# American Institute of Mathematical Sciences

2018, 38(4): 2029-2046. doi: 10.3934/dcds.2018082

## Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems

 1 School of Mathematical Sciences, Shanghai Jiao Tong University, Dongchuan Road 800, Shanghai 200240, China 2 Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, Maribor, SI-2000 Maribor, Slovenia

Received  March 2017 Revised  August 2017 Published  January 2018

In this paper we research global dynamics and bifurcations of planar piecewise smooth quadratic quasi-homogeneous but non-homogeneous polynomial differential systems. We present sufficient and necessary conditions for the existence of a center in piecewise smooth quadratic quasi-homogeneous systems. Moreover, the center is global and non-isochronous, which cannot appear in smooth quadratic quasi-homogeneous systems. Then the global structures of piecewise smooth quadratic quasi-homogeneous but non-homogeneous systems are obtained. Finally we investigate limit cycle bifurcations of the piecewise quadratic quasi-homogeneous center and give the maximal number of limit cycles bifurcating from periodic orbits of the center by applying the Melnikov method for piecewise smooth near-Hamiltonian systems.

Citation: Yilei Tang. Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2029-2046. doi: 10.3934/dcds.2018082
##### References:
 [1] A. Algaba, N. Fuentes and C. García, Center of quasihomogeneous polynomial planar systems, Nonlinear Anal. Real World Appl., 13 (2012), 419-431. doi: 10.1016/j.nonrwa.2011.07.056. [2] A. Algaba, E. Gamero and C. García, The integrability problem for a class of planar systems, Nonlinearity, 22 (2009), 396-420. doi: 10.1088/0951-7715/22/2/009. [3] A. A. Andronov, E. A. Leontovitch, I. I. Gordon and A. G. Maier, Qualitative Theory of Second-Order Dynamic Systems, Israel Program for Scientific Translations, John Wiley and Sons, New York, 1973. [4] A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966. [5] W. Aziz, J. Llibre and C. Pantazi, Centers of quasi-homogeneous polynomial differential equations of degree three, Adv. Math., 254 (2014), 233-250. doi: 10.1016/j.aim.2013.12.006. [6] I. S. Berezin and N. P. Zhidkov, Computing Methods, Volume Ⅱ, Pergamon Press, Oxford, 1965. [7] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications, Springer-Verlag, London, 2008. [8] M. di Bernardo, C. J. Budd, A. R. Champneys, P. Kowalczyk, A. Nordmark, G. Tost and P. Piiroinen, Bifurcations in nonsmooth dynamical systems, SIAM Review, 50 (2008), 629-701. doi: 10.1137/050625060. [9] C. Buzzi, C. Pessoa and J. Torregrosa, Piecewise linear perturbations of a linear center, Discrete Contin. Dyn. Syst., 33 (2013), 3915-3936. doi: 10.3934/dcds.2013.33.3915. [10] X. Chen, V. 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. [11] F. Dumortier, J. Llibre and J. C. Artés, Qualititive Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006. [12] A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Kluwer Academic, Dordrecht, 1988. [13] E. Freire, E. Ponce and F. Torres, Canonical discontinuous planar piecewise linear system, SIAM J. Appl. Dyn. Syst., 11 (2012), 181-211. doi: 10.1137/11083928X. [14] B. García, J. Llibre and J. S. Pérez del Río, Planar quasihomogeneous polynomial differential systems and their integrability, J. Differential Equations, 255 (2013), 3185-3204. doi: 10.1016/j.jde.2013.07.032. [15] L. Gavrilov, J. Giné and M. Grau, On the cyclicity of weight-homogeneous centers, J. Differential Equations, 246 (2009), 3126-3135. doi: 10.1016/j.jde.2009.02.010. [16] F. Giannakopoulos and K. Pliete, Planar system of piecewise linear differential equations with a line of discontinuity, Nonlinearity, 14 (2001), 1611-1632. doi: 10.1088/0951-7715/14/6/311. [17] J. Giné, M. Grau and J. Llibre, Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 33 (2013), 4531-4547. doi: 10.3934/dcds.2013.33.4531. [18] J. Giné, M. Grau and J. Llibre, Limit cycles bifurcating from planar polynomial quasi-homogeneous centers, J. Differential Equations, 259 (2015), 7135-7160. doi: 