- 9.7.1: In each of 1 through 6 an autonomous system is expressed in polar c...
- 9.7.2: In each of 1 through 6 an autonomous system is expressed in polar c...
- 9.7.3: In each of 1 through 6 an autonomous system is expressed in polar c...
- 9.7.4: In each of 1 through 6 an autonomous system is expressed in polar c...
- 9.7.5: In each of 1 through 6 an autonomous system is expressed in polar c...
- 9.7.6: In each of 1 through 6 an autonomous system is expressed in polar c...
- 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 system dx/dt = y + xf(r)/r, dy/dt = x + yf(r)/r h...
- 9.7.9: Determine the periodic solutions, if any, of the system dx dt = y +...
- 9.7.10: Using Theorem 9.7.2, show that the linear autonomous system dx/dt =...
- 9.7.11: In each of 11 and 12 show that the given system has no periodic sol...
- 9.7.12: In each of 11 and 12 show that the given system has no periodic sol...
- 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 equation u (1 1 3u2 )u + u = 0 is often called the Rayleigh13 e...
- 9.7.16: Consider the system of equations x = x + y x(x2 + y2 ), y = x + y y...
- 9.7.17: Consider the van der Pol system x = y, y = x + (1 x2 )y, where now ...
- 9.7.18: 18 and 19 extend the consideration of the RosenzweigMacArthur preda...
- 9.7.19: 18 and 19 extend the consideration of the RosenzweigMacArthur preda...
- 9.7.20: There are certain chemical reactions in which the constituent conce...
- 9.7.21: The system x = 3(x + y 1 3 x3 k), y = 1 3 (x + 0.8y 0.7) is a speci...
Solutions for Chapter 9.7: Periodic Solutions and Limit Cycles
Full solutions for Elementary Differential Equations and Boundary Value Problems | 9th 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).
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Invert A by row operations on [A I] to reach [I A-I].
lA-II = 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.
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.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
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).
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).