Abstract
In Żoła̧dek (Nonlinearity 8:843–860, 1995) the existence of 11 small amplitude limit cycles in a perturbation of some special cubic plane vector field with center was demonstrated. Here we present a new and corrected proof of that result.
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Notes
- 1.
When n > 3 there exist reversible centers but not rationally, only algebraically, i.e., with algebraic map \(\Psi\) (see [3]). Also the class Darboux type integrals should be replaced the with so-called Liouvillian first integrals of the form \(\int M\Omega,\) where M is an integrating multiplier of the Darboux form and \(\Omega\) is a polynomial 1-form associated with the vector field.
- 2.
More precisely, by estimating the cyclicity of the so-called limit sets (like a center or a focus or a polycycle) one can prove the existence of an upper bound for the number of limit cycles for any vector field of degree n). Their details are in [4].
- 3.
This is not the case for the so-called p: −q resonant complex center problem. For p = 1, q = 2 and n = 2 some components of the corresponding center variety are without moduli (see [6]).
- 4.
In [10] the authors refer to the paper [11] as the one where the case CD 4, 5 is studied; they evidently have mixed up two my papers. They make calculations with the help of some computer programs. But their result contradicts analogous computer calculations of the focus quantities in first order with respect to ɛ made (but not published) by C. Christopher; he found ten small amplitude limit cycles.
In [13] I analyzed the Melnikov integrals using only pen and a sheet of paper, here the MAPLE program has turned out useful.
We note also that Christopher in [2] studied Poincaré–Lyapunov quantities in second order with respect to parameters for perturbations of another component CD 4, 6 of the cubic center variety; he has found 11 cycles. In [9] an example of a cubic family with 13 limit cycles is presented.
- 5.
We can describe the formulas ( 23.23)–( 23.24) in terms of some algebraic correspondence. In the space \(\mathbb{C}^{5} = \mathbb{C}^{2} \times \mathbb{C}^{2} \times \mathbb{C},\) with the coordinates \(\left (u,v\right ),\ \left (x,y\right )\) and t, we define a complex algebraic surface \(\mathcal{S}\) by the formulas: \(F(x,y) = t,\ \left (ux\right )^{4} = t,\ \left (vx\right )^{4} = t\left (x^{4} + 4x^{4} + 4y\right ).\) Then the intersections \(\mathcal{S}_{t}\) of \(\mathcal{S}\) with the hyperplanes \(\left \{t =\mathrm{ const}\right \}\) define a family of correspondences between the curves \(\Sigma _{t}\) and \(\Pi _{t}\) via the projections of the curves \(\mathcal{S}_{t}\) to the corresponding two-dimensional spaces \(\mathbb{C}_{u,v}^{2}\) and \(\mathbb{C}_{x,y}^{2}.\) This correspondence is a morphism (in one direction) although the analogous correspondence defined by the same projections of \(\mathcal{S}\) to \(\mathbb{C}_{u,v}^{2}\) and \(\mathbb{C}_{x,y}^{2}\) is not a morphism.
Moreover, change ( 23.23) applies to the general case, i.e., for arbitrary a, as well. Then Eq. ( 23.24) is replaced with 10u 4 − 4v 5 + 5v 4 + 4at −1∕4 u 5 = t.
- 6.
One can notice that, in our analysis, we have not considered one more cycle on the curve \(\Pi _{t}\) (in the x,y coordinates); namely, a cycle vanishing (for t → 1) at the point \(p_{6} = \left (1: 0: 0\right )\) at infinity. Indeed, with t = 1 + 10s from Eq. ( 23.32 ) we find the local equation for \(\Pi _{t}\) of the form 2x 14 y = sx 20 + …, i.e., \(x_{3}^{5}\left (2x_{2}+\ldots \right ) = s\) (where x = x 1 ∕x 3 and y = x 2 ∕x 3 ). The corresponding vanishing cycle is \(\{x_{3} = s^{1/8}e^{\mathrm{i}\theta },\ x_{2} \approx \frac{1} {2}s^{3/8}e^{-5\mathrm{i}\theta }:\ 0 \leq \theta \leq 2\pi \},\) i.e., \(\{x = s^{-1/8}e^{-\mathrm{i}\theta },\ y \approx \frac{1} {2}s^{1/4}e^{-6\mathrm{i}\theta }\}.\) In the u,v coordinates we get \(\left \{u = s^{1/8}e^{\mathrm{i}\theta },\mbox{ }v \approx 1 + s^{1/4}e^{2\mathrm{i}\theta }\right \}.\)
We see that the projection of this cycle onto the v-plane surrounds two times the ramification points v 4,5 ≈ 1 ± s 1∕2 . Therefore, this cycle is a combination of the cycles \(\left (\mathbf{j}^{{\prime}}-\mathbf{j}\right ) \otimes \left (\mathbf{5} -\mathbf{4}\right ).\)
- 7.
This grading reflects the anti-symmetry of the unperturbed form \(\Omega\) with respect to the reflection x ↦ − x.
- 8.
One can check directly that the functions \(I_{\tilde{\xi }_{l,\pm }}^{\tilde{\Gamma }} + \frac{1} {2}(1 \pm \mathrm{ i})I_{\tilde{\xi }_{l,\pm }}^{\Delta }\) are single valued near f = 1 and the functions \(I_{\tilde{\xi }}^{\Delta }\) are holomorphic in \(s^{1/4} = \left (\left (f - 1\right )/10\right )^{1/4}.\)
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Acknowledgements
This work was supported by the Polish OPUS Grant No 2012/05/B/ST1/03195.
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Żoła̧dek, H. (2016). The CD45 Case Revisited. In: Toni, B. (eds) Mathematical Sciences with Multidisciplinary Applications. Springer Proceedings in Mathematics & Statistics, vol 157. Springer, Cham. https://doi.org/10.1007/978-3-319-31323-8_23
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