Alligood K., Sauer T., Yorke J.A. Chaos: An Introduction to Dynamical Systems
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C HALLENGE 7
x
z
+
z
-
F
+
F
-
I
IIIII
IV
Figure 7.23 Phase plane structure for the Van der Pol system.
The plane separates into four regions according to the sign of
˙
x and
˙
z. The direction of
the flow (which you provide, see Step 2) implies a circulation of solution trajectories
around the origin.
crosses F
⫹
, moves into region II, and then intersects the negative z-axis, denoted
z
⫺
. Then the solution moves through region III, through region IV, and back to
the half-line z
⫹
. Conclude that the solution repeatedly cycles through regions I,
II, III, and IV, in that order.
Define the map T on z
⫹
, as follows: For a point (0,a
⫹
)onz
⫹
, draw the
trajectory X through (0,a
⫹
), and let T(a
⫹
) be the first intersection (after a
⫹
)of
X with z
⫹
; that is, the intersection that occurs after X has cycled once through
regions I–IV. The definition of T is illustrated in Figure 7.24.
Step 3 Show that the map T is one-to-one, and that a number a
⫹
is
fixed under T if and only if the solution trajectory through (0,a
⫹
) is a periodic
solution of (7.51). Explain why an orbit of (7.51) is periodic if and only if the
corresponding orbit of (7.50) is periodic.
The map T is another example of the Poincar
´
e map, first introduced in
Chapter 2. In Steps 4–10 we show that T has an attracting fixed point.
Step 4 Let a
⫹
⬎ 0. Show that because of symmetry, if the solution through
(0,a
⫹
) intersects z
⫺
at (0, ⫺a
⫹
), then T(a
⫹
) ⫽ a
⫹
.
Notice that (7.49) bears some resemblance to the linear oscillator
¨
x ⫹ x ⫽ 0.
The coefficient of the middle term
˙
x goes from negative to positive with increasing
317
D IFFERENTIAL E QUATIONS
a
+
T(a
+
)
x
z
+
Figure 7.24 The first-return map
T
.
The point T(a
⫹
) is defined to be the intersection of the solution through a
⫹
with
the positive z-axis.
x, resulting in a variable “damping” effect. We define an energy function, as
follows:
E(x, z) ⫽ (1 2)(x
2
⫹ z
2
).
Starting with the point (0,a
⫹
), we want to measure the change in energy
over a segment X
0
of X from (0,a
⫹
) to the first crossing of X with z
⫺
, denoted
by a
⫺
. We define this change as follows: Let
h(a
⫹
) ⫽ E(0,a
⫺
) ⫺ E(0,a
⫹
) ⫽ (1 2)[(a
⫺
)
2
⫺ (a
⫹
)
2
].
According to Step 4, the problem reduces to showing that there exists a real
number c such that h(c) ⫽ 0.
Step 5 Show that
˙
E ⫽
x
3
3
⫺ x
˙
z,
and so
h(a
⫹
) ⫽
X
0
x
3
3
⫺ x
˙
zdt.
Let r be the x-coordinate of the point where the solution X intersects F
⫹
,
and let (q, 0) be the point on the positive x-axis where F
⫹
intersects the axis. For
this example, q ⫽
3. (See Figure 7.25.)
318
C HALLENGE 7
F
+
F
-
q
r
s
q
+
q
-
X
1
X
2
X
3
a
+
a
-
Figure 7.25 A schematic view of a Van der Pol trajectory.
Theinitialvalueis(0,a
⫹
); the intersection with the negative z-axis is a
⫺
.The
trajectory between is the union X
0
of three pieces denoted X
1
, X
2
,andX
3
.
Step 6 Show that if r ⬍ q, then
X
0
x
3
3
⫺ x
˙
zdt ⬎ 0.
Refer to Figure 7.25. Let q
⫹
(respectively, q
⫺
) be the first point (respectively,
second point) where X intersects the line x ⫽ q. For r ⬎ q, we write the path X
0
as X
1
傼 X
2
傼 X
3
, where X
1
goes from (0,a
⫹
)toq
⫹
, X
2
goes from q
⫹
to q
⫺
, and
X
3
goes from q
⫺
to (0,a
⫺
).
Step 7 Show:
X
1
dE ⬎ 0,
X
2
dE ⬍ 0, and
X
3
dE ⬎ 0.
