Limit cycles with stability of the first level

A classification of the singularities of(X, f)

“…This shrinking is illustrated in Figure 6. As pointed out in [7] and illustrated in [5], the main theorem on stability is applicable to the case of limit cycles. However, by the corollary given above, the limit cycles with stability of the first level can be analyzed.…”

“…The main theorem of stability as given in [4] was later explicitly shown to apply to systems with a limit cycle with stability of the first level [5]. In a recent article [7] Kallina has pointed out that the theorem given earlier should be reworded to distinguish the two cases: The closed test curve a (1) around a singular point, (2) enclosing a limit cycle.…”

“…If the main theorem [5,7] is applied to detect the stability property of this limit cycle, then by the reverse implication of the corollary given above along a test curve lying outside the limit cycle, various regions indicating both stability and instability will be expected. In Figure 7 taken from [4], ~ and cra are corn- s I Figure 7 )B d pared, and by the first corollary it is concluded that this limit cycle has a stability of higher order than the first.…”

“…The Case of Limit Cycles. In [5], an example of a limit cycle with stability of the first level was given. By use of the characteristic orientations, various types of limit cycles can be characterized.…”

“…A limit cycle given as stable from the outside can, after shrinking, be viewed as a focus or a possible node. A limit cycle of this nature is with stability I and 1 of the first level, because only Win1 Wm2 can characterize its stability [5].…”