10.1016/j.jde.2015.08.014. [19] A. Goriely, Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations, J. Math. Phys., 37 (1996), 1871-1893. doi: 10.1063/1.531484. [20] M. Han and L. Sheng, Bifurcation of limit cycles in piecewise smooth systems via Melnikov function, J. Appl. Anal. Comput., 5 (2015), 809-815. [21] M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differential Equation, 248 (2010), 2399-2416. doi: 10.1016/j.jde.2009.10.002. [22] Y. Hu, On the integrability of quasihomogeneous systems and quasidegenerate infinity systems, Adv. Difference Eqns. , (2007), Art ID 98427, 10 pp. [23] M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin-Heidelberg, 2000. [24] Yu. A. Kuznetsov, S. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Internat. J. Bifur. Chaos, 13 (2003), 2157-2188. doi: 10.1142/S0218127403007874. [25] W. Li, J. Llibre, J. Yang and Z. Zhang, Limit cycles bifurcating from the period annulus of quasi-homegeneous centers, J. Dyn. Diff. Eqns., 21 (2009), 133-152. doi: 10.1007/s10884-008-9126-1. [26] F. Liang, M. Han and V. Romanovski, Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop, Nonlinear Anal., 75 (2012), 4355-4374. doi: 10.1016/j.na.2012.03.022. [27] H. Liang, J. Huang and Y. Zhao, Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems, Nonlinear Dynam., 78 (2014), 1659-1681. doi: 10.1007/s11071-014-1541-8. [28] J. Llibre and E. Ponce, Three nested limit cycles in discontinuous piecewise linear differential systems with two zones, Dynam. Contin. Discrete Impuls. Systems. Ser. B Appl. Algorithms, 19 (2012), 325-335. [29] J. Llibre and X. Zhang, Polynomial first integrals for quasihomogeneous polynomial differential systems, Nonlinearity, 15 (2002), 1269-1280. doi: 10.1088/0951-7715/15/4/313. [30] 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. [31] J. Reyn, Phase Portraits of Planar Quadratic Systems, Mathematics and Its Applications, 583, Springer, New York, 2007. [32] Y. Tang, L. Wang and X. Zhang, Center of planar quintic quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 35 (2015), 2177-2191. [33] L. Wei and X. Zhang, Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems, Discrete Contin. Dyn. Syst., 36 (2016), 2803-2825. [34] Y. Xiong and M. Han, Planar quasi-homogeneous polynomial systems with a given weight degree, Discrete Contin. Dyn. Syst., 36 (2016), 4015-4025. doi: 10.3934/dcds.2016.36.4015. [35] J. Yu and L. Zhang, Center of planar quasi-homogeneous polynomial differential systems, Preprint. [36] Y. Zou, T. Kupper and W. J. Beyn, Generalized Hopf bifurcation for planar Filippov systems continuous at the origin, J. Nonlinear Science, 16 (2006), 159-177. doi: 10.1007/s00332-005-0606-8.

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##### References:
 [1] A. Algaba, N. Fuentes and C. García, Center of quasihomogeneous polynomial planar systems, Nonlinear Anal. Real World Appl., 13 (2012), 419-431. doi: 10.1016/j.nonrwa.2011.07.056. [2] A. Algaba, E. Gamero and C. García, The integrability problem for a class of planar systems, Nonlinearity, 22 (2009), 396-420. doi: 10.1088/0951-7715/22/2/009. [3] A. A. Andronov, E. A. Leontovitch, I. I. Gordon and A. G. Maier, Qualitative Theory of Second-Order Dynamic Systems, Israel Program for Scientific Translations, John Wiley and Sons, New York, 1973. [4] A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966. [5] W. Aziz, J. Llibre and C. Pantazi, Centers of quasi-homogeneous polynomial differential equations of degree three, Adv. Math., 254 (2014), 233-250. doi: 10.1016/j.aim.2013.12.006. [6] I. S. Berezin and N. P. Zhidkov, Computing Methods, Volume Ⅱ, Pergamon Press, Oxford, 1965. [7] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications, Springer-Verlag, London, 2008. [8] M. di Bernardo, C. J. Budd, A. R. Champneys, P. Kowalczyk, A. Nordmark, G. Tost and P. Piiroinen, Bifurcations in nonsmooth dynamical systems, SIAM Review, 50 (2008), 629-701. doi: 10.1137/050625060. [9] C. Buzzi, C. Pessoa and J. Torregrosa, Piecewise linear perturbations of a linear center, Discrete Contin. Dyn. Syst., 33 (2013), 3915-3936. doi: 10.3934/dcds.2013.33.3915. [10] X. Chen, V. 