 5.6.1: For = 0.1, y(0) = 1.1, v(0) = 0. 0.911.1 0.1 0.1 0.2 0.3 0 1 2 3 4 ...
 5.6.2: For = 0.4, y(0) = 1.1, v(0) = 0. 1 1 1 1 0 1 2 3 4 y v Poincare ret...
 5.6.3: For = 0.1, y(0) = 1.6, v(0) = 0. 1 1 1 1 0 1 2 3 4 y v Poincare ret...
 5.6.4: For = 0.5, y(0) = 1.6, v(0) = 0. 1 12 1 2 0 1 2 3 4 y v Poincare re...
 5.6.5: For = 0.1, (0) = .2, v(0) = 0. 1 1 1 2 01 3 4 v Poincare return map...
 5.6.6: For = 0.5, (0) = .2, v(0) = 0. 100 200 300 400 500 2 2 0 1 2 3 4 v ...
 5.6.7: For = 0.1, (0) = 1.06, v(0) = 0. 1 1 2 1 1 0 1 2 3 4 v Poincare ret...
 5.6.8: For = 0.5, (0) = 1.06, v(0) = 0. 600 400 200 3 1 1 3 0 1 2 3 4 v Po...
 5.6.9: The Poincare return map for a solution of the periodically forced h...
 5.6.10: Describe and sketch the Poincare return map for the solution of the...
Solutions for Chapter 5.6: PERIODIC FORCING OF NONLINEAR SYSTEMS AND CHAOS
Full solutions for Differential Equations 00  4th Edition
ISBN: 9780495561989
Solutions for Chapter 5.6: PERIODIC FORCING OF NONLINEAR SYSTEMS AND CHAOS
Get Full SolutionsDifferential Equations 00 was written by and is associated to the ISBN: 9780495561989. Chapter 5.6: PERIODIC FORCING OF NONLINEAR SYSTEMS AND CHAOS includes 10 full stepbystep solutions. This textbook survival guide was created for the textbook: Differential Equations 00, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Since 10 problems in chapter 5.6: PERIODIC FORCING OF NONLINEAR SYSTEMS AND CHAOS have been answered, more than 15669 students have viewed full stepbystep solutions from this chapter.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Column space C (A) =
space of all combinations of the columns of A.

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.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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

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

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.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

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.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.