Global-in-time existence of unbounded separatrices and blow-up time of solutions for Liénard systems whose restoring force is an odd power function

Abstract

The existence, uniqueness, and number of limit cycles in Liénard dynamics have been intensely studied since the 1940s because they relate to Hilbert's 16th problem. Limit cycles sharply distinguish the nature of adjacent orbits. Moreover, they are essential for understanding dynamical systems. In contrast, the importance of unbounded separatrices is less well recognized even though they play an identical role. Therefore, this study focuses on the Liénard dynamical system and investigates whether all orbits, including unbounded separatrices, exist globally in time. The blow-up times of solutions that cannot be extended up to negative or positive infinity are also discussed. The result allows classification of all solutions by the time for which they exist. For example, for the van der Pol oscillator known to have the sole limit cycle, all solutions that pass through any point in the region bounded by that limit cycle and the two solutions that correspond to the unbounded separatrices continue to exist for all time, whereas the other solutions have positive or negative blow-up times. This fact cannot be judged solely from the shapes of the orbits obtained by simulation. Phase plane analysis is applied to prove the main theorem. Furthermore, several examples and their plane figures are provided to facilitate the understanding of the proof.