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. [11] F. Dumortier, J. Llibre and J. C. Artés, Qualititive Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006. [12] A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Kluwer Academic, Dordrecht, 1988. [13] E. Freire, E. Ponce and F. Torres, Canonical discontinuous planar piecewise linear system, SIAM J. Appl. Dyn. Syst., 11 (2012), 181-211. doi: 10.1137/11083928X. [14] B. García, J. Llibre and J. S. Pérez del Río, Planar quasihomogeneous polynomial differential systems and their integrability, J. Differential Equations, 255 (2013), 3185-3204. doi: 10.1016/j.jde.2013.07.032. [15] L. Gavrilov, J. Giné and M. Grau, On the cyclicity of weight-homogeneous centers, J. Differential Equations, 246 (2009), 3126-3135. doi: 10.1016/j.jde.2009.02.010. [16] F. Giannakopoulos and K. Pliete, Planar system of piecewise linear differential equations with a line of discontinuity, Nonlinearity, 14 (2001), 1611-1632. doi: 10.1088/0951-7715/14/6/311. [17] J. Giné, M. Grau and J. Llibre, Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 33 (2013), 4531-4547. doi: 10.3934/dcds.2013.33.4531. [18] J. Giné, M. Grau and J. Llibre, Limit cycles bifurcating from planar polynomial quasi-homogeneous centers, J. Differential Equations, 259 (2015), 7135-7160. doi: 10.1016/j.jde.2015.08.014. [19] A. Goriely, Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations, J. Math. Phys., 37 (1996), 1871-1893. doi: 10.1063/1.531484. [20] M. Han and L. Sheng, Bifurcation of limit cycles in piecewise smooth systems via Melnikov function, J. Appl. Anal. Comput., 5 (2015), 809-815. [21] M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differential Equation, 248 (2010), 2399-2416. doi: 10.1016/j.jde.2009.10.002. [22] Y. Hu, On the integrability of quasihomogeneous systems and quasidegenerate infinity systems, Adv. Difference Eqns. , (2007), Art ID 98427, 10 pp. [23] M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin-Heidelberg, 2000. [24] Yu. A. Kuznetsov, S. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Internat. J. Bifur. Chaos, 13 (2003), 2157-2188. doi: 10.1142/S0218127403007874. [25] W. Li, J. Llibre, J. Yang and Z. Zhang, Limit cycles bifurcating from the period annulus of quasi-homegeneous centers, J. Dyn. Diff. Eqns., 21 (2009), 133-152. doi: 10.1007/s10884-008-9126-1. [26] F. Liang, M. Han and V. Romanovski, Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop, Nonlinear Anal., 75 (2012), 4355-4374. doi: 10.1016/j.na.2012.03.022. [27] H. Liang, J. Huang and Y. Zhao, Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems, Nonlinear Dynam., 78 (2014), 1659-1681. doi: 10.1007/s11071-014-1541-8. [28] J. Llibre and E. Ponce, Three nested limit cycles in discontinuous piecewise linear differential systems with two zones, Dynam. Contin. Discrete Impuls. Systems. Ser. B Appl. Algorithms, 19 (2012), 325-335. [29] J. Llibre and X. Zhang, Polynomial first integrals for quasihomogeneous polynomial differential systems, Nonlinearity, 15 (2002), 1269-1280. doi: 10.1088/0951-7715/15/4/313. [30] 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. [31] J. Reyn, Phase Portraits of Planar Quadratic Systems, Mathematics and Its Applications, 583, Springer, New York, 2007. [32] Y. Tang, L. Wang and X. Zhang, Center of planar quintic quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 35 (2015), 2177-2191. [33] L. Wei and X. Zhang, Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems, Discrete Contin. Dyn. Syst., 36 (2016), 2803-2825. [34] Y. Xiong and M. Han, Planar quasi-homogeneous polynomial systems with a given weight degree, Discrete Contin. Dyn. Syst., 36 (2016), 4015-4025. doi: 10.3934/dcds.2016.36.4015. [35] J. Yu and L. Zhang, Center of planar quasi-homogeneous polynomial differential systems, Preprint. [36] Y. Zou, T. Kupper and W. J. Beyn, Generalized Hopf bifurcation for planar Filippov systems continuous at the origin, J. Nonlinear Science, 16 (2006), 159-177. doi: 10.1007/s00332-005-0606-8.