Step 8 Show that
X
1
˙
Edt and
X
3
˙
Edt are decreasing functions of a
⫹
.
[Hint: For example,
X
1
˙
Edt ⫽
3
0
x
3
3
⫺ x
dz
dx
dx ⫽
3
0
x
3
3
⫺ x
⫺x
z ⫺ (
x
3
3
⫺ x)
dx.]
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D IFFERENTIAL E QUATIONS
Step 9 Show that
X
2
˙
Edtmonotonically decreases to ⫺
⬁
as a
⫹
→
⬁
.
[Hint: The integral
X
2
˙
Edt⫽⫺
X
2
x
3
3
⫺ x
xdt.
By uniqueness, the solutions must move farther to the right, and the terms (
x
3
3
⫺ x)
and x in the integral must increase, as a
⫹
is increased.]
Step 10 Conclude that there exists a unique positive number c such that
h(c) ⫽ 0, and that c is a fixed point of T.
Step 11 Show that c is an attracting fixed point of T. [Hint: Show that
uniqueness of solutions implies that if a
⫹
⬍ c,then⫺a
⫺
⬍ c. Show that the
change in E from a
⫹
to T(a
⫹
) is greater than 0 for a
⫹
⬍ c, and less than 0 for
a
⫹
⬎ c.]
Postscript. An attracting periodic orbit, or limit cycle, is the first distinctly non-
linear phenomenon we have seen for differential equations. Although Figure 7.9 shows
periodic orbits circling a Lyapunov stable equilibrium, the cycles are “neutrally stable”:
They do not attract nearby initial conditions. The periodic orbits of the undamped pen-
dulum in Figure 7.14 and the predator-prey system in Figure 7.22 are also not attracting.
In Challenge 7 we have explored the Van der Pol equation in detail because it has a
rigorously-provable attracting periodic orbit.
Attracting periodic orbits are interesting subjects of study because they exemplify
the notion that certain periodic oscillations in nature can be “locked in”, so as to be fairly
impervious to drift. If a small amount of noise were to push the current state away from a
particular periodic orbit in the neutrally stable examples of the previous paragraph, there is
no mechanism to return it to the original cycle. An attracting cycle, on the other hand, has
such a mechanism built in. This is consistent with overwhelming evidence that periodic
cycles, like your heartbeat, can oscillate more or less unperturbed, even in systems exposed
to external disturbances. Chapter 8 is dedicated to the study of periodic cycles. Lab Visit
8 examines experimental evidence for the usefulness of the limit cycle paradigm in the
context of neurons.
320
E XERCISES
E XERCISES
7.1. The differential equation governing a linear oscillator, such as a spring moving
according to Hooke’s law, is
¨
x ⫹ kx ⫽ 0.
(a) Convert to a system of first-order equations.
(b) Sketch the phase plane.
7.2. Consider the second-order equation
¨
x ⫹ 3
˙
x ⫺ 4x ⫽ 0.
(a) Convert to a system of first-order equations.
(b) Sketch the phase plane.
7.3. Consider the system
˙
x ⫽ 2x ⫺ y
˙
y ⫽ x
2
⫹ 4y.
Find equilibria and classify them as asymptotically stable, stable, or unstable.
7.4. Consider the equation
˙
x ⫽ a(x ⫺ b)(x ⫺ c), where b ⬍ c, with initial condition
x(0) ⫽ x
0
.
(a) Sketch the slope field and phase portrait. For each initial condition x
0
,
use this qualitative information to determine the long-term behavior of the
solutions. Treat the cases a ⬍ 0, a ⫽ 0, and a ⬎ 0 separately.
(b) Solve the equation and initial value problem by separation of variables and
partial fractions. Verify that the behavior of this solution is consistent with the
qualitative information in (a).
(c) Show that if a ⬎ 0andx
0
⬎ c,orifa ⬍ 0andx
0
⬍ b, the solutions blow
up in finite time. Find the largest interval [0,T) on which the solution exists.
Does this example contradict the existence theorem?
(d) What happens if b ⫽ c?
