Ch 9.7: Periodic Solutions and Limit Cycles

1. Ch 9.7: Periodic Solutions and Limit Cycles • In this section we discuss further the possible existence of periodic solutions of second order autonomous systems x' = f(x) • Such solutions satisfy the relation x(t + T) = x(t) for all t and for some nonnegative constant T called the period. • Periodic solutions are often important in physical problems because they represent phenomena that occur repeatedly. • In many situations a periodic solution represents a “final state” toward which all “neighboring” solutions tend as the transients due to the initial conditions die out.

2. Nonconstant Periodic Solutions • Thus a periodic solution satisfies x(t + T) = x(t) for all t and for some nonnegative constant T. • Note that a constant solution x = x0 is periodic for any T. • In this section, the periodic solutions that are discussed refer to nonconstant periodic solutions. • In this case the period T is positive and is usually chosen as the smallest positive number for which x(t + T) = x(t) is valid.

3. Linear Autonomous Systems • Recall: The solutions of a linear autonomous system x' = Ax are periodic if and only if the eigenvalues are pure imaginary. • Thus if the eigenvalues of A are pure imaginary, then every solution of x' = Ax is periodic, while if the eigenvalues of A are not pure imaginary, then there are no periodic solutions. • The predator-prey equations discussed in Section 9.5, although nonlinear, behave similarly: All solutions in the first quadrant are periodic. See graph below.

4. Example 1: Nonlinear System (1 of 8) • Consider the nonlinear autonomous system • It can be shown that (0, 0) is the only critical point and that this system is almost linear near the origin. • The corresponding linear system has eigenvalues 1  i, and hence the origin is an unstable spiral point.

5. Example 1: Unstable Spiral Point (2 of 8) • Thus the origin is an unstable spiral point, and hence any solution that starts near the origin in the phase plane will spiral away from the origin. • Since there are no other critical points, we might think that all solutions of our nonlinear system correspond to trajectories that spiral out to infinity. • However, we will show that this is incorrect, because far away from the origin the trajectories are directed inward.

6. Example 1: Polar Coordinates (3 of 8) • Our nonlinear system can be written as • Then • Using polar coordinates x = rcos and y = rsin, note that • Thus

7. Example 1: Critical Points for Equation of Radius (4 of 8) • The critical points (for r  0) of are r = 0 (the origin) and r = 1, which corresponds to the unit circle in the phase plane. • Note that dr/dt > 0 if r < 1 and dr/dt < 0 if r > 1. Thus inside the unit circle, the trajectories are directed outward, while outside the unit circle they are directed inward. • The circle r = 1 appears to be a limiting trajectory for system. • We next determine an equation for .

8. Example 1: Equation for Angle (5 of 8) • Recall our nonlinear system: • Then • Using polar coordinates x = rcos and y = rsin, note that • It follows that

9. Example 1: A Solution to Polar Equations (6 of 8) • Our original nonlinear system is therefore equivalent to the system • One solution to this system is where t0 is an arbitrary constant. • As t increases, a point on this solution trajectory moves clockwise around the unit circle.

10. Example 1: General Solution to Polar Equations (6 of 8) • Other solutions of can be found by separation of variables: For r  0 and r  1, and after using a partial fraction expansion and some algebra, where c0 and t0 are arbitrary constants. • Note that c0 = 0 yields r = 1,  = -t + t0, as before.

11. Example 1: Initial Value Problem in Polar Form (8 of 8) • The solution satisfying the initial value problem is given by • We have the following two cases: • If  < 1, then r  1 from the inside as t  . • If  > 1, then r  1 from the outside as t  . • See phase portrait on right.

12. Limit Cycle • In the previous example, the circle r = 1 not only corresponds to periodic solutions of the system but it also attracts other nonclosed trajectories that spiral toward it as t  . • In general, a closed trajectory in the phase plane such that other nonclosed trajectories spiral toward it, either from the inside or the outside, as t  , is called a limit cycle.

