Hirsch M., Smale S., Devaney R., Differential Equations, Dynamical Systems, and an Introduction to Chaos

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296 Chapter 13 Applications in Mechanics

To fully understand the ﬂow on , we introduce the angular variable ψ in

each u

–plane via

√

2sinψ

√

2cosψ.

The torus is now parameterized by θ and ψ.Inθψ coordinates, the system

becomes

θ =

√

2cosψ

ψ =

√

cosψ.

The circles C

are now given by ψ =±π/2. Eliminating time from this

equation, we ﬁnd

dψ

dθ

Thus all nonequilibrium solutions have constant slope 1/2 when viewed in θψ

coordinates. See Figure 13.4.

Now recall the collision-ejection solutions described in Section 13.4. Each

of these solutions leaves the origin and then returns along a ray θ =θ

∗

in con-

ﬁguration space. The solution departs with v

> 0 (and so u

> 0) and returns

with v

< 0(u

< 0). In our new four-dimensional coordinate system, it follows

that this solution forms an unstable curve associated to the equilibrium point

(0,θ

∗

√

2,0) and a stable curve associated to (0,θ

∗

,−

√

2,0). See Figure 13.5.

⫹

⫺

Figure 13.4 Solutions on 

in θψ coordinates. Recall that

θ and ψ are both defined

mod 2π , so opposite sides of

this square are identified to

form a torus.

13.8 Exploration: Other Central Force Problems 297

∗

Figure 13.5 A collision-ejection solution in the region

r > 0 leaving and returning to  and a connecting orbit

on the collision surface.

What happens to nearby noncollision solutions? Well, they come close to

the “lower” equilibrium point with θ =θ

∗

=−

√

2, then follow one of two

branches of the unstable curve through this point up to the “upper” equilib-

rium point θ =θ

∗

√

2, and then depart near the unstable curve leaving

this equilibrium point. Interpreting this motion in conﬁguration space, we see

that each near-collision solution approaches the origin and then retreats after

θ either increases or decreases by 2π units. Of course, we know this already,

since these solutions whip around the origin in tight ellipses.

13.8 Exploration: Other Central Force

Problems

In this exploration, we consider the (non-Newtonian) central force problem

for which the potential energy is given by

U (X ) =

|X|

where ν > 1. The primary goal is to understand near-collision solutions.

1. Write this system in polar coordinates (r,θ,v

) and state explicitly the

formulas for total energy and angular momentum.

2. Using a computer, investigate the behavior of solutions of this system

when h<0 and  =0.

298 Chapter 13 Applications in Mechanics

3. Blow up the singularity at the origin via the change of variables

ν/2

, u

ν/2

and an appropriate change of timescale and write down the new system.

4. Compute the vector ﬁeld on the collision surface  determined in r =0

by the total energy relation.

5. Describe the bifurcation that occurs on  when ν =2.

6. Describe the structure of 

h,

for all ν > 1.

7. Describe the change of structure of 

h,

that occurs when ν passes

through the value 2.

8. Describe the behavior of solutions in 

h,

when ν > 2.

9. Suppose 1 < ν < 2. Describe the behavior of solutions as they pass close

to the singularity at the origin.

10. Using the fact that solutions of this system are preserved by rotations

about the origin, describe the behavior of solutions in 

h,

when h<0

and  =0.

13.9 Exploration: Classical Limits of

Quantum Mechanical Systems

In this exploration we investigate the anisotropic Kepler problem. This is a

classical mechanical system with two degrees of freedom that depends on a

parameter μ. When μ =1 the system reduces to the Newtonian central force

system discussed in Section 13.4. When μ > 1 some anisotropy is introduced

into the system, so that we no longer have a central force ﬁeld. We still have

some collision-ejection orbits, as in the central force system, but the behavior

of nearby orbits is quite different from those when μ =1.

The anisotropic Kepler problem was ﬁrst introduced by Gutzwiller as a

classical mechanical approximation to certain quantum mechanical systems.

In particular, this system arises naturally when one looks for bound states of an

electron near a donor impurity of a semiconductor. Here the potential is due

to an ordinary Coulomb ﬁeld, while the kinetic energy becomes anisotropic

because of the electronic band structure in the solid. Equivalently, we can view

this system as having an anisotropic potential energy function. Gutzwiller

suggests that this situation is akin to an electron whose mass in one direction

is larger than in other directions. For more background on the quantum

mechanical applications of this work, refer to [22].

