## Errata and Other Infelicities in the Text

### DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS.

• Page xiv, line 1: I would like to thank... should be We would like to thank...

• Page 4, line 16: increase without bound

• Page 13, line 10 (and elsewhere on this page): The partial of f with respect to x0 is really the partial of f with respect to the "variable" phi.

• Page 27, last line should read: That is, two nonzero vectors V and W...

• Page 38, Exercise 13 should read: A( alpha V1 + beta V2) = alpha A V1 + beta A V2 .

• Page 58, very interesting way to enumerate the phase portraits in Figure 3.9, don't you think?

• Page 59, Exercise 9 should read complex eigenvalues with nonzero imaginary part.

• Page 66: Note that, in the proof of the Theorem, the "only if" part is missing. This part follows immediately, however, from the assertion at the top of page 66 regarding the fact that, if the eigenvalues are negative, then the conjugacy takes curves tending to the equilibrium point to other such curves. So if there is only one negative eigenvalue, then we only get a single curve of such solutions, but if there are two eigenvalues with negative real parts, we get a whole open set of such solutions and so there is no conjugacy.

• Page 86, last line: first component of V should be x1 + iy1.

• Page 88, line 18 Section 5.1, not 5.2.

• Page 88, last line: W2j should be E2j

• Page 102. Fifth line of the proof should read "Since V2 is open...".

• Page 111, line 3: in the k1 component of the solution replace ck with ck1.

• Page 116. In the general solution Y(t) at the bottom of the page, the terms yi(t) should each be multiplied by omegai

• Page 118: The formulas for theta1 and theta2 are wrong, they should read theta1 = theta1(0) - omega1(t), a minus sign rather than a plus sign in each of these cases. This does not affect the Poincare map, however.

• Page 144, Equation 3 (as well as just above) should read: uk+1(t) instead of ur+1(t). Similarly, uk(s) instead of ur(s). Most importantly, inside the integral should read F(uk(s)), not just uk(s).

• Page 147. Remove the first K in the Theorem.

• Page 164, last equation: theta' should be 1, not -1.

• Page 169, line 11: hs(y) should be hs(y).

• Page 177, line 13: The bifurcation should have gone "the other way", i.e., with a < a0.

• Page 178, line 3: The second term in the numerator should be GxGxa, but of course this does not change anything since this entire term is zero because Gx = 0

• Page 192, line 6: x + or - 1 should read x = + or -1, the equals sign is missing.

• Page 195, the example: Technically, we have not shown that linearization implies that the origin is unstable when epsilon > 0. However, this follows immediately from the fact that the x, y-plane is invariant and the system is actually linear on this plane for all epsilon.

• Page 196, line 2 and line 6: L' should be L'/2.

• Page 204, line -6: the words "isolated minimum" should be replaced by "isolated critical point that is a minimum."

• Page 242 line -10: The proof follows from Corollary 5 in 10.6, not Corollary 6.

• Page 251 line -10: The expression xMx + yMy should read xMx + yNy.

• Page 251 line -3: lies over mu should read to the left of nu.

• Page 295 last line: v_theta should be u_r

• Page 297, Exploration 13.8: The potential energy U(X) should be the negative of what is written, i.e., U(X) = -1 / |X|^v.

• Page 326, problem 9: This problem should read "show that the system is not chaotic in the region where x, y, and z are all positive."
• Page 332, figure 15.4: The labels of the graphs should be y = f(x) and y = f2(x) instead of y = xf(x) and y = xf2(x)

• Page 333, figure 15.5: The captions should read fc(x) rather than fc(x). Also, there is a missing parenthesis in the first displayed equation.

• Page 335, figure 15.6: The captions should read fc(x) rather than fc(x).

• Page 337, figure 15.7: In the caption the minus sign should be an equals sign.

• Page 347, line 19: tn+1 should be sn+1.

• Page 348, line -4: the superscripts should be n+1, not n.

• Page 379. The third equation in the Chua circuit system should read z' = - by, not -bz.

Special thanks to the following people for catching the above errors:

• Jamil Abreu
• Larry Baker
• Jeff Haag and his students at Humboldt State
• Kresimir Josic and his students at the University of Houston
• John Josef Leth
• Benjamin Lotto
• Tiennyu Ma
• Luis Fernando Mello
The text passage is a list of errata and corrections for a textbook on differential equations, dynamical systems, and chaos. It contains no mathematical questions or answers to such questions.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
## Errata and Other Infelicities in the Text

### DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS.

• Page xiv, line 1: I would like to thank... should be We would like to thank...

