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PSI
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