# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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r'= jx(1 — r2 cos2 9)r sin2 9. (21)

Again, consider an annular region R given by ^ < r < /2, where ^ is small and r^ is large. When r = rv the linear term on the right side of Eq. (21) dominates, and / > 0 except on the x-axis, where sin 9 = 0 and consequently r' = 0 also. Thus, trajectories are entering R at every point on the circle r = r1 except possibly for those on the x-axis, where the trajectories are tangent to the circle. When r = ^, the cubic term on the right side of Eq. (21) is the dominant one. Thus r' < 0 except for points on the x-axis where r' = 0 and for points near the y-axis where r2 cos2 9 < 1 and the linear term makes r' > 0. Thus, no matter how large a circle is chosen, there will be points on it (namely, the points on or near the y-axis) where trajectories are leaving R. Therefore, the Poincare-Bendixson theorem is not applicable unless we consider more complicated regions.

It is possible to show, by a more intricate analysis, that the van der Pol equation does have a unique limit cycle. However, we will not follow this line of argument further, but turn instead to a different approach in which we plot numerically computed solutions. Experimental observations indicate that the van der Pol equation has a stable periodic solution whose period and amplitude depend on the parameter fi. By looking at graphs of trajectories in the phase plane and of u versus t we can gain some understanding of this periodic behavior.

Figure 9.7.2 shows two trajectories of the van der Pol equation in the phase plane for i = 0.2. The trajectory starting near the origin spirals outward in the clockwise

FIGURE 9.7.2 Trajectories of the van der Pol equation (17) for i = 0.2.

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Chapter 9. Nonlinear Differential Equations and Stability

FIGURE 9.7.3 Plots of u versus t for the trajectories in Figure 9.7.2.

direction; this is consistent with the behavior of the linear approximation near the origin. The other trajectory passes through (—3, 2) and spirals inward, again in the clockwise direction. Both trajectories approach a closed curve that corresponds to a stable periodic solution. In Figure 9.7.3 we show the plots of u versus t for the solutions corresponding to the trajectories in Figure 9.7.2. The solution that is initially smaller gradually increases in amplitude, while the larger solution gradually decays. Both solutions approach a stable periodic motion that corresponds to the limit cycle. Figure 9.7.3 also shows that there is a phase difference between the two solutions as they approach the limit cycle. The plots of u versus t are nearly sinusoidal in shape, consistent with the nearly circular limit cycle in this case.

Figures 9.7.4 and 9.7.5 show similar plots for the case fi = 1. Trajectories again move clockwise in the phase plane, but the limit cycle is considerably different from a

FIGURE 9.7.4 Trajectories of the van der Pol equation (17) for n = 1.

9.7 Periodic Solutions and Limit Cycles

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circle. The plots of u versus t tend more rapidly to the limiting oscillation, and again show a phase difference. The oscillations are somewhat less symmetric in this case, rising somewhat more steeply than they fall.

Figure 9.7.6 shows the phase plane for fi = 5. The motion remains clockwise, and the limit cycle is even more elongated, especially in the y direction. In Figure 9.7.7 is a plot of u versus t. Although the solution starts far from the limit cycle, the limiting oscillation is virtually reached in a fraction of a period. Starting from one of its extreme values on the x-axis in the phase plane, the solution moves toward the other extreme position slowly at first, but once a certain point on the trajectory is reached, the remainder of the transition is completed very swiftly. The process is then repeated

FIGURE 9.7.6 Trajectories of the van der Pol equation (17) for n = 5.

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Chapter 9. Nonlinear Differential Equations and Stability

FIGURE 9.7.7 Plot of u versus t for the outward spiralling trajectory in Figure 9.7.6.

in the opposite direction. The waveform of the limit cycle, as shown in Figure 9.7.7, is quite different from a sine wave.

These graphs clearly show that, in the absence of external excitation, the van der Pol oscillator has a certain characteristic mode of vibration for each value of fi. The graphs of u versus t show that the amplitude of this oscillation changes very little with i, but the period increases as i increases. At the same time, the waveform changes from one that is very nearly sinusoidal to one that is much less smooth.

The presence of a single periodic motion that attracts all (nearby) solutions, that is, a stable limit cycle, is one of the characteristic phenomena associated with nonlinear differential equations.

PROBLEMS In each of Problems 1 through 6 an autonomous system is expressed in polar coordinates.

a Determine all periodic solutions, all limit cycles, and determine their stability characteristics.

1. dr/dt = r2(1 - r2), 69/dt = 1

2. dr/dt = r(1 - r)2, 69/dt =-1

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