 9.8.1: 1 through 3 ask you to fill in some of the details of the analysis ...
 9.8.2: 1 through 3 ask you to fill in some of the details of the analysis ...
 9.8.3: 1 through 3 ask you to fill in some of the details of the analysis ...
 9.8.4: Use the Liapunov function V(x, y, z) = x2 + y2 + z2 to show that th...
 9.8.5: Consider the ellipsoidV(x, y, z) = rx2 + y2 + (z 2r)2 = c > 0.(a) C...
 9.8.6: 6 through 10 suggest some further investigations of the Lorenz equa...
 9.8.7: 6 through 10 suggest some further investigations of the Lorenz equa...
 9.8.8: 6 through 10 suggest some further investigations of the Lorenz equa...
 9.8.9: 6 through 10 suggest some further investigations of the Lorenz equa...
 9.8.10: 6 through 10 suggest some further investigations of the Lorenz equa...
 9.8.11: The Rssler19 System. The systemx = y z, y = x + ay, z = b + z(x c),...
 9.8.12: The Rssler19 System. The systemx = y z, y = x + ay, z = b + z(x c),...
 9.8.13: The Rssler19 System. The systemx = y z, y = x + ay, z = b + z(x c),...
 9.8.14: The Rssler19 System. The systemx = y z, y = x + ay, z = b + z(x c),...
 9.8.15: The Rssler19 System. The systemx = y z, y = x + ay, z = b + z(x c),...
Solutions for Chapter 9.8: Chaos and Strange Attractors: The Lorenz Equations
Full solutions for Elementary Differential Equations  10th Edition
ISBN: 9780470458327
Solutions for Chapter 9.8: Chaos and Strange Attractors: The Lorenz Equations
Get Full SolutionsSince 15 problems in chapter 9.8: Chaos and Strange Attractors: The Lorenz Equations have been answered, more than 11401 students have viewed full stepbystep solutions from this chapter. Elementary Differential Equations was written by and is associated to the ISBN: 9780470458327. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 9.8: Chaos and Strange Attractors: The Lorenz Equations includes 15 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations, edition: 10.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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.

Outer product uv T
= column times row = rank one matrix.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).