1. Boyce/DiPrima 9th ed, Ch 9.7: Periodic Solutions and Limit Cycles Elementary Differential Equations and Boundary Value Problems, 9th edition, by William E. Boyce and Richard C. DiPrima, ©2009 by John Wiley & Sons, Inc.
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 the 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 it 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. Applying Theorem (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 may not be the best possible result.

17. Annular Region and Theorem 9.7.2 (2 of 2)
Note that
However, Theorem does not apply since the annular region r > (1/2)½ is not simply connected.
Thus we cannot use Theorem 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. 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 R at t = t0 cannot leave but must stay in R for all t > t0, and is either a periodic solution or approaches one as t  .

20. Example 2: 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 2: 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 2: Theorems 9.7.1 and 9.7.2 (3 of 13)
With regard to periodic solutions, Theorems and provide only partial information.
From Theorem 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 2: 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 2: Theorem (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 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 2: 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 2: 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 2: 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 2: 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 2: 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 2: 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 2: 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 2: 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.