 6.1: In 1 through 8, nd the critical point or points of the given autono...
 6.2: In 1 through 8, nd the critical point or points of the given autono...
 6.3: In 1 through 8, nd the critical point or points of the given autono...
 6.4: In 1 through 8, nd the critical point or points of the given autono...
 6.5: In 1 through 8, nd the critical point or points of the given autono...
 6.6: In 1 through 8, nd the critical point or points of the given autono...
 6.7: In 1 through 8, nd the critical point or points of the given autono...
 6.8: In 1 through 8, nd the critical point or points of the given autono...
 6.9: In 9 through 12, nd each equilibrium solution x.t/ $ x0 of the give...
 6.10: In 9 through 12, nd each equilibrium solution x.t/ $ x0 of the give...
 6.11: In 9 through 12, nd each equilibrium solution x.t/ $ x0 of the give...
 6.12: In 9 through 12, nd each equilibrium solution x.t/ $ x0 of the give...
 6.13: Solve each of the linear systems in 13 through 20 to determine whet...
 6.14: Solve each of the linear systems in 13 through 20 to determine whet...
 6.15: Solve each of the linear systems in 13 through 20 to determine whet...
 6.16: Solve each of the linear systems in 13 through 20 to determine whet...
 6.17: Solve each of the linear systems in 13 through 20 to determine whet...
 6.18: Solve each of the linear systems in 13 through 20 to determine whet...
 6.19: Solve each of the linear systems in 13 through 20 to determine whet...
 6.20: Solve each of the linear systems in 13 through 20 to determine whet...
 6.21: Verify that .0; 0/ is the only critical point of the system in Exam...
 6.22: Separate variables in Eq. (20) to derive the solution in (21).
 6.23: In 23 through 26, a system dx=dtDF.x;y/, dy=dt D G.x;y/ is given. S...
 6.24: In 23 through 26, a system dx=dtDF.x;y/, dy=dt D G.x;y/ is given. S...
 6.25: In 23 through 26, a system dx=dtDF.x;y/, dy=dt D G.x;y/ is given. S...
 6.26: In 23 through 26, a system dx=dtDF.x;y/, dy=dt D G.x;y/ is given. S...
 6.27: Let .x.t/;y.t// be a nontrivial solution of the nonautonomous syste...
 6.28: 28 through 30 deal with the systemdx dt D F.x;y/;dy dt D G.x;y/in a...
 6.29: 28 through 30 deal with the systemdx dt D F.x;y/;dy dt D G.x;y/in a...
 6.30: 28 through 30 deal with the systemdx dt D F.x;y/;dy dt D G.x;y/in a...
 6.31: In 29 through 32, nd all critical points of the given system, and i...
 6.32: In 29 through 32, nd all critical points of the given system, and i...
 6.33: BifurcationsThe term bifurcation generally refers to something spli...
 6.34: BifurcationsThe term bifurcation generally refers to something spli...
 6.35: BifurcationsThe term bifurcation generally refers to something spli...
 6.36: BifurcationsThe term bifurcation generally refers to something spli...
 6.37: BifurcationsThe term bifurcation generally refers to something spli...
 6.38: BifurcationsThe term bifurcation generally refers to something spli...
Solutions for Chapter 6: Nonlinear Systems and Phenomena
Full solutions for Differential Equations: Computing and Modeling  5th Edition
ISBN: 9780321816252
Solutions for Chapter 6: Nonlinear Systems and Phenomena
Get Full SolutionsChapter 6: Nonlinear Systems and Phenomena includes 38 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Differential Equations: Computing and Modeling was written by and is associated to the ISBN: 9780321816252. This textbook survival guide was created for the textbook: Differential Equations: Computing and Modeling, edition: 5. Since 38 problems in chapter 6: Nonlinear Systems and Phenomena have been answered, more than 1785 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

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.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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.