Question

Exercise 4.1.6: Sketch a phase portrait that shows an unstable limit cycle.

The image I included below is meant to help with the exercise above. It is the background information given.

This new kind of friction then gives us a new differential equation A simulation of this equation results in the trajectories shown in Figure 4.8, right. spring friction Figure 4.8: Upper: spring force and friction force for the Rayleigh clarinet model. Lower Left: Two representative trajectories for this model. Lower Right: All trajectories, from any initial condition except (0, O), approach the red loop asymptotically Consider the closed orbit shown in red. Note an interesting fact about it, which we have not seen before: if you choose an initial condition that is not on the red loop, the ensuing trajectory will get closer and closer to the red loop, and will approach it as t oo. This is true whether you are inside the red loop or outside it; all trajectories, with the exception of the one point at (0, 0), approach the red loop arbitrarily closely In other words, the red loop fits the definition of an attractor. It is our first example of a closed orbit attractor, or periodic attractor. A third name for these is based on the idea that just as an equilibrium point is a limit point, the red loop is a limit cycle, and so these are called limit cycle attractors.1 Note that another name for the red loop is a stable limit cycle. It is stable in exactly the same sense as a stable equilibrium point: if you perturb the system off the cycle, the behavior returns to the cycle. So it really is an attractor Exercise 4.1.6 Sketch a phase portrait that shows an unstable limit cycle

Answer 1

The phase potrait of the unstable limit cycle is as under Consider the closed orbit shown in red. Now. note that this is an unstable limit cycle, because, if you choose an initial condition that is not on ed lcop, the ensuing trajectory gets farther and Farther from the red lop as tThis is true hether we are inside the ed loopo, outside it. Note that the initial condition should not be on red loop and the sketch demonstrates two dictinct nitial conditions that are close to the red loop but rot on it and with time their trojectories moves avay from the loop