October  2019, 39(10): 5729-5742. doi: 10.3934/dcds.2019251

On the maximal saddle order of $ p:-q $ resonant saddle

1. 

Department of Mathematics, Jinan University, Guangzhou 510632, China

2. 

School of Mathematics(Zhuhai), Sun Yat-Sen University, Zhuhai 519082, China

3. 

School of Mathematical Sciences, Peking University, Beijing 100871, China

* Corresponding author: donggf@jnu.edu.cn

Received  July 2018 Revised  March 2019 Published  July 2019

In this paper, we obtain some estimations of the saddle order which is the sole topological invariant of the non-integrable resonant saddles of planar polynomial vector fields of arbitrary degree $n$. Firstly, we prove that, for any given resonance $p:-q$, $(p, q)=1$, and sufficiently big integer $n$, the maximal saddle order can grow at least as rapidly as $n^2$. Secondly, we show that there exists an integer $k_0$, which grows at least as rapidly as $3n^2/2$, such that $L_{k_0}$ does not belong to the ideal generated by the first $k_0-1$ saddle values $L_1, L_2, \cdots, L_{k_0-1}$, where $L_{k}$ represents the $k$-th saddle value of the given system. In particular, if $p=1$ (or $q=1$), we obtain a sharper result that $k_0$ can grow at least as rapidly as $2 n^2$. These results are valid for both cases of real and complex resonant saddles.

Citation: Guangfeng Dong, Changjian Liu, Jiazhong Yang. On the maximal saddle order of $ p:-q $ resonant saddle. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5729-5742. doi: 10.3934/dcds.2019251
References:
[1]

J. Bai and Y. Liu, A class of planar degree n (even number) polynomial systems with a fine focus of order $n^2-n$, Chinese Sci. Bull., 12 (1992), 1063–1065 (in Chinese).Google Scholar

[2]

N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Math. USSR Sb., 100 (1954), 397-413. Google Scholar

[3]

Y. N. Bibikov, Local Theory of Nonlinear Analytic Ordinary Differential Equations, Lecture Notes in Mathematics, 702, Springer-Verlag, New York, 1979. Google Scholar

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C. Camacho and P. Sad, Topological classification and bifurcations of holomorphic flows with resonances in $\mathbb{C}^2$, Invent. math., 67 (1982), 447-472. doi: 10.1007/BF01398931. Google Scholar

[5]

X. ChenJ. GinéV. G. Romanovski and D. S. Shafer, The $1: -q$ resonant center problem for certain cubic Lotka-Volterra systems, Appl. Math. Comput., 218 (2012), 11620-11633. doi: 10.1016/j.amc.2012.05.045. Google Scholar

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D. DolićaninJ. GinéR. Oliveira and V. G. Romanovski, The center problem for a 2:-3 resonant cubic Lotka-Volterra system, Appl. Math. Comput., 220 (2013), 12-19. doi: 10.1016/j.amc.2013.06.007. Google Scholar

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G. DongC. Liu and J. Yang, The complexity of generalized center problem, Qual. Theory Dyn. Syst., 14 (2015), 11-23. doi: 10.1007/s12346-015-0131-6. Google Scholar

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G. Dong and J. Yang, On the saddle order of polynomial differential systems at a resonant singular point, J. Math. Anal. Appl., 423 (2015), 1557-1569. doi: 10.1016/j.jmaa.2014.10.085. Google Scholar

[9]

M. DukarićB. Ferčec and J. Giné, The solution of the $1:-3$ resonant center problem in the quadratic case, Appl. Math. Comput., 237 (2014), 501-511. doi: 10.1016/j.amc.2014.03.147. Google Scholar

[10]

B. FerčecX. Chen and V. G. Romanovski, Intergrability conditions for complex systems with homogeneous quintic nonlinearities, J. Appl. Anal. Comput., 1 (2011), 9-20. Google Scholar

[11]

B. FerčecJ. GinéM. Mencinger and R. Oliveira, The center problem for a 1:-4 resonant quadratic system, J. Math. Anal. Appl., 420 (2014), 1568-1591. doi: 10.1016/j.jmaa.2014.06.060. Google Scholar

[12]

A. FronvilleA. Sadovski and H. Zoladek, Solution of the $1:-2$ resonant center problem in the quadratic case, Fund. Math., 157 (1998), 191-207. Google Scholar

[13]

J. Giné, Higher order limit cycle bifurcations from non-degenerate centers, Appl. Math. Comput., 218 (2012), 8853-8860. doi: 10.1016/j.amc.2012.02.044. Google Scholar

[14]

J. Giné and J. Llibre, Integrability conditions of a resonant saddle in generalized Liénard-like complex polynomial differential systems, Chaos Solitons Fractals, 96 (2017), 130-131. doi: 10.1016/j.chaos.2017.01.014. Google Scholar

[15]

J. Giné and C. Valls, Integrability conditions for complex kukles systems, Dyn. Syst., 32 (2017), 211-220. doi: 10.1080/14689367.2016.1181715. Google Scholar

[16]