• Page 4, line 16: increase without bound

• Page 13, line 10 (and elsewhere on this page): The partial of f with respect to x0 is really the partial of f with respect to the "variable" phi.

• Page 27, last line should read: That is, two nonzero vectors V and W...

• Page 38, Exercise 13 should read: A( alpha V1 + beta V2) = alpha A V1 + beta A V2 .

• Page 58, very interesting way to enumerate the phase portraits in Figure 3.9, don't you think?

• Page 59, Exercise 9 should read complex eigenvalues with nonzero imaginary part.

• Page 66: Note that, in the proof of the Theorem, the "only if" part is missing. This part follows immediately, however, from the assertion at the top of page 66 regarding the fact that, if the eigenvalues are negative, then the conjugacy takes curves tending to the equilibrium point to other such curves. So if there is only one negative eigenvalue, then we only get a single curve of such solutions, but if there are two eigenvalues with negative real parts, we get a whole open set of such solutions and so there is no conjugacy.

• Page 86, last line: first component of V should be x1 + iy1.

• Page 88, line 18 Section 5.1, not 5.2.

• Page 88, last line: W2j should be E2j

• Page 102. Fifth line of the proof should read "Since V2 is open...".

• Page 111, line 3: in the k1 component of the solution replace ck with ck1.

• Page 116. In the general solution Y(t) at the bottom of the page, the terms yi(t) should each be multiplied by omegai

• Page 118: The formulas for theta1 and theta2 are wrong, they should read theta1 = theta1(0) - omega1(t), a minus sign rather than a plus sign in each of these cases. This does not affect the Poincare map, however.

• Page 144, Equation 3 (as well as just above) should read: uk+1(t) instead of ur+1(t). Similarly, uk(s) instead of ur(s). Most importantly, inside the integral should read F(uk(s)), not just uk(s).

• Page 147. Remove the first K in the Theorem.

• Page 164, last equation: theta' should be 1, not -1.

• Page 169, line 11: hs(y) should be hs(y).

• Page 177, line 13: The bifurcation should have gone "the other way", i.e., with a < a0.

• Page 178, line 3: The second term in the numerator should be GxGxa, but of course this does not change anything since this entire term is zero because Gx = 0

• Page 192, line 6: x + or - 1 should read x = + or -1, the equals sign is missing.

• Page 195, the example: Technically, we have not shown that linearization implies that the origin is unstable when epsilon > 0. However, this follows immediately from the fact that the x, y-plane is invariant and the system is actually linear on this plane for all epsilon.

• Page 196, line 2 and line 6: L' should be L'/2.

• Page 204, line -6: the words "isolated minimum" should be replaced by "isolated critical point that is a minimum."

• Page 242 line -10: The proof follows from Corollary 5 in 10.6, not Corollary 6.

• Page 251 line -10: The expression xMx + yMy should read xMx + yNy.

• Page 251 line -3: lies over mu should read to the left of nu.

• Page 295 last line: v_theta should be u_r

• Page 297, Exploration 13.8: The potential energy U(X) should be the negative of what is written, i.e., U(X) = -1 / |X|^v.

• Page 326, problem 9: This problem should read "show that the system is not chaotic in the region where x, y, and z are all positive."
• Page 332, figure 15.4: The labels of the graphs should be y = f(x) and y = f2(x) instead of y = xf(x) and y = xf2(x)

• Page 333, figure 15.5: The captions should read fc(x) rather than fc(x). Also, there is a missing parenthesis in the first displayed equation.

• Page 335, figure 15.6: The captions should read fc(x) rather than fc(x).

• Page 337, figure 15.7: In the caption the minus sign should be an equals sign.

• Page 347, line 19: tn+1 should be sn+1.

• Page 348, line -4: the superscripts should be n+1, not n.

• Page 379. The third equation in the Chua circuit system should read z' = - by, not -bz.

Special thanks to the following people for catching the above errors:

• Jamil Abreu
• Larry Baker
• Jeff Haag and his students at Humboldt State
• Kresimir Josic and his students at the University of Houston
• John Josef Leth
• Benjamin Lotto
• Tiennyu Ma
• Luis Fernando Mello
user
The text passage is a list of errata and corrections for a textbook on differential equations, dynamical systems, and chaos. It contains no mathematical questions or answers to such questions.

TYPE: NO_QUESTION