New criterions on nonexistence of periodic orbits of planar dynamical systems and their applications. (English) Zbl 07923955
Summary: Characterizing existence or not of periodic orbit is a classical problem, and it has both theoretical importance and many real applications. Here, several new criterions on nonexistence of periodic orbits of the planar dynamical system \(\dot{x}=y,\, \dot{y}=-g(x)-f(x,y)y\) are obtained and by examples shows that these criterions are applicable, but the known ones are invalid to them. Based on these criterions, we further characterize the local topological structures of its equilibrium, which also show that one of the classical results by Andreev (Am Math Soc Transl 8:183-207, 1958) on local topological classification of the degenerate equilibrium is incomplete. Finally, as another application of these results, we classify the global phase portraits of a planar differential system, which comes from the third question in the list of the 33 questions posed by A. Gasull and also from a mechanical oscillator under suitable restriction to its parameters.
MSC:
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
| 34K18 | Bifurcation theory of functional-differential equations |
Keywords:
limit cycle; nonexistence; nilpotent equilibrium; Andreev topological classification; global phase portraitReferences:
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