13. Stability of Closed Trajectories • If all trajectories that start near a closed trajectory spiral toward the closed trajectory as t  , both from the inside and the outside, then the limit cycle is asymptotically stable. • In this case, since the closed trajectory is itself a periodic orbit rather than an equilibrium point, this type of stability is often called orbital stability. • If the trajectories on one side spiral toward a closed trajectory , while those on the other side spiral away as t  , then the closed trajectory is semistable. • If the trajectories on both sides of a closed trajectory spiral away as t  , then the closed trajectory is unstable. • Closed trajectories for which other trajectories neither approach nor depart from are called stable.

14. Theorem 9.7.1 • Consider the autonomous system • Let F and G have continuous first partial derivatives in a domain D in the xy-plane. • A closed trajectory of the system must necessarily enclose at least one critical (equilibrium) point. • If encloses only one critical point, the critical point cannot be a saddle point. • Note: It follows that in any region not containing a critical point, there cannot be a closed trajectory within that region.

15. Theorem 9.7.2 • Consider the autonomous system • Let F and G have continuous first partial derivatives in a simply connected domain D in the xy-plane. • If Fx + Gy has the same sign throughout D, then there is no closed trajectory of the system lying entirely within D. • Note: A simply connected domain in the xy-plane is a domain with no holes. • Also, If Fx + Gy changes sign in D, then no conclusion can be drawn.

16. Example 2: Applying Theorem 9.7.2 (1 of 2) • Consider again the nonlinear autonomous system • Then • Thus Fx + Gy > 0 on 0  r < (1/2)½, so there is no closed trajectory in this simply connected circular disk. • From Example 1, there is no closed trajectory in r < 1. • Thus the information given in Theorem 9.7.2 may not be the best possible result.

17. Example 2: Annular Region and Theorem 9.7.2 (2 of 2) • Note that • However, Theorem 9.7.2 does not apply since the annular region r > (1/2)½ is not simply connected. • Thus we cannot use Theorem 9.7.2 to conclude that there is no closed trajectory lying entirely within r > (1/2)½. • In fact, from Example 1, we know that r = 1 is a closed trajectory for the system that lies entirely within r > (1/2)½.

18. Theorem 9.7.3 (Poincaré-Bendixson) • Consider the autonomous system • Let F and G have continuous first partial derivatives in a domain D in the xy-plane. • Let D1 be a bounded subdomain in D, and let R be the region that consists of D1 plus its boundary (all points of R are in D). • Suppose that R contains no critical point of the system. • If there exists a constant t0 such that x = (t), y =  (t) is a solution of the system that exists and stays in R for all t > t0, then x = (t), y =  (t) either is a periodic solution (closed trajectory) or spirals toward a closed trajectory as t  . • In either case, the system has a periodic solution in R.

19. Example 3: Applying Theorem 9.7.3 • Consider again the nonlinear autonomous system • Since the origin is a critical point, it must be excluded from R. • Consider the region R defined by 0.5  r  2. • Recall from Example 1 that dr/dt = r(1-r) for 0.5  r  2. • For r = 0.5, dr/dt > 0 and hence r increases, while for r = 2, dr/dt < 0 and hence r decreases. • Thus a trajectory that crosses the boundary of R is entering R. • Consequently, any solution that starts in Rat t = t0 cannot leave but must stay in R for all t > t0, and is either a periodic solution orapproaches one as t  .

20. Example 4: Van der Pol Equation (1 of 13) • The van der Pol equation describes the current u in a triode oscillator: • If  = 0, then the equation reduces to u''+ u = 0, whose solutions are sine or cosine waves of period 2. • If  > 0, then -(1– u2) is the resistance coefficient. • For large , the resistance term is positive and acts to reduce the amplitude of the response. • For small , the resistance term is negative and causes the response to grow. • This suggests that perhaps there is a solution of intermediate size that other solutions approach as t increases.

21. Example 4: Unstable Critical Point (2 of 13) • Let x = u and y = u'.Then the van der Pol equation becomes • The only critical point is the origin. This system is almost linear, with linear approximation whose eigenvalues are [  (2 – 4)½]/2. • Thus the origin is an unstable spiral point for 0 <  < 2, and an unstable node for   2. In all cases, a solution that starts near the origin grows as t increases.