(e) A chemical reaction Y+Z
→X proceeds at a rate proportional to the product
of the concentration of the reactants Y and Z. Each time the reaction occurs, one
molecule of species X appears and one each of Y and Z disappears. Therefore the
reaction is modeled by the differential equation
˙
x ⫽ a(b ⫺ x)(c ⫺ x), x(0) ⫽ 0,
where x(t) is the relative molar concentration of species X at time t,andb and
c are the relative molar concentrations of Y and Z, respectively, at time t ⫽ 0.
The constant a is called the rate constant of the reaction. Which parts of the
above analysis (which combinations of a and x
0
) correspond to this physical
problem, and which do not? Describe the eventual outcome of the reaction in
these cases.
321
D IFFERENTIAL E QUATIONS
7.5. Sketch phase portraits for the following linear systems.
(a)
˙
x ⫽ 3x ⫺ y
˙
y ⫽ 2x ⫹ 4y
(b)
˙
x ⫽⫺2x ⫹ 3y
˙
y ⫽ 7x ⫺ 6y
(c)
˙
x ⫽ 3x ⫺ y
˙
y ⫽ x ⫹ 3y
˙
z ⫽⫺2z
7.6. Show by examples that there are equilibria that are attracting but not stable, and
equilibria that are stable but not attracting.
7.7. Verify that if the independent variable t does not appear in the right side of (7.29),
then solutions do not cross in the phase portrait.
7.8. Consider the second-order equation
¨
x ⫹ x ⫹ x
3
⫽ 0.
(a) Find a Lyapunov function for the equilibrium x ⫽
˙
x ⫽ 0. Verify your answer.
(b) Convert the equation to a first-order system and sketch the phase plane.
7.9. Let
¨
x ⫹
˙
x ⫹ x ⫹ x
3
⫽ 0. Show that the equilibrium (0, 0) is globally asymptotically
stable.
7.10. Each graph in Figure 7.26 represents the graph of a potential function P(x). In
each case, sketch the phase plane for the system
¨
x ⫹ f(x) ⫽ 0, where f(x) ⫽ P
(x).
Identify the equilibria and discuss their stability.
7.11. Let
¨
x ⫹ b
˙
x ⫹ f(x) ⫽ 0, where b is a positive constant and f(x) ⫽ P
(x)whereP(x)
is shown in Figure 7.26. Write the equation as a first-order system and sketch the
phase plane.
7.12. Figure 7.14(b) shows the phase portrait of the undamped pendulum. Graph the six
qualitatively different solutions x(t) as functions of t.
322
E XERCISES
P(x)
x
x
1
x
2
x
P(x)
x
1
x
2
(a) (b)
Figure 7.26 Potential functions for Exercise 7.10.
In part (b), P
(x
2
) ⫽ 0.
7.13. Use the method of nullclines to determine the global behavior of solutions of the
following competing species models. Describe sets of initial conditions that evolve
to distinct final states.
(a)
˙
x ⫽ 3x(1 ⫺ x) ⫺ xy
˙
y ⫽ 5y(1 ⫺ y) ⫺ 2xy
(b)
˙
x ⫽ 2x(1 ⫺ x) ⫺ xy
˙
y ⫽ y(1 ⫺ y) ⫺ 3xy
7.14. Consider the system
˙
x ⫽⫺z ⫹
x
3
3
⫺ x
˙
z ⫽ x.
(a) By comparing with (7.51), explain why this system can be called the
backwards-time Van der Pol system.
(b) Show that V(x, z) ⫽ x
2
⫹ z
2
is a Lyapunov function.
(c) Find the largest possible open disk contained in the basin of attraction of
the origin.
(d) What can you conclude from this about the solutions of (7.51)?
323
D IFFERENTIAL E QUATIONS
7.15. Find two different solutions of the differential equation
˙
x ⫽
0ifx ⬍ 0
3
x if x ⱖ 0
with initial condition x(0) ⫽ 0. Explain why this does not contradict Theorem
7.14.
7.16. Sketch the phase plane in (x,
˙
x) for the system
¨
x ⫽ x ⫺ ax
3
for various values of a.
Find all stable and unstable equilibrium points.
7.17. The system
˙
x
1
⫽⫺3x
1
˙
x
2
⫽⫺8x
2
1
⫹ 2x
2
has its only equilibrium point at the origin.
(a) Find the solution for initial condition (x
0
,y
0
).
