Abstract
This paper deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincare-Bendixson regions by using transversal curves, that enables us to prove the existence of a limit cycle that has been numerically detected. We apply our results to several known systems, like the Brusselator one or some Lienard systems, to prove the existence of the limit cycles and to locate them very precisely in the phase space. Our method, combined with some other classical tools can be applied to obtain sharp bounds for the bifurcation values of a saddle-node bifurcation of limit cycles, as we do for the Rychkov system.
Highlights
We consider real planar polynomial differential systems of the form x = dx/dt = P (x, y), y = dy/dt = Q(x, y), (1.1)where P (x, y) and Q(x, y) are real polynomials
For a given vector field, when it is not very near of a bifurcation, the limit cycles can usually be detected by numerical methods
In this work we present a new procedure to prove the existence of a limit cycle once it is numerically detected
Summary
As we have already stated our aim is to find transversal curves which define Poincare–Bendixson annular regions and to prove the existence of limit cycles, as well as to locate them. We prove that the different level of difficulty is hidden in the sizes of the respective Fourier coefficients of the two limit cycles, see Theorem 4.1 This theorem shows that our approach for detecting strictly transversal closed curves always works in finitely many steps. The upper bound is proved by constructing a polynomial function in (x, y) of very high degree such that its total derivative with respect to the vector field does not change sign This method is proposed and already developed for general classical Lienard systems by Cherkas( [2]) and by Giacomini-Neukirch ( [8, 9]). This corollary can be useful to construct curves without contact to a piece Γ of solution of (1.1), not closed, which are “parallel” to it and such the flow crosses them either towards Γ or in the opposite direction, as desired
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