22. Example 4: Theorems 9.7.1 and 9.7.2 (3 of 13) • With regard to periodic solutions, Theorems 9.7.1 and 9.7.2 provide only partial information. • From Theorem 9.7.1 we conclude that if there are closed trajectories, then they must enclose the origin. • To apply Theorem 9.7.2, we first calculate • It follows that closed trajectories, if there are any, are not contained in the strip |x| < 1, where Fx + Gy > 0. • To apply Theorem 9.7.3, we introduce polar coordinates to obtain the following equation for r:

23. Example 4: Theorems 9.7.1 and 9.7.2 (4 of 13) • We have the following equation for r: • Consider the annular region R given by r1  r  r2, where r1 is small and r2 is large. • When r = r1, the linear term in the equation for r' dominates, and r' > 0 except on the x-axis, where sin = 0, hence r' = 0. • Thus the trajectories are entering R at every point on the circle r = r1, except possibly those on the x-axis, where trajectories are tangent to the circle.

24. Example 4: Theorem 9.7.3 (5 of 13) • We have the following equation for r: and R given by r1  r  r2, where r1 is small and r2 is large. • When r = r2, the cubic term in the equation for r' dominates, and r' < 0 except on the x-axis, where r' = 0, and for points near the y-axis where r2cos2 < 1, and hence 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 Theorem 9.7.3 is not applicable unless we consider more complicated regions. • It is possible to show that this system does have a unique limit cycle, but we will not pursue that here.

25. Example 4: Numerical Solutions (6 of 13) • We next plot numerically computed solutions. • Experimental observations show that the van der Pol equation has an asymptotically stable periodic solution whose period and amplitude depend on the parameter . • Graphs of trajectories in the phase plane and of u versus t can provide some understanding of periodic behavior.

26. Example 4: Phase Portrait ( = 0.2)(7 of 13) • The graph below shows two trajectories when  = 0.2. • The trajectory starting near the origin spirals outward in the clockwise 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.

27. Example 4: Limit Cycle ( = 0.2) (8 of 13) • Given below are the graphs for the two trajectories previously mentioned, along with corresponding graphs of u versus t. • The solution solution that is initially smaller gradually increases in amplitude, while larger solution gradually decays. • Both solutions approach a stable periodic motion that corresponds to the limit cycle.

28. Example 4: Phase Difference ( = 0.2) (9 of 13) • Given below are the graphs for the two trajectories previously mentioned, along with corresponding graphs of u versus t. • The graph of u versus t 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.

29. Example 4: Solution Graphs ( = 1)(10 of 13) • The graphs below show similar plots for the case  = 1. • Trajectories again move clockwise in the phase plane, but the limit cycle is considerably different from a circle. • The graphs of u versus t tend more rapidly to the limiting oscillation than before, and again show a phase difference. • The oscillations are somewhat less symmetric in this case, rising somewhat more steeply than the fall.

30. Example 4: Phase Portrait ( = 5)(11 of 13) • The graph below shows a phase portrait for the case  = 5. • Trajectories again move clockwise in the phase plane. • Although solution starts far from the limit cycle, the limiting oscillation is virtually reached in a fraction of the period. • Starting from one of its extreme values on the x axis, the solution moves toward other extreme slowly at first, but once a certain point is reached, the rest of the transition is completed swiftly. The process is repeated in the opposite direction.

31. Example 4: Solution Graphs ( = 5)(12 of 13) • Given below is a graph of u versus t for the case  = 5, along with the phase portrait discussed on the previous slide. • Note that the waveform of the limit cycle is quite different from a sine wave.

32. Example 4: Discussion(13 of 13) • Recall that the van der Pol equation is • The graphs on the previous slides show that, in the absence of external excitation, the van der Pol oscillator has a certain characteristic mode of vibration for each value of . • The graphs of u versus t show that the amplitude of oscillator changes very little with , but period increases as  increases. • At the same time, the waveform changes from one that is nearly sinusoidal to one that is much less smooth. • The presence of a single periodic motion that attracts all nearby solutions (asymptotically stable limit cycle), is one of the characteristics associated with nonlinear equations.