- Page 474, Equation (9):
-
(Spotted by Paul Rydeen)
The notation $x(t), y(t)$ is inconsistent with that of the previous
cases. Either $x, y$ in equation (9)
should be changed to $x_1(t), x_2(t)$, or the
axis labels in Figure 8.26 should be changed to $x, y$.
- Pages 475–476:
-
(Spotted by Dr. Neofytos Komninos)
Near the top of page 476 we have:
“Clearly, the low state is asymptotically stable…”.
This is in reference to the motion of the undamped pendulum (equation (10)
on page 475). The correct statement should be:
“Clearly, the low state is stable but not
asymptotically stable”.
Further comment:
That erroneous statement has crept in possibly under the influence of the
nonsensical illustration in Figure 8.27 on page 475.
A static figure like that cannot convey the
concept of asymptotic stability as it is attempting to do. Ignore it.
- Page 479, Exercise 14:
-
(Spotted by Paul Rydeen)j
The answer in the back of the book has
“proper”
and
“improper”
switched. It should be
\begin{align*}
d^2 - 4mk < 0 & \Rightarrow \text{ spiral point (asymptotically stable)} \\
d^2 - 4mk = 0 & \Rightarrow {\it\text{ improper node }} \text{(asymptotically stable)} \\
d^2 - 4mk > 0 & \Rightarrow {\it\text{ proper node }} \text{(asymptotically stable)}
\end{align*}
- Page 479, Equations (15) and (16):
-
The discussion under Linearizing Nonlinear Systems
introduces the linearized system (16) corresponding to
an equilibrium point $(x_0, y_0)$ of the nonlinear system (15).
Then it makes the statement:
“The trajectories of the linearized system approximate the
trajectories of the nonlinear system (15) in a small region
near $(\xi, \eta)$.”
There are two errors in that statement.
- A minor error:
The
“small region near $(\xi, \eta)$”
should be
“small region near $(x_0,y_0)$”.
- A major error:
The statement is guaranteed to be true if the linearized system (16)
has no eigenvalues on the imaginary axis in the complex plane.
Otherwise, the linearized system
provides no information about the nonlinear system.
This is the subject of the celebrated
Hartman–Grobman Theorem.
It is alarming how often that
essential “fine print” on eigenvalues
is omitted/ignored in published articles
and books, especially in the engineering literature. Be on guard!
To the book's credit, the
linearization theorem is revisited in Section 8.3 where
boxes on page 483 correctly state the linearization theorem and
point out the need for the assumption on the eigenvalues.
- Page 479, Exercises 18–21:
-
(Spotted by Paul Rydeen)
The answers are given in terms of $x$ and $y$ rather than
$\eta$ and $\xi$.
- Page 479, Exercise #18:
-
(Spotted by Dr. Neofytos Komninos)
The answer in the back of the book says $\dot{x}=y$. It should say
$\dot{x}=x$.
- Page 479, Exercise #19:
-
The linearized system has eigenvalues $\pm i$. Since these
lie on the imaginary axis in the complex plane, the linearization theorem
provides no information about stability—see the previous comment
with reference to the
Hartman–Grobman Theorem.
The answer in the back of the book is correct, however
it does not follow from the linearization theorem. It has to be
proved through independent means.
- Page 479, Exercise #20:
-
The answer in the back of the book gives the linearized
equation correctly, but its analysis of the cases is incorrect.
Furthermore, the equation $\dot{y} = \epsilon(x^2-1)y - x$ is not
Van der Pol's equation, despite the problem's statement.
Van der Pol's equation would be
$\dot{y} = -\epsilon(x^2-1)y - x$.
- Page 483, the boxed Theorem 8.2
-
Where it says
“… of the almost system (6) are of the same type …”,
the “almost system (6)”
should be
the “almost linear system (8)”.
- Page 484, Example 2:
-
(Spotted by Dr. Neofytos Komninos)
The analysis is correct up to equation (10) but the conclusion
that the eigenvalues are $\lambda_1=-1$ and $\lambda_2=3$ is incorrect.
The correct eigenvalues are
$1 \pm i\sqrt{2}$, and consequently the equilibrium is an unstable spiral.
- Page 486, Exercise #2:
-
(Spotted by Paul Rydeen)
The eigenvalues of the linearized system are $\lambda=\pm i$, and therefore,
according to Table 8.2, the origin is either a center point or a spiral
of indeterminate stability.
The book’s answer is correct regarding stability but it neglects
to mention the two types of point that this can be.
- Page 486, Exercise #5:
-
(Spotted by Paul Rydeen)
The system has an equilibrium at $(0,0)$ and another at
approximately $(-0.507, -2.118)$. The book's answer
addresses only the equilibrium at $(0,0)$. The other equilibrium
is a stable spiral.
- Page 486, Exercise #6:
-
(Spotted by Paul Rydeen)
The system has an equilibrium at $(0,0)$ which is
an asymptotically stable node as the book correctly states.
The system also has an equilibrium at $(2/3, 2/5)$ which
is a saddle point.
- Page 486, Exercise #7:
-
(Spotted by Paul Rydeen)
The book's answer is incorrect. In fact, $(0,0)$ is not
an equilibrium. The equilibria are $(1,1)$ which is an improper
asymptotically stable node, and $(-1,-1)$ which is a saddle (unstable).
- Page 486, Exercise #7:
-
(Spotted by Paul Rydeen)
Linearization about $(0,0)$ leads to eigenvalues $\lambda=\pm i$,
and therefore, according to Table 8.2, the origin is either
a center point or a spiral of indeterminate stability.
The book’s answer is correct regarding stability but it neglects
to mention the two types of point that this equilibrium can be.
The book also says that the stability of $(1/2,0)$ is the same
as the stability of $(0,0)$. But that's not correct since the
eigenvalues of the linearized system about $(1/2,0)$ are
$\pm \sqrt{3/2}$, implying that the equilibrium is a saddle.
- Page 487, Exercise #11:
-
(Spotted by Natalie Gubbay)
In the problem's statement just before part (a) we have
“…at $(0,0)$ and $(c/d,a/b)$
which represent the constant solutions
$x(t)\equiv0, y(t)\equiv0$
and
$x(t)\equiv c/d, y(t)\equiv a/b$.” In that statement the $x$'s should
be $r$'s, and the $y$'s should be $f$'s.
Similarly, the $x$ and $y$ in part (a) should be $r$ and $f$.
- Page 489, Equation (1):
-
(Spotted by Paul Rydeen)
In the second equation, $\dot{x}$ should be $\dot{y}$.
- Page 493, the boxed Historical Note:
-
It refers to “The Soviet mathematician Aleksandr Mikhailovich Liapunov
(1857–1918)…”. However Liapunov was not a Soviet
mathematician—he died before there was a Soviet Union. As the footnote
on page 488 correctly states, Liapunov was a Russian mathematician.
