Maximum Number of Limit Cycles for Generalized Kukles Polynomial Differential Systems

Abstract

We study the maximum number of limit cycles of the polynomial differential systems of the form

$$\begin{aligned} \dot{x}=-y+l(x), \,\dot{y}=x-f(x)-g(x)y-h(x)y^{2}-d_{0}y^{3}, \end{aligned}$$

where $l(x)=\varepsilon l^{1}(x)+\varepsilon ^{2}l^{2}(x),$$f(x)=\varepsilon f^{1}(x)+\varepsilon ^{2}f^{2}(x),$$g(x)=\varepsilon g^{1}(x)+\varepsilon ^{2}g^{2}(x),$$h(x)=\varepsilon h^{1}(x)+\varepsilon ^{2}h^{2}(x)$ and $d_{0}=\varepsilon d_{0}^{1}+\varepsilon ^{2}d_{0}^{2}$ where $l^{k}(x),$$f^{k}(x),$$g^{k}(x)$ and $h^{k}(x)$ have degree m,  $n_{1},$$n_{2}$ and $n_{3}$ respectively, $d_{0}^{k}\ne 0$ is a real number for each $k=1,2,$ and $\varepsilon$ is a small parameter. We provide an upper bound of the maximum number of limit cycles that the above system can have bifurcating from the periodic orbits of the linear centre $\dot{x}=-y,\, \dot{y}=x$ using the averaging theory of first and second order.

Appendix: Formulas

In this appendix we recall some formulas that will be used during the paper, see for more details [1, 5]. For $i\ge 0$ we have

${\displaystyle \int \limits _{0}^{2\pi }} \cos ^{i}\theta \sin ^{j}\theta d\theta \ne 0,$   if i and j even,

${\displaystyle \int \limits _{0}^{2\pi }} \cos ^{i}\theta \sin ^{j}\theta d\theta =\left\{ \begin{array}{ll} 0,&{}\quad \text {if }\,i ~~\text { or }~~j\, ~\text {odd,}\\ \frac{\pi \alpha _{k}}{2^{k-1}k!},&{}\quad \text {if }\, i=2k \,~~\text {and }~~j=0,\\ \frac{\pi \alpha _{k}}{2^{k}(k+1)!},&{}\quad \text {if } \, i=2k \, ~~\text {and} ~~ j=2,\\ \frac{3\pi \alpha _{k}}{2^{k+1}(k+2)!},&{}\quad \text {if } \,i=2k \, ~~\text { and }~~ j=4,\\ \frac{15\pi \alpha _{k}}{2^{k+2}(k+3)!}, &{}\quad \text {if } \, i=2k \, ~~\text {and}~~ \,j=6, \end{array} \right.$

where $\alpha _{i}=1\cdot 3\cdot 5\cdot \cdot \cdot (2i-1).$

${\displaystyle \int \limits _{0}^{2\pi }} \cos ^{i}\theta \sin ^{j}\theta \sin (2l\theta ) d\theta \ne 0,$    if i and j odd,

${\displaystyle \int \limits _{0}^{2\pi }} \cos ^{i}\theta \sin ^{j}\theta \sin ((2l+1)\theta )d\theta \ne 0,$   if i even and j odd,

${\displaystyle \int \limits _{0}^{2\pi }} \cos ^{i}\theta \sin ^{j}\theta \sin ((2l+1)\theta )d\theta =\left\{ \begin{array}{ll} 0,&{}\quad \text {if } i ~~\text { odd or }~~ j \,\text { even,}\\ \pi C_{i,l},&{} \quad \text {if }\, i~~\text { even~~and }~~ \, j=1,\, l\ge 0, \\ \pi K_{i,l},&{} \quad \text {if }\, i~~\text {even~~and }~~\,j=3,\, l\ge 0, \end{array} \right.$

where $C_{i,l}$ and $K_{i,l}$ are non-zero constants.