J. Giné and C. Valls, Integrability conditions of a resonant saddle perturbed with homogeneous quintic nonlinearities, Nonlinear Dynam., 81 (2015), 2021-2030. doi: 10.1007/s11071-015-2122-1. Google Scholar

[17]

Z. HuV. G. Romanovski and D. S. Shafer, $1:-3$ resonant center on $\mathbb{C}^{2}$ with homogeneous cubic nonlinearities, Comput. Math. Appl., 56 (2008), 1927-1940. doi: 10.1016/j.camwa.2008.04.009. Google Scholar

[18]

J. HuangF. WangL. Wang and J. Yang, A quartic system and a quintic system with fine focus of order 18, Bull. Sci. Math., 132 (2008), 205-217. doi: 10.1016/j.bulsci.2006.12.006. Google Scholar

[19]

Y. S. Ilyashenko and A. S. Pyartli, Materialization of resonances and divergence of normalizing series for polynomial differential equations, J. Sov. Math., 32 (1986), 300-313. doi: 10.1007/BF01106073. Google Scholar

[20]

P. Joyal and C. Rousseau, Saddle quantities and applications, J. Differential Equations, 78 (1989), 374-399. doi: 10.1016/0022-0396(89)90069-7. Google Scholar

[21]

H. Liang and J. Torregrosa, Parallelization of the Lyapunov constants and cyclicity for centers of planar polynomial vector fields, J. Differential Equations, 259 (2015), 6494-6509. doi: 10.1016/j.jde.2015.07.027. Google Scholar

[22]

H. Liang and J. Torregrosa, Weak-foci of high order and cyclicity, Qual. Theory Dyn. Syst., 16 (2017), 235-248. doi: 10.1007/s12346-016-0189-9. Google Scholar

[23]

C. Liu and Y. Li, The integrability of a class of cubic Lotka-Volterra systems, onlinear Anal. Real World Appl., 19 (2014), 67-74. doi: 10.1016/j.nonrwa.2014.02.007. Google Scholar

[24]

Y. Liu and J. Li, Theory of values of singular point in complex autonomous differential systems, Sci. China Ser. A, 33 (1990), 10-23. Google Scholar

[25]

Y. Qiu and J. Yang, On the focus order of planar polynomial differential equations, J. Differential Equations, 246 (2009), 3361-3379. doi: 10.1016/j.jde.2009.02.005. Google Scholar

[26]

V. G. Romanovski and D. S. Shafer, On the center problem for $p:-q$ resonant polynomial vector fields, Bull. Belg. Math. Soc. Simon Stevin, 15 (2008), 871-887. Google Scholar

[27]

A. P. Sadovskii, Center and foci for a class of cubic systems, Differ. Equ., 36 (2000), 1812-1818. doi: 10.1023/A:1017552612453. Google Scholar

[28] K. S. Sibirskii, Introduction to the Algebraic Theory of Invariants of Differential Equations, Manchester University Press, Manchester, 1988. Google Scholar
[29]

P. Yu and Y. Tian, Twelve limit cycles around a singular point in a planar cubic-degree polynomial system, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 2690-2705. doi: 10.1016/j.cnsns.2013.12.014. Google Scholar

[30]

H. Zoladek, The problem of center for resonant singular points of polynomials vector fields, J. Differential Equations, 137 (1997), 94-118. doi: 10.1006/jdeq.1997.3260. Google Scholar

show all references

References:
[1]

J. Bai and Y. Liu, A class of planar degree n (even number) polynomial systems with a fine focus of order $n^2-n$, Chinese Sci. Bull., 12 (1992), 1063–1065 (in Chinese).Google Scholar

[2]

N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Math. USSR Sb., 100 (1954), 397-413. Google Scholar

[3]

Y. N. Bibikov, Local Theory of Nonlinear Analytic Ordinary Differential Equations, Lecture Notes in Mathematics, 702, Springer-Verlag, New York, 1979. Google Scholar

[4]

C. Camacho and P. Sad, Topological classification and bifurcations of holomorphic flows with resonances in $\mathbb{C}^2$, Invent. math., 67 (1982), 447-472. doi: 10.1007/BF01398931. Google Scholar

[5]

X. ChenJ. GinéV. G. Romanovski and D. S. Shafer, The $1: -q$ resonant center problem for certain cubic Lotka-Volterra systems, Appl. Math. Comput., 218 (2012), 11620-11633. doi: 10.1016/j.amc.2012.05.045. Google Scholar

[6]

D. DolićaninJ. GinéR. Oliveira and V. G. Romanovski, The center problem for a 2:-3 resonant cubic Lotka-Volterra system, Appl. Math. Comput., 220 (2013), 12-19. doi: 10.1016/j.amc.2013.06.007. Google Scholar

[7]

G. DongC. Liu and J. Yang, The complexity of generalized center problem, Qual. Theory Dyn. Syst., 14 (2015), 11-23. doi: 10.1007/s12346-015-0131-6. Google Scholar

[8]

G. Dong and J. Yang, On the saddle order of polynomial differential systems at a resonant singular point, J. Math. Anal. Appl., 423 (2015), 1557-1569. doi: 10.1016/j.jmaa.2014.10.085. Google Scholar

