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3. dr/dt = r(r - 1)(r - 3), d9/dt = 1
4. dr/dt = r(1 - r)(r - 2), d9/dt =-1
5. dr/dt = sin n r, d9/dt = 1
6. dr/dt = r|r - 2|(r - 3), d9/dt = -1
7. If x = r cos 9, y = r sin 9, show that y(dx/dt) - x(dy/dt) = r2(d9/dt).
8. (a) Show that the system
dx/dt = -y + xf(r)/ r, dy/dt = x + yf(r )/r
has periodic solutions corresponding to the zeros of f (r). What is the direction of motion on the closed trajectories in the phase plane?
(b) Let f (r) = r(r - 2)2(r2 - 4r + 3). Determine all periodic solutions and determine their stability characteristics.
9. Determine the periodic solutions, if any, of the system
dx x,, dy y i i
= y + ----- (x2 + y2 - 2), -f = -x + (x2 + y2 - 2).
dt Vx2 + y2 dt Vx2 + y2
9.7 Periodic Solutions and Limit Cycles
10. Using Theorem 9.7.2, show that the linear autonomous system
dx/dt = a11 x + a12 y, dy/dt = a21 x + a22 y
does not have a periodic solution (other than x = 0, y = 0) if a11 + a22 = 0.
In each of Problems 11 and 12 show that the given system has no periodic solutions other than constant solutions.
11. dx/dt = x + y + x3 - y2, dy/dt =-x + 2y + x2 y + y3/3
12. dx/dt = 2x 3y xy2, dy/dt = y + x3 x2y
13. Prove Theorem 9.7.2 by completing the following argument. According to Greens theorem in the plane, if C is a sufficiently smooth simple closed curve, and if F and G are continuous and have continuous first partial derivatives, then
j[ F (x, y) dy G (x, y) dx] = yI [ Fx (x, y) + Gy (x, y)] dA,
where C is traversed counterclockwise and R is the region enclosed by C. Assume that x = 0(t), y = ?(t) is a solution of the system (15) that is periodic with period T. Let C
be the closed curve given by x = 0(t), y = ? (t) for 0 < t < T. Show that for this curve
the line integral is zero. Then show that the conclusion of Theorem 9.7.2 must follow.
? 14. (a) By examining the graphs of u versus t in Figures 9.7.3, 9.7.5, and 9.7.7 estimate the
period T of the van der Pol oscillator in these cases.
(b) Calculate and plot the graphs of solutions of the van der Pol equation for other values of the parameter i. Estimate the period T in these cases also.
(c) Plot the estimated values of T versus i. Describe how T depends on i.
? 15. The equation
u" i(1 1 u'2)u' + u = 0 is often called the Rayleigh equation.
(a) Write the Rayleigh equation as a system of two first order equations.
(b) Show that the origin is the only critical point of this system. Determine its type and whether it is stable or unstable.
(c) Let i = 1. Choose initial conditions and compute the corresponding solution of the system on an interval such as 0 < t < 20 or longer. Plot u versus t and also plot the trajectory in the phase plane. Observe that the trajectory approaches a closed curve (limit cycle). Estimate the amplitude A and the period T of the limit cycle.
(d) Repeat part (c) for other values of i, such as i = 0.2, 0.5, 2, and 5. In each case estimate the amplitude A and the period T.
(e) Describe how the limit cycle changes as i increases. For example, make a table of
values and/or plot A and T as functions of i.
? 16. The system
F = 3(x + y 3 x3 k), y = 3 (x + 0.8y 0.7)
is a special case of the Fitzhugh-Nagumo equations, which model the transmission of neural impulses along an axon. The parameter k is the external stimulus.
(a) For k = 0 show that there is one critical point. Find this point and show that it is an asymptotically stable spiral point. Repeat the analysis for k = 0.5 and show that the critical point is now an unstable spiral point. Draw a phase portrait for the system in each case.
(b) Find the value k0 where the critical point changes from asymptotically stable to unstable. Draw a phase portrait for the system for k = k0.
(c) For k > k0 the system exhibits an asymptotically stable limit cycle. Plot x versus t for k = k0 for several periods and estimate the value of the period T.
(d) The limit cycle actually exists for a small range of k below k0. Let kj be the smallest value of k for which there is a limit cycle. Find k1.
Chapter 9. Nonlinear Differential Equations and Stability
9.8 Chaos and Strange Attractors: The Lorenz Equations
ฎIn principle, the methods described in this chapter for second order autonomous systems can also be applied to higher order systems as well. In practice, there are several difficulties that arise in trying to do this. One problem is that there is simply a greater number of possible cases that can occur, and the number increases with the order of the system (and the dimension of the phase space). Another is the difficulty of graphing trajectories accurately in a phase space of higher than two dimensions; even in three dimensions it may not be easy to construct a clear and understandable plot of the trajectories, and it becomes more difficult as the number of variables increases. Finally, and this has only been clearly realized in the last few years, there are different and very complex phenomena that can occur, and do frequently occur, in systems of third and higher order that are not present in second order systems. Our goal in this section is to provide a brief introduction to some of these phenomena by discussing one particular third order autonomous system that has been intensively studied in recent years. In some respects the presentation here is similar to the treatment of the logistic difference equation in Section 2.9.