Integrability and limit cycles in cubic Kukles systems with a nilpotent singular point

Abstract

In this paper, integrability problem and bifurcation of limit cycles for cubic Kukles systems which are assumed to have a nilpotent origin are investigated. A complete classification is given on the integrability conditions and proven to have a total of 7 cases. Bifurcation of limit cycles is also discussed; six or seven limit cycles can be obtained by two different perturbation methods. Integrability problem and bifurcation of limit cycles for the cubic Kukles systems with a nilpotent origin have been completely solved.

Appendix

$$\begin{aligned} f_1&=4464 a_{11}^5 - 47460 a_{11}^4 b_{02} + 91260 a_{11}^3 b_{02}^2 \\&\quad -37155 a_{11}^2 b_{02}^3 \\&\quad - 3330 a_{11} b_{02}^4+ 2373 b_{02}^5+17520 a_{11}^5 \mu ^2\\&\quad - 59300 a_{11}^4 b_{02} \mu ^2 \\&\quad +42300 a_{11}^3 b_{02}^2 \mu ^2+ 11725 a_{11}^2 b_{02}^3 \mu ^2\\&\quad -13650 a_{11} b_{02}^4 \mu ^2 \\&\quad + 1765 b_{02}^5 \mu ^2 + 52500 a_{11}^3 b_{12}-195000 a_{11}^2 b_{02} b_{12} \\&\quad + 144375 a_{11} b_{02}^2 b_{12} - 1875 b_{02}^3 b_{12} +60500 a_{11}^3 \mu ^2 b_{12} \\&\quad - 187000 a_{11}^2 b_{02} \mu ^2 b_{12} +70375 a_{11} b_{02}^2 \mu ^2 b_{12} \\&\quad + 32125 b_{02}^3 \mu ^2 b_{12}+71250 a_{11} b_{12}^2 - 213750 b_{02} b_{12}^2 \\&\quad + 51250 a_{11} \mu ^2 b_{12}^2 -153750 b_{02} \mu ^2 b_{12}^2;\\ f_2&=456 a_{11}^3 - 1154 a_{11}^2 b_{02} + 87 a_{11} b_{02}^2 + 563 b_{02}^3 \\&\quad + 650 a_{11} b_{12} -1950 b_{02} b_{12};\\ f_3&=8 a_{11}^3 - 22 a_{11}^2 b_{02} + 11 a_{11} b_{02}^2 - b_{02}^3+ 10 a_{11} b_{12} \\&\quad -30 b_{02} b_{12};\\ f_4&=6867072 a_{11}^9 - 34328544 a_{11}^8 b_{02} + 60640928 a_{11}^7 b_{02}^2 \\&\quad -107964696 a_{11}^6 b_{02}^3+ 218203332 a_{11}^5 b_{02}^4 \\&\quad -200771496 a_{11}^4 b_{02}^5+ 40153167 a_{11}^3 b_{02}^6\\&\quad + 6353871 a_{11}^2 b_{02}^7 \\&\quad +11243353 a_{11} b_{02}^8 {-} 5166051 b_{02}^9 {+} 35008800 a_{11}^7 b_{12} \\&\quad -148154800 a_{11}^6 b_{02} b_{12} + 229543200 a_{11}^5 b_{02}^2 b_{12}\\&\quad -402053500 a_{11}^4 b_{02}^3 b_{12} + 643866750 a_{11}^3 b_{02}^4 b_{12} \\&\quad -401462650 a_{11}^2 b_{02}^5 b_{12} + 54228900 a_{11} b_{02}^6 b_{12} \\&\quad +12764400 b_{02}^7 b_{12} + 52100000 a_{11}^5 b_{12}^2 \\&\quad -192250000 a_{11}^4 b_{02} b_{12}^2 + 362750000 a_{11}^3 b_{02}^2 b_{12}^2\\&\quad -550250000 a_{11}^2 b_{02}^3 b_{12}^2 + 515843750 a_{11} b_{02}^4 b_{12}^2\\&\quad -126018750 b_{02}^5 b_{12}^2 + 11500000 a_{11}^3 b_{12}^3 \\&\quad -25375000 a_{11}^2 b_{02} b_{12}^3+ 230812500 a_{11} b_{02}^2 b_{12}^3 \\&\quad -305812500 b_{02}^3 b_{12}^3 - 18750000 a_{11} b_{12}^4\\&\quad + 56250000 b_{02} b_{12}^4;\\ f_5&=65616 a_{11}^5 - 189740 a_{11}^4 b_{02} + 77940 a_{11}^3 b_{02}^2 \\&\quad +84055 a_{11}^2 b_{02}^3- 51270 a_{11} b_{02}^4+ 4687 b_{02}^5 \\&\quad +189500 a_{11}^3 b_{12} - 553000 a_{11}^2 b_{02} b_{12} \\&\quad + 137125 a_{11} b_{02}^2 b_{12} +130375 b_{02}^3 b_{12} \\&\quad + 133750 a_{11} b_{12}^2 - 401250 b_{02} b_{12};\\ f_7&=2832 a_{11}^5 - 45980 a_{11}^4 b_{02} \\&\quad + 97380 a_{11}^3 b_{02}^2-43265 a_{11}^2 b_{02}^3 \\&\quad - 2040 a_{11} b_{02}^4+ 2449 b_{02}^5+ 51500 a_{11}^3 b_{12} \\&\quad -196000 a_{11}^2 b_{02} b_{12} \\&\quad + 153625 a_{11} b_{02}^2 b_{12}- 6125 b_{02}^3 b_{12}+73750 a_{11} b_{12}^2\\&\quad - 221250 b_{02} b_{12}^2;\\ f_8&=24864 a_{11}^5 - 65960 a_{11}^4 b_{02} + 14760 a_{11}^3 b_{02}^2 \\&\quad +39220 a_{11}^2 b_{02}^3 - 19455 a_{11} b_{02}^4 + 1423 b_{02}^5 \\&\quad + 65000 a_{11}^3 b_{12} -182500 a_{11}^2 b_{02} b_{12} \\&\quad + 28750 a_{11} b_{02}^2 b_{12}\\&\quad + 51250 b_{02}^3 b_{12}+40000 a_{11} b_{12}^2 - 120000 b_{02} b_{12}^2.\\ \end{aligned}$$

The expressions of $f_6$ and $f_9$ are too long to be given here.