Periodic Process, Closed Curve, Limit Cycle and Its Stability

The second piece in the series deals with the subjects mentioned in the above title.

1 Periodic (or cyclic) process 

Definition: A process each assumed value recurs after the same time lapse is called a periodic process. The formula is the following:
f(t) = f(t + T),
with t – time, f – a time function, T – the time, that is, the interval between two repetitions following each other.

This is taught at high school; therefore I did not find it necessary to give any literature references.

Experiment: A ball was rolled round and around along an edge. With any particular point of the edge chosen, after a certain period of time the ball got back to the same point. (Needless to say, the experiment was not perfect, as the ball could not be rolled around always at the same speed. Accordingly, the time it took for it to get back to the point of departure varied too. )

Explanation: Presumably, this does not need to be explained.

2. Closed curve of a phase space

Definition: A curve of the n-dimensional phase space each assumed value of which recurs when going along the curve is called a closed curve. The formula is the following:
g(y, x1…xn-1, t) = g(y, x1…xn-1, t + T),
with t, T – same as in point 1, g – a function of the variables in brackets, y – dependent variable, x1…xn-1 – n-1independent variable of the n-dimensional phase space.

Experiment: same as in point 1.

Explanation: It is closed, as departing any direction along the curve and going toward the same direction, one gets to the same section again after doing a certain distance. That is, the curve has closed and repeats .

3 Limit cycle

Definition: A closed curve of the n-dimensional phase space contained in a non- infinitely thin space section in which there is no other closed curve is called a limit cycle.

Experiment: The edge of a hill has such characteristics because, as already mentioned, it is a closed curve in points 1 and 2, while all the others were not. For whichever other point the ball was placed on, it did not get back to its point of departure. (The only exception being the stable singular point, but there it was not possible to move around, as it was merely a point.)

Definition: It is a cycle because it is cyclic (as mentioned in point 1, this is another term for periodic), and it is a separator because it separates the non-cyclic movements within and around it from each other.

4 Stable limit cycle

Definition: A limit cycle of the n-dimensional phase space contained in a non-infinitely thin space section from the points in which the moving point does not move away, due to the process to the system, beyond a given non-infinitely thin space section which contains the limit cycle is called, by Liapunoff’s definition, a stable limit cycle.
In the case of asymptotic stability, it goes toward the limit cycle and is called an asymptotic stable limit cycle.

Experiment: Here, the edge was a ring formed at the bottom of the pit, within which ring there was a small hillock. Placed on any point of the sidewall of the hillock or the pit, the balled moved onto the ring, which it did not leave when rolled.

Explanation: Similarly to the pit bottom earlier called a singular point, the ring ‘attracted’ the ball, that is, the ball rolled there.

 

5 Unstable limit cycle

Definition: A limit cycle of the n-dimensional phase space contained in a non-infinitely thin space section from the points in which the moving point moves away, due to the process to the system, beyond a given non-infinitely thin space section which contains the limit cycle is called, by Liapunoff’s definition, an unstable limit cycle.
In case of asymptotic instability, it goes away from the limit cycle and is called an asymptotic unstable limit cycle.

Experiment: Here, there was only an edge, and the ball, whichever direction taken from it, never rolled back onto the edge.

Explanation: The experiment explained everything.

 

6 Coordinate-values-dependent stable limit cycle

No literature reference can be given here, as this term was introduced by me in the study referred to in my previous piece.

Definition: A limit cycle which is stable, with the exception of certain coordinate values of the n-dimensional phase space, in the space section surrounding the singular formation. Similarly to the above, it can be Liapunoff-defined, or asymptotic.

Experiment: In the sidewall of the pit, there was a ring the ball was not able to roll through. Thus, when the ball started its journey from a point of the sidewall higher than the ring and traveled that direction, it did not make it to the bottom of the pit and, when rolled, it moved around the ring. However, when it started its journey from a lower point of the sidewall, it did not roll to the ring but to the bottom of the pit.

Explanation: The ball rolls from every point to the bottom of the pit, that is, to the singular point (thus it is stable regarding these points); it is unstable regarding the limit cycle, from where it does not go to that area. Regarding those points from which it gets there it is stable, as the ball stops there. (Thus it does not make it to the singular point at the bottom of the pit.

 

7 Global stability

Definition: In case the stable domain of the limit cycle is of infinite dimension regarding all the coordinates of the n-dimensional phase space, it s called globally stable.

Experiment: An infinitely huge hill and pit would have been needed here, which it was impossible to build.

Explanation: It is easy to see that nothing demonstrated earlier is altered in any way by the fact that everything is enormously huge here.

8 Remark

Here too global stability is related to the coordinate-values-dependent stable or unstable character of the examined limit cycle. This means that a limit cycle can be locally stable, but globally stable only depending on the coordinate values.

Literature:
Andronov-Haikin-Vitt: Theory of Oscillation, 1959, Moscow.

Dr. Endre Simonyi