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On the limit cycles of planar autonomous systems. (English) Zbl 0996.34026

Consider a general planar autonomous ordinary differential equation (ODE). This paper presents several results to prove either nonexistence or giving upper bounds on its number of limit cycles. These results do not assume any special structure for the ODE but their hypotheses seem difficult to be verified. Anyway, the authors present several concrete examples of application. Two of the main ideas used in the proofs are: (1) an application of the generalized Bendixson-Dulac criterion for nonsimply connected domains; (2) the existence of a Lyapunov function in a ball and a comparison between the trajectories of the ODE considered and a second ODE, outside this ball. The second ODE is integrable and of the form \(\dot x=P(y),\) \(\dot y=Q(x)\) and it is constructed from the first one.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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
37B25 Stability of topological dynamical systems
37C27 Periodic orbits of vector fields and flows
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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References:

[1] doi:10.1088/0951-7715/9/2/013 · Zbl 0886.58087 · doi:10.1088/0951-7715/9/2/013
[2] doi:10.1016/0362-546X(95)00061-Y · Zbl 0851.34029 · doi:10.1016/0362-546X(95)00061-Y
[5] doi:10.1016/0022-247X(90)90023-9 · Zbl 0706.34037 · doi:10.1016/0022-247X(90)90023-9
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