 8.2.1: F(x) = x2 2x
 8.2.2: F(x) = x5
 8.2.3: F(x) = sin x
 8.2.4: F(x) = x3 x
 8.2.5: F(x) = arctan x
 8.2.6: F(x) = 3x(1 x)
 8.2.7: F(x) = (/2)sin x
 8.2.8: F(x) = x2 3
 8.2.9: F(x) = 1/x
 8.2.10: F(x) = 1/x2
 8.2.11: F(x) = ex
 8.2.12: F(x) = x5 + 1
 8.2.13: F(x) = 1 x2
 8.2.14: F(x) = (/2) cos x
 8.2.15: F(x) = x + 1, if x < 3.5; 2x 8, if x 3.5.
 8.2.16: F(x) = 1 x3
 8.2.17: F(x) = x 2 1
 8.2.18: F(x) = x x2
 8.2.19: F(x) = 1/x
 8.2.20: F(x) = sin x
 8.2.21: F(x) = tan x
 8.2.22: F(x) = x + x3
 8.2.23: F(x) = x x3
 8.2.24: F(x) = ex1 (fixed point is 1)
 8.2.25: F(x) = e ex (fixed point is 1)
 8.2.26: Find all fixed points for Fc(x) = x2 + c for all values of c. Deter...
 8.2.27: How many fixed points does F(x) = tan x have? Are they attracting, ...
 8.2.28: What can you say about fixed points for Fc(x) = cex with c > 0? Wha...
 8.2.29: Consider the function T (x) = 4x, x < 1/2; 4 4x, x 1/2. Does T have...
 8.2.30: Recall from calculus that Newtons method is an iterative procedure ...
Solutions for Chapter 8.2: FIXED POINTS AND PERIODIC POINTS
Full solutions for Differential Equations 00  4th Edition
ISBN: 9780495561989
Solutions for Chapter 8.2: FIXED POINTS AND PERIODIC POINTS
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations 00, edition: 4. Since 30 problems in chapter 8.2: FIXED POINTS AND PERIODIC POINTS have been answered, more than 17190 students have viewed full stepbystep solutions from this chapter. Chapter 8.2: FIXED POINTS AND PERIODIC POINTS includes 30 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Differential Equations 00 was written by and is associated to the ISBN: 9780495561989.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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).

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).