Highest weak focus order for trigonometric Liénard equations
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Given a planar analytic differential equation with a critical point which is a weak focus of order k, it is well known that at most k limit cycles can bifurcate from it. Moreover, in case of analytic Liénard differential equations this order can be computed as one half of the multiplicity of an associated planar analytic map. By using this approach, we can give an upper bound of the maximum order of the weak focus of pure trigonometric Liénard equations only in terms of the degrees of the involved trigonometric polynomials. Our result extends to this trigonometric Liénard case a similar result known for polynomial Liénard equations.
Is part ofAnnali di Matematica Pura ed Applicata, 2020, vol.199, p. 1673-1684
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Gasull, Armengol; Giné, Jaume; Valls, Claudia (Elsevier, 2017)We give a complete algebraic characterization of the non-degenerated centers for planar trigonometric Liénard systems. The main tools used in our proof are the classical results of Cherkas on planar analytic Liénard systems ...
Gasull i Embid, Armengol; Giné, Jaume (Springer International Publishing, 2017)We characterize the local analytic integrability of weak saddles for complex Lienard systems, x˙ = y−F(x), y˙ = ax, 0 = a ∈ C, with F analytic at 0 and F(0) = F (0) = 0. We prove that they are locally integrable at the ...
Gasull i Embid, Armengol; Giné, Jaume (Springer, 2010)We prove that there are one-parameter families of planar differential equations for which the center problem has a trivial solution and on the other hand the cyclicity of the weak focus is arbitrarily high. We illustrate ...