[9]

M. DukarićB. Ferčec and J. Giné, The solution of the $1:-3$ resonant center problem in the quadratic case, Appl. Math. Comput., 237 (2014), 501-511. doi: 10.1016/j.amc.2014.03.147. Google Scholar

[10]

B. FerčecX. Chen and V. G. Romanovski, Intergrability conditions for complex systems with homogeneous quintic nonlinearities, J. Appl. Anal. Comput., 1 (2011), 9-20. Google Scholar

[11]

B. FerčecJ. GinéM. Mencinger and R. Oliveira, The center problem for a 1:-4 resonant quadratic system, J. Math. Anal. Appl., 420 (2014), 1568-1591. doi: 10.1016/j.jmaa.2014.06.060. Google Scholar

[12]

A. FronvilleA. Sadovski and H. Zoladek, Solution of the $1:-2$ resonant center problem in the quadratic case, Fund. Math., 157 (1998), 191-207. Google Scholar

[13]

J. Giné, Higher order limit cycle bifurcations from non-degenerate centers, Appl. Math. Comput., 218 (2012), 8853-8860. doi: 10.1016/j.amc.2012.02.044. Google Scholar

[14]

J. Giné and J. Llibre, Integrability conditions of a resonant saddle in generalized Liénard-like complex polynomial differential systems, Chaos Solitons Fractals, 96 (2017), 130-131. doi: 10.1016/j.chaos.2017.01.014. Google Scholar

[15]

J. Giné and C. Valls, Integrability conditions for complex kukles systems, Dyn. Syst., 32 (2017), 211-220. doi: 10.1080/14689367.2016.1181715. Google Scholar

[16]

J. Giné and C. Valls, Integrability conditions of a resonant saddle perturbed with homogeneous quintic nonlinearities, Nonlinear Dynam., 81 (2015), 2021-2030. doi: 10.1007/s11071-015-2122-1. Google Scholar

[17]

Z. HuV. G. Romanovski and D. S. Shafer, $1:-3$ resonant center on $\mathbb{C}^{2}$ with homogeneous cubic nonlinearities, Comput. Math. Appl., 56 (2008), 1927-1940. doi: 10.1016/j.camwa.2008.04.009. Google Scholar

[18]

J. HuangF. WangL. Wang and J. Yang, A quartic system and a quintic system with fine focus of order 18, Bull. Sci. Math., 132 (2008), 205-217. doi: 10.1016/j.bulsci.2006.12.006. Google Scholar

[19]

Y. S. Ilyashenko and A. S. Pyartli, Materialization of resonances and divergence of normalizing series for polynomial differential equations, J. Sov. Math., 32 (1986), 300-313. doi: 10.1007/BF01106073. Google Scholar

[20]

P. Joyal and C. Rousseau, Saddle quantities and applications, J. Differential Equations, 78 (1989), 374-399. doi: 10.1016/0022-0396(89)90069-7. Google Scholar

[21]

H. Liang and J. Torregrosa, Parallelization of the Lyapunov constants and cyclicity for centers of planar polynomial vector fields, J. Differential Equations, 259 (2015), 6494-6509. doi: 10.1016/j.jde.2015.07.027. Google Scholar

[22]

H. Liang and J. Torregrosa, Weak-foci of high order and cyclicity, Qual. Theory Dyn. Syst., 16 (2017), 235-248. doi: 10.1007/s12346-016-0189-9. Google Scholar

[23]

C. Liu and Y. Li, The integrability of a class of cubic Lotka-Volterra systems, onlinear Anal. Real World Appl., 19 (2014), 67-74. doi: 10.1016/j.nonrwa.2014.02.007. Google Scholar

[24]

Y. Liu and J. Li, Theory of values of singular point in complex autonomous differential systems, Sci. China Ser. A, 33 (1990), 10-23. Google Scholar

[25]

Y. Qiu and J. Yang, On the focus order of planar polynomial differential equations, J. Differential Equations, 246 (2009), 3361-3379. doi: 10.1016/j.jde.2009.02.005. Google Scholar

[26]

V. G. Romanovski and D. S. Shafer, On the center problem for $p:-q$ resonant polynomial vector fields, Bull. Belg. Math. Soc. Simon Stevin, 15 (2008), 871-887. Google Scholar

[27]

A. P. Sadovskii, Center and foci for a class of cubic systems, Differ. Equ., 36 (2000), 1812-1818. doi: 10.1023/A:1017552612453. Google Scholar

[28] K. S. Sibirskii, Introduction to the Algebraic Theory of Invariants of Differential Equations, Manchester University Press, Manchester, 1988. Google Scholar
[29]

P. Yu and Y. Tian, Twelve limit cycles around a singular point in a planar cubic-degree polynomial system, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 2690-2705. doi: 10.1016/j.cnsns.2013.12.014. Google Scholar

[30]

H. Zoladek, The problem of center for resonant singular points of polynomials vector fields, J. Differential Equations, 137 (1997), 94-118. doi: 10.1006/jdeq.1997.3260. Google Scholar

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