- 9.7.1: In each of 1 through 6, an autonomous system is expressed in polar ...
- 9.7.2: In each of 1 through 6, an autonomous system is expressed in polar ...
- 9.7.3: In each of 1 through 6, an autonomous system is expressed in polar ...
- 9.7.4: In each of 1 through 6, an autonomous system is expressed in polar ...
- 9.7.5: In each of 1 through 6, an autonomous system is expressed in polar ...
- 9.7.6: In each of 1 through 6, an autonomous system is expressed in polar ...
- 9.7.7: If x = r cos , y = r sin , show that y(dx/dt) x(dy/dt) = r2(d/dt).
- 9.7.8: (a) Show that the systemdx/dt = y + xf(r)/r, dy/dt = x + yf(r)/rhas...
- 9.7.9: Determine the periodic solutions, if any, of the systemdxdt = y + x...
- 9.7.10: Using Theorem 9.7.2, show that the linear autonomous systemdx/dt = ...
- 9.7.11: In each of 11 and 12, show that the given system has no periodic so...
- 9.7.12: In each of 11 and 12, show that the given system has no periodic so...
- 9.7.13: Prove Theorem 9.7.2 by completing the following argument. According...
- 9.7.14: (a) By examining the graphs of u versus t in Figures 9.7.3, 9.7.5, ...
- 9.7.15: The equationu (1 13u2)u+ u = 0is often called the Rayleigh13 equati...
- 9.7.16: Consider the system of equationsx= x + y x(x2 + y2), y= x + y y(x2 ...
- 9.7.17: . Consider the van der Pol systemx= y, y= x + (1 x2)y,where now we ...
- 9.7.18: Consider the systemx= x2.4 0.2x 2yx + 6, y= y0.25 + xx + 6.Observe ...
- 9.7.19: Consider the systemx= xa 0.2x 2yx + 6, y= y0.25 + xx + 6,where a is...
- 9.7.20: There are certain chemical reactions in which the constituent conce...
- 9.7.21: The systemx= 3(x + y 13 x3 k), y= 13 (x + 0.8y 0.7)is a special cas...
Solutions for Chapter 9.7: Periodic Solutions and Limit Cycles
Full solutions for Elementary Differential Equations and Boundary Value Problems | 10th Edition
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.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Invert A by row operations on [A I] to reach [I A-I].
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.
A symmetric matrix with eigenvalues of both signs (+ and - ).
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.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
= Xl (column 1) + ... + xn(column n) = combination of columns.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.
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.
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.
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).
Solvable system Ax = b.
The right side b is in the column space of A.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.