Existence of closed orbits for system $(I)$
The global phase portraits of system $(I)$
The global phase portraits of system $(III)$
The closed orbit of system $(I)$ and its perturbation
Parameter conditions of Figure 2
 Figure 2 Parameter conditions (1) $b_1>0$, $a_1>0$ and $\tilde{a}_1>0$ (2) $b_1>0$, $a_1>0$ and $\tilde{a}_1<0$ (3) $b_1>0$, $a_1<0$ and $\tilde{a}_1<0$ (4) $b_1>0$, $a_1<0$ and $\tilde{a}_1>0$ (5) $b_1<0$, $a_1>0$ and $\tilde{a}_1>0$ (6) $b_1<0$, $a_1>0$ and $\tilde{a}_1<0$ (7) $b_1<0$, $a_1<0$ and $\tilde{a}_1<0$ (8) $b_1<0$, $a_1<0$ and $\tilde{a}_1>0$
 Figure 2 Parameter conditions (1) $b_1>0$, $a_1>0$ and $\tilde{a}_1>0$ (2) $b_1>0$, $a_1>0$ and $\tilde{a}_1<0$ (3) $b_1>0$, $a_1<0$ and $\tilde{a}_1<0$ (4) $b_1>0$, $a_1<0$ and $\tilde{a}_1>0$ (5) $b_1<0$, $a_1>0$ and $\tilde{a}_1>0$ (6) $b_1<0$, $a_1>0$ and $\tilde{a}_1<0$ (7) $b_1<0$, $a_1<0$ and $\tilde{a}_1<0$ (8) $b_1<0$, $a_1<0$ and $\tilde{a}_1>0$
Parameter conditions of Figure 3
 Figure 3 Parameter conditions (1) $a_{31}<0$, $b_3<0$, $a_{31}\ge 2b_3$, $a_{32}<0$ and $\tilde{a}_{31}>2$ (2) $a_{31}<0$, $b_3<0$, $a_{31}\ge 2b_3$, $a_{32}<0$ and $0<\tilde{a}_{31}\le 2$ (3) $a_{31}<0$, $b_3<0$, $a_{31}\ge 2b_3$, $a_{32}<0$ and $\tilde{a}_{31}<0$ (4) $a_{31}<0$, $b_3<0$, $a_{31}\ge 2b_3$, $a_{32}>0$ and $\tilde{a}_{31}>2$ (5) $a_{31}<0$, $b_3<0$, $a_{31}\ge 2b_3$, $a_{32}>0$ and $0<\tilde{a}_{31}\le 2$ (6) $a_{31}<0$, $b_3<0$, $a_{31}\ge 2b_3$, $a_{32}>0$ and $\tilde{a}_{31}<0$ (7) $a_{31}<0$, $b_3<0$, $a_{31}< 2b_3$, $a_{32}>0$ and $\tilde{a}_{31}>2$ (8) $a_{31}<0$, $b_3<0$, $a_{31}< 2b_3$, $a_{32}>0$ and $0<\tilde{a}_{31}\le 2$ (9) $a_{31}<0$, $b_3<0$, $a_{31}< 2b_3$, $a_{32}>0$ and $\tilde{a}_{31}<0$ (10) $a_{31}<0$, $b_3<0$, $a_{31}< 2b_3$, $a_{32}<0$ and $\tilde{a}_{31}>2$ (11) $a_{31}<0$, $b_3<0$, $a_{31}< 2b_3$, $a_{32}<0$ and $0<\tilde{a}_{31}\le 2$ (12) $a_{31}<0$, $b_3<0$, $a_{31}< 2b_3$, $a_{32}<0$ and $\tilde{a}_{31}<0$ (13) $a_{31}>0$, $b_3>0$, $a_{31}\le 2b_3$, $a_{32}<0$ and $\tilde{a}_{31}>2$ (14) $a_{31}>0$, $b_3>0$, $a_{31}\le 2b_3$, $a_{32}<0$ and $0<\tilde{a}_{31}\le 2$ (15) $a_{31}>0$, $b_3>0$, $a_{31}\le 2b_3$, $a_{32}<0$ and $\tilde{a}_{31}<0$ (16) $a_{31}>0$, $b_3>0$, $a_{31}\le 2b_3$, $a_{32}>0$ and $\tilde{a}_{31}>2$ (17) $a_{31}>0$, $b_3>0$, $a_{31}\le 2b_3$, $a_{32}>0$ and $0<\tilde{a}_{31}\le 2$ (18) $a_{31}>0$, $b_3>0$, $a_{31}\le 2b_3$, $a_{32}>0$ and $\tilde{a}_{31}<0$ (19) $a_{31}>0$, $b_3>0$, $a_{31}> 