� � � � � � � � � Section 5. Graphing Systems 5A. The Phase Plane 5A-1. Find the critical points of each of the following non-linear autonomous systems. x � = x 2 − y 2 x = 1 − x + y a) � b) � y = x − xy y = y + 2x 2 5A-2. Write each of the following equations as an equivalent first-order system, and find the critical points. a) x�� + a(x2 − 1)x� + x = 0 b) x�� − x� + 1 − x2 = 0 5A-3. In general, what can you say about the relation between the trajectories and the critical points of the system on the left below, and those of the two systems on the right? x = f (x, y) x � = −f (x, y) x = g(x, y) a) y = g(x, y) y � = −g(x, y) b) y � = −f (x, y) 5A-4. Consider the autonomous system x = f (x, y) x(t); solution : x = . y = g(x, y) y(t) a) Show that if x1(t) is a solution, then x2(t) = x1(t − t0) is also a solution. What is the geometric relation between the two solutions? b) The existence and uniqueness theorem for the system says that if f and g are contin-uously differentiable everywhere, there is one and only one solution x(t) satisfying a given initial condition x(t0) = x0. Using this and part (a), show that two trajectories cannot intersect anywhere. (Note that if two trajectories intersect at a point (a, b), the corresponding solutions x(t) which trace them out may be at (a, b) at different times. Part (a) shows how to get around this difficulty.) 5B. Sketching Linear Systems x = −x 5B-1. Follow the Notes (GS.2) for graphing the trajectories of the system � y = −2y . dya) Eliminate t to get one ODE = F (x, y). Solve it and sketch the solution curves. dx b) Solve the original system (by inspection, or using eigenvalues and eigenvectors), obtaining the equations of the trajectories in parametric form: x = x(t), y = y(t). Using these, put the direction of motion on your solution curves for part (a). What new trajectories are there which were not included in the curves found in part (a)? c) How many trajectories are needed to cover a typical solution curve found in part (a)? Indicate them on your sketch. 1� � � � � �� � � � 2 18.03 EXERCISES d) If the system were x� = x, y� = 2y instead, how would your picture be modified? (Consider both parts (a) and (b).) 5B-2. Answer the same questions as in 5B-1 for the system x = y, y� = x. (For part (d), use −y and −x as the two functions on the right.) 5B-3. Answer the same question as in 5B-1a,b for the system x� = y, y = −2x. For part (b), put in the direction of motion on the curves by making use of the vector field corresponding to the system. 5B-4. For each of the following linear systems, carry out the graphing program in Notes GS.4; that is, (i) find the eigenvalues of the associated matrix and from this determine the geometric type of the critical point at the origin, and its stability; (ii) if the eigenvalues are real, find the associated eigenvectors and sketch the corre-sponding trajectories, showing the direction of motion for increasing t; then draw in some nearby trajectories; (iii) if the eigenvalues are complex, obtain the direction of motion and the approximate shape of the spiral by sketching in a few vectors from the vector field defined by the system. x = 2x − 3y x � = 2x x = −2x − 2y a) � b) c) � y = x − 2y y � = 3x + y y = −x − 3y x = x − 2y x � = x + y d) � e) � y = x + y y = −2x − y 5B-5. For the damped spring-mass system modeled by the ODE mx + cx � + kx = 0, m, c, k > 0 , a) write it as an equivalent first-order linear system; b) tell what the geometric type of the critical point at (0, 0) is, and determine its stability, in each of the following cases. Do it by the methods of Sections GS.3 and GS.4, and check the result by physical intuition. (i) c = 0 (ii) c � 0; m, k � 1. (iii) Can the geometric type be a saddle? Explain. 5C. Sketching Non-linear Systems 5C-1. For the following system, the origin is clearly a critical point. Give its geometric type and stability, and sketch some nearby trajectories of the system. x = x − y + xy y � = 3x − 2y − xy � 2 x = x + 2x 2 − y 5C-2. Repeat 5C-1 for the system y = x − 2y + x 3� � � � � 3 SECTION 5. GRAPHING SYSTEMS 3 x � = 2x + y + xy 5C-3. Repeat 5C-1 for the system � y = x − 2y − xy 5C-4. For the following system, carry out the program outlined in Notes GS.6 for sketching trajectories — find the critical points, analyse each, draw in nearby trajectories, then add some other trajectories compatible with the ones you have drawn; when necessary, put in a vector from the vector field to help. x = 1 − y � 2 2 y = x − y � 2 x = x − x − xy 5C-5. Repeat 5C-4 for the system 2 y � = 3y − xy − 2y 5D. Limit Cycles 5D-1. In Notes LC, Example 1, a) Show that (0, 0) is the only critical point (hint: show that if (x, y) is a non-zero critical point, then y/x = −x/y; derive a contradiction). b) Show that (cos t, sin t) is a solution; it is periodic: what is its trajectory? c) Show that all other non-zero solutions to the system get steadily closer to the solution in part (b). (This shows the solution is an asymptotically stable limit cycle, and the only periodic solution to the system.) 5D-2. Show that each of these systems has no closed trajectories in the region R (this is the whole xy-plane, except in part (c)). � 2 x = 2x + x 2 + y� 3 2 2x = x + x 3 + yx � = x + y 2 2a) b) c) y � = x − y y � = y + x 3 + y 3 y � = 1 + x − y R = half-plane x < −1 x = ax + bx2 − 2cxy + dy2 find the condition(s) on the six constants that d) � : y = ex + f x 2 − 2bxy + cy 2 guarantees no closed trajectories in the xy-plane 5D-3. Show that Lienard’s equation (Notes LC, (6)) has no periodic solution if either a) u(x) > …
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