1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 423 4 2 y 0 -2 -4 -4 -2 0 2 4 x Figure 10 .5 Saddle Point If a particle starts at a point o ...

424 Ordinary Differential Equations Now suppose that .A = a + (3i where (3 #- 0 is an eigenvalue of the real matrix A and w = u ...

Applications of Systems of Equations EXAMPLE 5 An Asymptotically Unstable Critical Point at the Origin The eigenvalues of the sy ...

426 Ordinary Differential Equations In examining the stability of the critical point at the origin for system (4), dz/ dt = Az, ...

Applications of Systems of Equations 427 g(x, y) = g(x, y) + gx(x, y)(x - x) + gy(x, y)(y - y) + R 2 (x, y) where Ri(x, y)/ J(x ...

428 Ordinary Differential Equations EXAMPLE 7 Determining Critical Points and Type of Stability for a Nonlinear System Find all ...

Applications of Systems of Equations 429 Translating the critical point (1, 1) to the origin by letting X = x - 1 and Y = y - 1 ...

430 Ordinary Differential Equations 2 1 y 0 -1 -1 0 1 2 x Figure 10.8b Phase-Plane Portrait for System (11) EXAMPLE 8 Determinin ...

Applications of Systems of Equations 431 Calculating first partial derivatives, we find fx = -3x^2 , fy = -1, 9x = 1, and gy = - ...

432 Ordinary Differential Equations E X ERCISES 10.2 I n exercises 1 -8 calculate the eigenvalues of the given linear sys- tem. ...

Applications of Systems of Equations Consider the nonlinear autonomous system x' = y + ax^3 + axy^2 (14) y' = -x + ay^3 + ayx^ ...

434 Ordinary Differential Equations We analyzed the linear a rms race model (1) earlier. Suppose that the term representing t he ...

Applications of Systems of Equations EXAMPLE 9 A Nonlinear Modified Richardson's Arms Race Model Consider the modified arms race ...

436 Ordinary Differential Equations At (2, 3) the associated linear system is x' = fx(2, 3)(x - 2) + fy(2, 3)(y- 3) = -8(x - 2) ...

Applications of Systems of Equations 437 SOLUTION Mathematical Analysis The functions fx = -4x, fy = 3, 9x = 8, and gy = -2y are ...

438 Ordinary Differential Equations display the graph on t he rectangle R where 0 :::; x :::; 5 and 0 :::; y :::; 5. The resulti ...

Applications of Systems of Equations 439 5 4 3 y 2 1 0 0 1 2 3 4 5 x Figure 10.12 Phase-Plane Portrait for System (8) jcomments ...

440 Ordinary Differential Equations EXERCISES 10.3 Consider the nonlinear Richardson's arms race model dx = 5y-x^2 + 5 dt (9) ...

Applications of Systems of Equations Consider the nonlinear arms race model (11) dx 2 dt = 9y-x - 9 dy 2 - = 4x-y - 8. dt a. F ...

442 Ordinary Differential Equations c. (i) Find the critical points in the first quadrant of the system dx dt = 3y-x -1 (13) dy ...