

category itself. Its pathway to chaos consists of a sequence of periodic

and chaotic regions. With changing the control parameter, chaotic

regions become larger and larger and eventually fill the entire

space.

In the global bifurcations, the trajectories approach simple attract-

ors within some control parameter range. With further change of the

control parameter, these trajectories become increasingly complicated

and in the end, exhibit chaotic motion. Global bifurcations are parti-

tioned into crises and chaotic transients. Crises include sudden

changes in the size of chaotic attractors, sudden appearances of the

chaotic attractors, and sudden destructions of chaotic attractors and

their basins. In chaotic transients, typical trajectories initially behave

in an apparently chaotic manner for some time, but then move to

some other region of the phase space. This movement may asymptot-

ically approach a non-chaotic attractor.

Unfortunately, there is no simple rule for determining the condi-

tions at which chaos appears in a given flow. Moreover, the same

system may become chaotic in different ways depending on its par-

ameters. Hence, attentive analysis is needed for every particular

system.

7.6 MEASURING CHAOS

As it was noticed in in Section 7.1, it is important to understand

whether randomness of an empirical time series is caused by noise or

by the chaotic nature of the underlying deterministic process. To

address this problem, let us introduce the Lyapunov exponent. The

major property of a chaotic attractor is exponential divergence of its

84 Nonlinear Dynamical Systems

nearby trajectories. Namely, if two nearby trajectories are separated

by distance d0 at t Â¼ 0, the separation evolves as

d(t) Â¼ d0 exp (lt) (7:6:1)

The parameter l in (7.6.1) is called the Lyapunov exponent. For the

rigorous definition, consider two points in the phase space, X0 and

X0 Ã¾ Dx0 , that generate two trajectories with some flow (7.3.2). If the

function Dx(X0 , t) defines evolution of the distance between these

points, then

1 jDx(X0 , t)j

l Â¼ lim ln , t ! 1, Dx0 ! 0 (7:6:2)

jDx0 j

t

When l < 0, the system is asymptotically stable. If l Â¼ 0, the system

is conservative. Finally, the case with l > 0 indicates chaos since the

system trajectories diverge exponentially.

The practical receipt for calculating the Lyapunov exponent is as

follows. Consider n observations of a time series x(t): x(tk ) Â¼ xk , k Â¼ 1,

. . . , n. First, select a point xi and another point xj close to xi . Then

calculate the distances

d0 Â¼ jxi Ã€ xj j, d1 Â¼ jxiÃ¾1 Ã€ xjÃ¾1 j, . . . , dn Â¼ jxiÃ¾n Ã€ xjÃ¾n j (7:6:3)

If the distance between xiÃ¾n and xjÃ¾n evolves with n accordingly with

(7.6.1), then

1 dn

l(xi ) Â¼ ln (7:6:4)

n d0

The value of the Lyapunov exponent l(xi ) in (7.6.4) is expected to be

sensitive to the choice of the initial point xi . Therefore, the average

value over a large number of trials N of l(xi ) is used in practice

1X N

lÂ¼ l(xi ) (7:6:5)

N iÂ¼1

Due to the finite size of empirical data samples, there are limitations

on the values of n and N, which affects the accuracy of calculating the

Lyapunov exponent. More details about this problem, as well as other

chaos quantifiers, such as the Kolmogorov-Sinai entropy, can be

found in [5] and references therein.

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