# American Institute of Mathematical Sciences

December  2019, 24(12): 6541-6552. doi: 10.3934/dcdsb.2019153

## Cyclicity of $(1,3)$-switching FF type equilibria

 1 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China 2 Department de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

* Corresponding author: Weinian Zhang

Received  January 2019 Published  July 2019

Fund Project: The first author is supported by NSFC #11871355. The second author has been partially supported by the Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and MDM-2014-0445, the Agència de Gestió d'Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The third author is supported by NSFC #11726623 and #11771307

Hilbert's 16th Problem suggests a concern to the cyclicity of planar polynomial differential systems, but it is known that a key step to the answer is finding the cyclicity of center-focus equilibria of polynomial differential systems (even of order 2 or 3). Correspondingly, the same question for polynomial discontinuous differential systems is also interesting. Recently, it was proved that the cyclicity of $(1, 2)$-switching FF type equilibria is at least 5. In this paper we prove that the cyclicity of $(1, 3)$-switching FF type equilibria with homogeneous cubic nonlinearities is at least 3.

Citation: Xingwu Chen, Jaume Llibre, Weinian Zhang. Cyclicity of $(1,3)$-switching FF type equilibria. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6541-6552. doi: 10.3934/dcdsb.2019153
##### References:
 [1] 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. Sbornik, 30 (1952), 181–196 (in Russian); Transl. Amer. Math. Soc., 100 (1954), 19pp. Google Scholar [2] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008. Google Scholar [3] N. N. Bogoliubov, On Some Statistical Methods in Mathematical Physics, Akademiya Nauk Ukrainsko$\mathop {\text{i}}\limits^ \vee$, 1945. Google Scholar [4] N. N. Bogoliubov and N. Krylov, The Application of Methods of Nonlinear Mechanics in the Theory of Stationary Oscillations, Ukrainian Academy of Science, 1934.Google Scholar [5] X. Cen, J. Llibre and M. Zhang, Periodic solutions and their stability of some higher-order positively homogenous differential equations, Chaos, Solitons & Fractals, 106 (2018), 285-288. doi: 10.1016/j.chaos.2017.11.032. Google Scholar [6] X. Chen and Z. Du, Limit cycles bifurcate from centers of discontinuous quadratic systems, Comput. Math. Appl., 59 (2010), 3836-3848. doi: 10.1016/j.camwa.2010.04.019. Google Scholar [7] X. Chen, J. Llibre and W. Zhang, Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems, Discrete and Continuous Dynamical Systems-Series B, 22 (2017), 3953-3965. doi: 10.3934/dcdsb.2017203. Google Scholar [8] X. Chen, V. G. Romanovski and W. Zhang, Degenerate Hopf bifurcations in a family of FF-type switching systems, J. Math. Anal. Appl., 432 (2015), 1058-1076. doi: 10.1016/j.jmaa.2015.07.036. Google Scholar [9] X. Chen and W. Zhang, Normal forms of planar switching systems, Disc. Cont. Dyn. Syst., 36 (2016), 6715-6736. doi: 10.3934/dcds.2016092. Google Scholar [10] C. Christopher and C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2007. Google Scholar [11] B. Coll, A. Gasull and R. Prohens, Degenerate Hopf bifurcations in discontinuous planar systems, J. Math. Anal. Appl., 253 (2001), 671-690. doi: 10.1006/jmaa.2000.7188. Google Scholar [12] A. F. Filippov, Differential Equation with Discontinuous Right-Hand Sides, Kluwer Academic, Amsterdam, 1988. doi: 10.1007/978-94-015-7793-9. Google Scholar [13] A. Gasull and J. Torregrosa, Center-focus problem for discontinuous planar differential equations, Int. J. Bifurc. Chaos, 13 (2003), 1755-1765. doi: 10.1142/S0218127403007618. Google Scholar [14] J. Giné, M. Grau and J. Llibre, Averaging theory at any order for computing periodic orbits, Physica D, 250 (2013), 58-65. doi: 10.1016/j.physd.2013.01.015. Google Scholar [15] J. K. Hale and H. Hoçak, Dynamics and Bifurcations, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-1-4612-4426-4. Google Scholar [16] M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Diff. Equa., 248 (2010), 2399-2416. doi: 10.1016/j.jde.2009.10.002. Google Scholar [17] Yu. Ilyashenko, Centennial history of Hilbert's $16$th problem, Bull. (New Series) Amer. Math. Soc., 39 (2002), 301-354. doi: 10.1090/S0273-0979-02-00946-1. Google Scholar [18] M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103843. Google Scholar [19] J. Li, Hilbert's $16$th problem and bifurcations of planar polynomial vector fields, Int. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 47-106. doi: 10.1142/S0218127403006352. Google Scholar [20] J. Llibre, D. D. Novaes and M. A. Teixeira, Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearity, 27 (2014), 563-583. doi: 10.1088/0951-7715/27/3/563. Google Scholar [21] J. Llibre, D. D. Novaes and M. A. Teixeira, On the birth of limit cycles for non–smooth dynamical systems, Bull. Sci. math., 139 (2015), 229-244. doi: 10.1016/j.bulsci.2014.08.011. Google Scholar [22] J. Llibre, D. D. Novaes and M. A. Teixeira, Maximum number of limit cycles for certain piecewise linear dynamical systems, Nonlinear Dynamics, 82 (2015), 1159-1175. doi: 10.1007/s11071-015-2223-x. Google Scholar [23] O. Makarenkov and J. S. W. Lamb, Dynamics and bifurcations of nonsmooth systems: A survey, Physica D, 241 (2012), 1826-1844. doi: 10.1016/j.physd.2012.08.002. Google Scholar [24] P. Patou, Sur le mouvement d'un système soumis à des forces à courte période, Bull. Soc. Math. France, 56 (1928), 98-139. Google Scholar [25] D. J. W. Simpson, Bifurcations in Piecewise–Smooth Continuous Systems, World Scientific Series on Nonlinear Science A, vol 69, World Scientific, Singapore, 2010. doi: 10.1142/7612. Google Scholar [26] H. Żoldek, Eleven small limit cycles in a cubic vector field, Nonlinearity, 8 (1995), 843-860. doi: 10.1088/0951-7715/8/5/011. Google Scholar

