By Stephen Lynch
This re-creation has been completely up-to-date and extended to incorporate extra purposes, examples, and workouts, all with options. It includes new chapters on neural networks and simulation.
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I (t) = - The periodic expression ~ sin (t)- ~ cos(t) is called the steady state, and the term Hie-! is called the transient. Note that the transient decays to zero as t -+ oo. Example 10. t. time and substitute for ~ to obtain d 2I di 1 dE L-+R-+-I=-. dt2 dt c dt The second-order differential equation for a certain RLC circuit is given by d 2I dt 2 di + 5 dt + 6I = Solve this differential equation given that I (0) . 10sm(t). = i (0) = 0 (a passive circuit). 4. Existence and Uniqueness Theorem 27 Solution.
At each Consider the slope of the trajectories: The slopes are given by point (x, y) in the plane. Manifolds: There is one linearly independent eigenvector, (0, I) T. Therefore, the critical point is a stable degenerate node. The stable manifold Es is the y axis. 1 0. * x Phase portraits of nonlinear planar autonomaus systems will be considered in the next chapter, where stable and unstable manifolds do not necessarily lie on straight lines. However, all is not lost as the manifolds for certain critical points are tangent to the eigenvectors of the linearized system at that point.
10)o A critical point that is not stable is called an unstable critical point. 4. Existence and Uniqueness Theorem 29 Example 12. i = x 2 - 1. Solutions. (a) There is one critical point at xo = 0. i > 0. Therefore, xo is an unstable critical point. Solutions starting either side of xo are repelled from it. (b) There is one critical point at xo = 0. i < 0. Solutions starting either side of xo are attracted towards it. The critical point is stable. (c) There are two critical points, one at X! = -1 and the other at x2 = 1.