The anisotropic Kepler system is given by



−μx

(μx

)

3/2

Exercises 299



−y

(μx

)

3/2

where μ is a parameter that we assume is greater than 1.

1. Show that this system is conservative with potential energy given by

U (x,y) =

−1



μx

and write down an explicit formula for total energy.

2. Describe the geometry of the energy surface 

for energy h<0.

3. Restricting to the case of negative energy, show that the only solutions

that meet the zero velocity curve and are straight-line collision-ejection

orbits for the system lie on the x- and y-axes in conﬁguration space.

4. Show that angular momentum is no longer an integral for this system.

5. Rewrite this system in polar coordinates.

6. Using a change of variables and time rescaling as for the Newtonian

central force problem (Section 13.7), blow up the singularity and write

down a new system without any singularities at r =0.

7. Describe the structure of the collision surface  (the intersection of 

with r =0 in the scaled coordinates). In particular, why would someone

call this surface a “bumpy torus?”

8. Find all equilibrium points on  and determine the eigenvalues of the

linearized system at these points. Determine which equilibria on  are

sinks, sources, and saddles.

9. Explain the bifurcation that occurs on  when μ =9/8.

10. Find a function that is nondecreasing along all nonequilibrium solutions

in .

11. Determine the fate of the stable and unstable curves of the saddle points

in the collision surface. Hint: Rewrite the equation on this surface to

eliminate time and estimate the slope of solutions as they climb up .

12. When μ >9/8, describe in qualitative terms what happens to solutions

that approach collision close to the collision-ejection orbits on the x-axis.

In particular, how do they retreat from the origin in the conﬁguration

space? How do solutions approach collision when traveling near the

y-axis?

EXERCISES

1. Which of the following force ﬁelds on R

are conservative?

(a) F(x, y) =(−x

, −2y

)

300 Chapter 13 Applications in Mechanics

(b) F(x, y) =(x

−y

,2xy)

2. Prove that the equation

(1+ cos θ)

determines a hyperbola, parabola, and ellipse when  > 1,  =1, and

 < 1, respectively.

3. Consider the case of a particle moving directly away from the origin at

time t =0 in the Newtonian central force system. Find a speciﬁc formula

for this solution and discuss the corresponding motion of the particle. For

which initial conditions does the particle eventually reverse direction?

4. In the Newtonian central force system, describe the geometry of 

h,

when h>0 and h =0.

5. Let F(X ) be a force ﬁeld on

. Let X

, X

be points in R

and let Y (s)

be a path in

with s

≤s ≤s

, parametrized by arc length s, from X

. The work done in moving a particle along this path is deﬁned to be

the integral



F(y(s))·y



(s)ds

where Y



(s) is the (unit) tangent vector to the path. Prove that the force

ﬁeld is conservative if and only if the work is independent of the path. In

fact, if F =−grad V , then the work done is V (X

)−V (X

6. Describe solutions to the non-Newtonian central force system given by



=−

|X|

7. Discuss solutions of the equation



|X|

This equation corresponds to a repulsive rather than attractive force at

the origin.

8. This and the next two problems deal with the two-body problem. Let the

potential energy be

U =

−X

Exercises 301

and

grad

(U ) =



∂U

∂x

∂U

∂x

∂U

∂x



Show that the equations for the two-body problem can be written



=−grad

(U ).

9. Show that the total energy K +U of the system is a constant of the motion,

where

K =





10. Deﬁne the angular momentum of the system by

 =m

×V

)+m

×V

)

and show that  is also a ﬁrst integral.

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The Lorenz System

So far, in all of the differential equations we have studied, we have not encoun-

tered any “chaos.” The reason is simple: The linear systems of the ﬁrst few

chapters always exhibit straightforward, predictable behavior. (OK, we may

see solutions wrap densely around a torus as in the oscillators of Chapter 6, but

this is not chaos.) Also, for the nonlinear planar systems of the last few chap-

ters, the Poincaré-Bendixson theorem completely eliminates any possibility of

chaotic behavior. So, to ﬁnd chaotic behavior, we need to look at nonlinear,

higher dimensional systems.