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##### References:
 [1] 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. Sbornik, 30 (1952), 181–196 (in Russian); Transl. Amer. Math. Soc., 100 (1954), 19pp. Google Scholar [2] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008. Google Scholar [3] N. N. Bogoliubov, On Some Statistical Methods in Mathematical Physics, Akademiya Nauk Ukrainsko$\mathop {\text{i}}\limits^ \vee$, 1945. Google Scholar [4] N. N. Bogoliubov and N. Krylov, The Application of Methods of Nonlinear Mechanics in the Theory of Stationary Oscillations, Ukrainian Academy of Science, 1934.Google Scholar [5] X. Cen, J. Llibre and M. Zhang, Periodic solutions and their stability of some higher-order positively homogenous differential equations, Chaos, Solitons & Fractals, 106 (2018), 285-288. doi: 10.1016/j.chaos.2017.11.032. Google Scholar [6] X. Chen and Z. Du, Limit cycles bifurcate from centers of discontinuous quadratic systems, Comput. Math. Appl., 59 (2010), 3836-3848. doi: 10.1016/j.camwa.2010.04.019. Google Scholar [7] X. Chen, J. Llibre and W. Zhang, Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems, Discrete and Continuous Dynamical Systems-Series B, 22 (2017), 3953-3965. doi: 10.3934/dcdsb.2017203. Google Scholar [8] X. Chen, V. G. Romanovski and W. Zhang, Degenerate Hopf bifurcations in a family of FF-type switching systems, J. Math. Anal. Appl., 432 (2015), 1058-1076. doi: 10.1016/j.jmaa.2015.07.036. Google Scholar [9] X. Chen and W. Zhang, Normal forms of planar switching systems, Disc. Cont. Dyn. Syst., 36 (2016), 6715-6736. doi: 10.3934/dcds.2016092. Google Scholar [10] C. Christopher and C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2007. Google Scholar [11] B. Coll, A. Gasull and R. Prohens, Degenerate Hopf bifurcations in discontinuous planar systems, J. Math. Anal. Appl., 253 (2001), 671-690. doi: 10.1006/jmaa.2000.7188. Google Scholar [12] A. F. Filippov, Differential Equation with Discontinuous Right-Hand Sides, Kluwer Academic, Amsterdam, 1988. doi: 10.1007/978-94-015-7793-9. Google Scholar [13] A. Gasull and J. Torregrosa, Center-focus problem for discontinuous planar differential equations, Int. J. Bifurc. Chaos, 13 (2003), 1755-1765. doi: 10.1142/S0218127403007618. Google Scholar [14] J. Giné, M. Grau and J. Llibre, Averaging theory at any order for computing periodic orbits, Physica D, 250 (2013), 58-65. doi: 10.1016/j.physd.2013.01.015. Google Scholar [15] J. K. Hale and H. Hoçak, Dynamics and Bifurcations, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-1-4612-4426-4. Google Scholar [16] M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Diff. Equa., 248 (2010), 2399-2416. doi: 10.1016/j.jde.2009.10.002. Google Scholar [17] Yu. Ilyashenko, Centennial history of Hilbert's $16$th problem, Bull. (New Series) Amer. Math. Soc., 39 (2002), 301-354. doi: 10.1090/S0273-0979-02-00946-1. Google Scholar [18] M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103843. Google Scholar [19] J. Li, Hilbert's $16$th problem and bifurcations of planar polynomial vector fields, Int. