 9.8.1: a. Show that the eigenvalues of the linear system (8), valid near t...
 9.8.2: a. Show that the linear approximation valid near the critical point...
 9.8.3: N a. By solving equation (11) numerically, show that the real part ...
 9.8.4: Use the Liapunov function V( x, y, z) = x2 + y2 + z2 to show that t...
 9.8.5: Consider the ellipsoid V( x, y, z) = rx2 + y2 + ( z 2r)2 = c > 0. a...
 9.8.6: For r = 28, plot x versus t for the cases shown in Figures 9.8.2 an...
 9.8.7: For r = 28, plot the projections in the xy and xzplanes, respecti...
 9.8.8: G a. For r = 21, plot x versus t for the solutions starting at init...
 9.8.9: For certain r intervals, or windows, the Lorenz equations exhibit a...
 9.8.10: Now consider values of r slightly larger than those in 9. G a. Plot...
 9.8.11: C a. Show that there are no critical points when c < 1/_2, there is...
 9.8.12: a. Let c = 1.3. Find the critical points and the corresponding eige...
 9.8.13: The limit cycle found in comes into existence as a result of a Hopf...
 9.8.14: a. Let c = 3. Find the critical points and the corresponding eigenv...
 9.8.15: a. Let c = 3.8. Find the critical points and the corresponding eige...
Solutions for Chapter 9.8: Chaos and Strange Attractors: The Lorenz Equations
Full solutions for Elementary Differential Equations and Boundary Value Problems  11th Edition
ISBN: 9781119256007
Solutions for Chapter 9.8: Chaos and Strange Attractors: The Lorenz Equations
Get Full SolutionsChapter 9.8: Chaos and Strange Attractors: The Lorenz Equations includes 15 full stepbystep solutions. Since 15 problems in chapter 9.8: Chaos and Strange Attractors: The Lorenz Equations have been answered, more than 12392 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.