In this chapter we investigate the system that is, without doubt, the most

famous of all chaotic differential equations, the Lorenz system from meteo-

rology. First formulated in 1963 by E. N. Lorenz as a vastly oversimpliﬁed

model of atmospheric convection, this system possesses what has come to

be known as a “strange attractor.” Before the Lorenz model started making

headlines, the only types of stable attractors known in differential equations

were equilibria and closed orbits. The Lorenz system truly opened up new

horizons in all areas of science and engineering, because many of the phe-

nomena present in the Lorenz system have later been found in all of the

areas we have previously investigated (biology, circuit theory, mechanics, and

elsewhere).

In the ensuing 40 years, much progress has been made in the study of

chaotic systems. Be forewarned, however, that the analysis of the chaotic

behavior of particular systems like the Lorenz system is usually extremely

difﬁcult. Most of the chaotic behavior that is readily understandable arises

from geometric models for particular differential equations, rather than from

303

304 Chapter 14 The Lorenz System

the actual equations themselves. Indeed, this is the avenue we pursue here. We

will present a geometric model for the Lorenz system that can be completely

analyzed using tools from discrete dynamics. Although this model has been

known for some 30 years, it is interesting to note the fact that this model was

only shown to be equivalent to the Lorenz system in the year 1999.

14.1 Introduction to the Lorenz System

In 1963, E. N. Lorenz [29] attempted to set up a system of differential equations

that would explain some of the unpredictable behavior of the weather. Most

viable models for weather involve partial differential equations; Lorenz sought

a much simpler and easier-to-analyze system.

The Lorenz model may be somewhat inaccurately thought of as follows.

Imagine a planet whose “atmosphere” consists of a single ﬂuid particle. As on

earth, this particle is heated from below (and hence rises) and cooled from

above (so then falls back down). Can a weather expert predict the “weather”

on this planet? Sadly, the answer is no, which raises a lot of questions about

the possibility of accurate weather prediction down here on earth, where we

have quite a few more particles in our atmosphere.

A little more precisely, Lorenz looked at a two-dimensional ﬂuid cell that

was heated from below and cooled from above. The ﬂuid motion can be

described by a system of differential equations involving inﬁnitely many vari-

ables. Lorenz made the tremendous simplifying assumption that all but three

of these variables remained constant. The remaining independent variables

then measured, roughly speaking, the rate of convective “overturning” (x),

and the horizontal and vertical temperature variation (y and z, respectively).

The resulting motion led to a three-dimensional system of differential equa-

tions that involved three parameters: the Prandtl number σ , the Rayleigh

number r, and another parameter b that is related to the physical size of the

system. When all of these simpliﬁcations were made, the system of differential

equations involved only two nonlinear terms and was given by



= σ (y − x)



= rx − y − xz



= xy − bz.

In this system all three parameters are assumed to be positive and, moreover,

σ >b+ 1. We denote this system by X



= L(X). In Figure 14.1, we have

displayed the solution curves through two different initial conditions P

(0, 2, 0) and P

= (0, −2, 0) when the parameters are σ = 10, b = 8/3, and

14.1 Introduction to the Lorenz System 305

Figure 14.1 The Lorenz attractor. Two solutions with initial conditions P

(0, 2, 0) and P

= (0, −2, 0).

r = 28. These are the original parameters that led to Lorenz’s discovery. Note

how both solutions start out very differently, but eventually have more or less

the same fate: They both seem to wind around a pair of points, alternating at

times which point they encircle. This is the ﬁrst important fact about the Lorenz

system: All nonequilibrium solutions tend eventually to the same complicated

set, the so-called Lorenz attractor.

There is another important ingredient lurking in the background here. In

Figure 14.1, we started with two relatively far apart initial conditions. Had we

started with two very close initial conditions, we would not have observed the

“transient behavior” apparent in Figure 14.1. Rather, more or less the same

picture would have resulted for each solution. This, however, is misleading.

When we plot the actual coordinates of the solutions, we see that these two

solutions actually move quite far apart during their journey around the Lorenz

attractor. This is illustrated in Figure 14.2, where we have graphed the x

coordinates of two solutions that start out nearby, one at (0, 2, 0), the other (in

gray) at (0, 2. 01, 0). These graphs are nearly identical for a certain time period,

but then they differ considerably as one solution travels around one of the lobes

of the attractor while the other solution travels around the other. No matter

how close two solutions start, they always move apart in this manner when

they are close to the attractor. This is sensitive dependence on initial conditions,

one of the main features of a chaotic system.

We will describe in detail the concept of an attractor and chaos in this

chapter. But ﬁrst, we need to investigate some of the more familiar features of

the system.