 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
ISBN: 9780470458310
Solutions for Chapter 9.7: Periodic Solutions and Limit Cycles
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470458310. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 10. Chapter 9.7: Periodic Solutions and Limit Cycles includes 21 full stepbystep solutions. Since 21 problems in chapter 9.7: Periodic Solutions and Limit Cycles have been answered, more than 16979 students have viewed full stepbystep solutions from this chapter.

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!).

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

Indefinite matrix.
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.

Lucas numbers
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.

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

Network.
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.

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.

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.