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 47-106. doi: 10.1142/S0218127403006352. Google Scholar [20] J. Llibre, D. D. Novaes and M. A. Teixeira, Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearity, 27 (2014), 563-583. doi: 10.1088/0951-7715/27/3/563. Google Scholar [21] J. Llibre, D. D. Novaes and M. A. Teixeira, On the birth of limit cycles for non–smooth dynamical systems, Bull. Sci. math., 139 (2015), 229-244. doi: 10.1016/j.bulsci.2014.08.011. Google Scholar [22] J. Llibre, D. D. Novaes and M. A. Teixeira, Maximum number of limit cycles for certain piecewise linear dynamical systems, Nonlinear Dynamics, 82 (2015), 1159-1175. doi: 10.1007/s11071-015-2223-x. Google Scholar [23] O. Makarenkov and J. S. W. Lamb, Dynamics and bifurcations of nonsmooth systems: A survey, Physica D, 241 (2012), 1826-1844. doi: 10.1016/j.physd.2012.08.002. Google Scholar [24] P. Patou, Sur le mouvement d'un système soumis à des forces à courte période, Bull. Soc. Math. France, 56 (1928), 98-139. Google Scholar [25] D. J. W. Simpson, Bifurcations in Piecewise–Smooth Continuous Systems, World Scientific Series on Nonlinear Science A, vol 69, World Scientific, Singapore, 2010. doi: 10.1142/7612. Google Scholar [26] H. Żoldek, Eleven small limit cycles in a cubic vector field, Nonlinearity, 8 (1995), 843-860. doi: 10.1088/0951-7715/8/5/011. Google Scholar
Numbers of positive simple zeros of $f_1, ..., f_5$
 $\#Z_+(f_1)$ $f_1\!\!\equiv \!0$ $\#Z_+(f_2)$ $f_2\!\!\equiv\! 0$ $\#Z_+(f_3)$ $f_3\!\!\equiv\! 0$ $\#Z_+(f_4)$ $f_4\equiv 0$ $\#Z_+(f_5)$ $0$ $\lambda_{11}\!\!=\!0$ 1 $C_2$ 1 $C_3$ $2$ $C_{41}$ $2$ $C_{42}$ $2$ $C_{43}$ $2$
 $\#Z_+(f_1)$ $f_1\!\!\equiv \!0$ $\#Z_+(f_2)$ $f_2\!\!\equiv\! 0$ $\#Z_+(f_3)$ $f_3\!\!\equiv\! 0$ $\#Z_+(f_4)$ $f_4\equiv 0$ $\#Z_+(f_5)$ $0$ $\lambda_{11}\!\!=\!0$ 1 $C_2$ 1 $C_3$ $2$ $C_{41}$ $2$ $C_{42}$ $2$ $C_{43}$ $2$
Number of positive simple zeros of $f_6$
 condition for $f_4\equiv 0$ condition for $f_5\equiv 0$ $\#Z_+(f_6)$ $C_{41}$ $C_{411}$ $3$ $C_{42}$ $C_{421}$ $2$ $C_{422}$ $2$ $C_{423}$ $2$ $C_{43}$ $C_{431}$ $3$ $C_{432}$ $2$ $C_{433}$ $3$
 condition for $f_4\equiv 0$ condition for $f_5\equiv 0$ $\#Z_+(f_6)$ $C_{41}$ $C_{411}$ $3$ $C_{42}$ $C_{421}$ $2$ $C_{422}$ $2$ $C_{423}$ $2$ $C_{43}$ $C_{431}$ $3$ $C_{432}$ $2$ $C_{433}$ $3$
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