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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

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