(b) Find the set of all initial conditions whose solutions approach the origin as
t
→
⬁
. (This is the stable manifold of (0, 0)).
(c) Find the set of all initial conditions whose solutions approach the origin as
t
→ ⫺
⬁
. (This is the unstable manifold of (0, 0)).
(d) Describe the asymptotic behavior of solutions starting at an arbitrary initial
condition as t
→
⬁
.Dothesamefort
→ ⫺
⬁
.
7.18. Consider the system
˙
x ⫽⫺x
3
⫹ x
2
y
˙
y ⫽⫺y
3
⫺ x
3
Describe the asymptotic behavior of the solution with initial condition (x
0
,y
0
) ⫽
(1, 2).
324
L AB V ISIT 7
☞ L AB V ISIT 7
Fly vs. Fly
D
URING THE PAST ten years a competition has developed between two
mosquito species in North America. The tiger mosquito Aedes albopictus,which
occupies much of Asia, was unknown in the United States before being observed
in Houston, Texas in 1985, apparently as the result of imports of used automobile
tires.
The invasion of the tiger mosquito has caused alarm in the United States
public health community because it is a vector for dengue fever. Dengue is a viral
disease known as “breakbone fever” in Africa, a reflection of the level of pain
that accompanies it. Even in its mildest form it causes intense headaches and
extreme muscle spasms. The severe form, known as dengue hemorrhagic fever,
causes internal bleeding, coma and shock, killing over 10% of those afflicted. Due
in part to reduced mosquito eradication programs, Latin America experienced a
sixty-fold increase in cases of dengue hemorrhagic fever in 1989–1994, compared
with the previous five-year period.
A key scientific question is to address the potential success of the new
mosquito’s invasion into North American mosquito habitats. Here we describe
an experiment, which was designed to gauge the possibility that the new mosquito
species will be able to coexist with or displace currently-existing mosquito species
that are less susceptible to carrying viruses, and less aggressive biters. An example
of a native species, Aedes triseriatus, was used in the study.
Two common mosquito habitats were simulated, treeholes and automobile
tires. The critical time for development is the larval stage; the species that can
compete best for resources at this stage will have a competitive survival advantage.
In Experiment A, one-day-old larvae of both species were put in plastic cups
containing leaf litter and 100 ml of stagnant water taken from holes in maple
trees, which represents a typical habitat for mosquito larvae. Experiment B was the
same except that “treehole fluid” was replaced with stagnant water from discarded
tires.
The parameters in the competing species model (7.47) were estimated as the
result of these experiments. The nullclines derived from the estimated parameters
Livdahl, T. P., Willey, M. S. 1991. Propects for an invasion: Competition between Aedes
albopictus and native Aedes triseriatus. Science 253:189–91.
325
D IFFERENTIAL E QUATIONS
Figure 7.27 Nullclines of two mosquito species A. triseriatus and A. albo-
pictus.
Hatched regions depict 95% confidence levels for the nullclines for Experiment A
(treehole habitat) and Experiment B (tire habitat). Circles denote the equilibria of
the competing species equations. The relative slopes of the nullclines for Experiment
A indicate that the equilibrium is stable. Experiment B points toward extinction of
A. triseriatus.
are graphed in Figure 7.27, along with confidence limits based on the estimates.
The true nullclines lie within the shaded regions with 95% certainty. The units in
the axes of the figure differ from our assumptions for (7.47). If we rescale units so
that the x-nullcline has an x-intercept of 1 and the y-nullcline has an y-intercept
of 1, then we can interpret the nullclines exactly as in (7.47).
In any case, Exercise T7.18 shows how to interpret the experimentally-
measured nullclines shown in the Figure. In Experiment A (treehole habitat),
there is an asymptotically stable state of coexisting nonzero populations. In Ex-
periment B (tire habitat), there is no equilibrium of coexisting populations, and
all initial populations are attracted to the horizontal axis, which represents ex-
tinction of A. triseriatus.
The significance of the predictions that follow from the experiment has
increased since 1991. Now ten years since it appeared in Houston, the tiger
mosquito has been identified in half of the United States, and as far north as
Chicago. It is now the dominant mosquito in many urban areas, including New
Orleans and several Florida cities.
Corroboration of the laboratory result has come from ecological studies.
Coexistence of the two species within treehole communities is suggested by a
326