- 4.9.1: Find the bifurcation points of each of the following systems of equ...
- 4.9.2: Find the bifurcation points of each of the following systems of equ...
- 4.9.3: Find the bifurcation points of each of the following systems of equ...
- 4.9.4: Find the bifurcation points of each of the following systems of equ...
- 4.9.5: Find the bifurcation points of each of the following systems of equ...
- 4.9.6: In each of 6-8, show that more than one equilibrium solutions bifur...
- 4.9.7: In each of 6-8, show that more than one equilibrium solutions bifur...
- 4.9.8: In each of 6-8, show that more than one equilibrium solutions bifur...
- 4.9.9: Consider the system of equations kl = 3~x1 -Sex2 + X: x2 = 2~x1 - E...
- 4.9.10: Show that are eigenvectors of the matrix with eigenvalues and - res...
Solutions for Chapter 4.9: Introduction to bifurcation theory
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics | 3rd 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).
Tv = Av + Vo = linear transformation plus shift.
Upper triangular systems are solved in reverse order Xn to Xl.
peA) = det(A - AI) has peA) = zero matrix.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
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.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
A directed graph that has constants Cl, ... , Cm associated with the edges.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. 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.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.