$\begin{array}{l} {\displaystyle \int \limits _{0}^{2\pi }} \cos ^{i}\theta \sin ^{j}\theta \sin (2l\theta )d\theta \\ \quad =\left\{ \begin{array}{ll} 0, &{}\quad \text {if } \,i \,~~ \text {or}~~ \, j \text {even,}\\ \frac{\pi (2k+1)\alpha _{k}}{2^{k}(k+2)!},&{}\quad \text {if }\, i=2k+1, \, j=1 \, ~~\text {and }~~\, l=1,\\ \frac{\pi k(2k+1)\alpha _{k}}{2^{k-1}(k+3)!},&{}\quad \text {if }\, i=2k+1, \,j=1\,~~\text { and }~~\,l=2,\\ \frac{3\pi (2k+1)\alpha _{k}}{2^{k+1}(k+3)!},&{}\quad \text {if }\, i=2k+1,\, j=3\, ~~\text { and }~~\, l=1,\\ \frac{3\pi (k-1)(2k+1)\alpha _{k}}{2^{k}(k+4)!},&{}\quad \text {if }\, i=2k+1\, \, j=3\,~~\text { and }~~\, l=2,\\ \pi \overset{\sim }{C}_{i,l},&{}\quad \text {if }\, i\, ~\text { odd~~and }~~\, j=1,\,l\ge 0,\\ \pi \overset{\sim }{K}_{i,l},&{}\quad \text {if }\,i\, ~\text {odd~~ and }~~\,j=3,\,l\ge 0, \end{array} \right. \end{array}$

where $\overset{\sim }{\text { }C}_{i,l}$ and $\overset{\sim }{K}_{i,l}$ are non-zero constants. ${\int _{0}^{\theta }} \cos ^{i}t\sin tdt=\frac{1}{i+1}\left( 1-\cos ^{i+1}\theta \right) ,$

${\displaystyle \int \limits _{0}^{\theta }} \cos ^{2i+1}tdt= {\displaystyle \sum \limits _{l=0}^{i}} \gamma _{i,l}\sin (2l+1)\theta ,$ see [5, p. 153].

${\displaystyle \int \limits _{0}^{\theta }} \cos ^{2i+1}t\sin ^{2}tdt=\frac{-1}{2i+3}\cos ^{2i+2}\theta \sin \theta +\frac{1}{\left( 2i+3\right) } {\displaystyle \sum \limits _{l=0}^{i}} \gamma _{i,l}\sin (2l+1)\theta ,$ see [5, p. 151]

${\displaystyle \int \limits _{0}^{\theta }} \cos ^{2i}tdt=\frac{1}{2^{2i}}\left( \begin{array}{l} 2i\\ i \end{array} \right) \theta + {\displaystyle \sum \limits _{l=1}^{i}} \beta _{i,l}\sin (2l\theta ),$ see [5, p. 153],

where $\gamma _{i,l}=\frac{1}{2^{2i}}\left( \begin{array}{l} 2i+1\\ i-l \end{array} \right) \frac{1}{2l+1},$$\beta _{i,l}=\frac{1}{2^{2i}}\left( \begin{array}{l} 2i\\ i+l \end{array} \right) \frac{1}{l}.$

${\displaystyle \int \limits _{0}^{\theta }} \cos ^{i}t\sin ^{3}tdt=\frac{2}{(i+1)(i+3)}-\frac{1}{i+1}\cos ^{i+1} \theta +\frac{1}{i+3}\cos ^{i+3}\theta ,$

${\displaystyle \int \limits _{0}^{\theta }} \cos ^{2}t\sin ^{2}tdt=\frac{1}{32}\left( 4\theta -\sin 4\theta \right) ,$

${\displaystyle \int \limits _{0}^{\theta }} \sin ^{4}tdt=\frac{1}{32}\left( 12\theta -8\sin 2\theta +\sin 4\theta \right) ,$

${\displaystyle \int \limits _{0}^{\theta }} \cos ^{4}tdt=\frac{1}{32}\left( 12\theta +8\sin 2\theta +\sin 4\theta \right) .$