2b_3$, $a_{32}<0$ and $\tilde{a}_{31}>2$ (20) $a_{31}>0$, $b_3>0$, $a_{31}> 2b_3$, $a_{32}<0$ and $0<\tilde{a}_{31}\le 2$ (21) $a_{31}>0$, $b_3>0$, $a_{31}> 2b_3$, $a_{32}<0$ and $\tilde{a}_{31}<0$ (22) $a_{31}>0$, $b_3>0$, $a_{31}> 2b_3$, $a_{32}>0$ and $\tilde{a}_{31}>2$ (23) $a_{31}>0$, $b_3>0$, $a_{31}> 2b_3$, $a_{32}>0$ and $0<\tilde{a}_{31}\le 2$ (24) $a_{31}>0$, $b_3>0$, $a_{31}> 2b_3$, $a_{32}>0$ and $\tilde{a}_{31}<0$ (25) $a_{31}>0$, $b_3<0$, $a_{32}<0$ and $\tilde{a}_{31}>2$ (26) $a_{31}>0$, $b_3<0$, $a_{32}<0$ and $0<\tilde{a}_{31}\le 2$ (27) $a_{31}>0$, $b_3<0$, $a_{32}<0$ and $\tilde{a}_{31}<0$ (28) $a_{31}>0$, $b_3<0$, $a_{32}>0$ and $\tilde{a}_{31}>2$ (29) $a_{31}>0$, $b_3<0$, $a_{32}>0$ and $0<\tilde{a}_{31}\le 2$ (30) $a_{31}>0$, $b_3<0$, $a_{32}>0$ and $\tilde{a}_{31}<0$ (31) $a_{31}<0$, $b_3>0$, $a_{32}<0$ and $\tilde{a}_{31}>2$ (32) $a_{31}<0$, $b_3>0$, $a_{32}<0$ and $0<\tilde{a}_{31}\le 2$ (33) $a_{31}<0$, $b_3>0$, $a_{32}<0$ and $\tilde{a}_{31}<0$ (34) $a_{31}<0$, $b_3>0$, $a_{32}>0$ and $\tilde{a}_{31}>2$ (35) $a_{31}<0$, $b_3>0$, $a_{32}>0$ and $0<\tilde{a}_{31}\le 2$ (36) $a_{31}<0$, $b_3>0$, $a_{32}>0$ and $\tilde{a}_{31}<0$
 Figure 3 Parameter conditions (1) $a_{31}<0$, $b_3<0$, $a_{31}\ge 2b_3$, $a_{32}<0$ and $\tilde{a}_{31}>2$ (2) $a_{31}<0$, $b_3<0$, $a_{31}\ge 2b_3$, $a_{32}<0$ and $0<\tilde{a}_{31}\le 2$ (3) $a_{31}<0$, $b_3<0$, $a_{31}\ge 2b_3$, $a_{32}<0$ and $\tilde{a}_{31}<0$ (4) $a_{31}<0$, $b_3<0$, $a_{31}\ge 2b_3$, $a_{32}>0$ and $\tilde{a}_{31}>2$ (5) $a_{31}<0$, $b_3<0$, $a_{31}\ge 2b_3$, $a_{32}>0$ and $0<\tilde{a}_{31}\le 2$ (6) $a_{31}<0$, $b_3<0$, $a_{31}\ge 2b_3$, $a_{32}>0$ and $\tilde{a}_{31}<0$ (7) $a_{31}<0$, $b_3<0$, $a_{31}< 2b_3$, $a_{32}>0$ and $\tilde{a}_{31}>2$ (8) $a_{31}<0$, $b_3<0$, $a_{31}< 2b_3$, $a_{32}>0$ and $0<\tilde{a}_{31}\le 2$ (9) $a_{31}<0$, $b_3<0$, $a_{31}< 2b_3$, $a_{32}>0$ and $\tilde{a}_{31}<0$ (10) $a_{31}<0$, $b_3<0$, $a_{31}< 2b_3$, $a_{32}<0$ and $\tilde{a}_{31}>2$ (11) $a_{31}<0$, $b_3<0$, $a_{31}< 2b_3$, $a_{32}<0$ and $0<\tilde{a}_{31}\le 2$ (12) $a_{31}<0$, $b_3<0$, $a_{31}< 2b_3$, $a_{32}<0$ and $\tilde{a}_{31}<0$ (13) $a_{31}>0$, $b_3>0$, $a_{31}\le 2b_3$, $a_{32}<0$ and $\tilde{a}_{31}>2$ (14) $a_{31}>0$, $b_3>0$, $a_{31}\le 2b_3$, $a_{32}<0$ and $0<\tilde{a}_{31}\le 2$ (15) $a_{31}>0$, $b_3>0$, $a_{31}\le 2b_3$, $a_{32}<0$ and $\tilde{a}_{31}<0$ (16) $a_{31}>0$, $b_3>0$, $a_{31}\le 2b_3$, $a_{32}>0$ and $\tilde{a}_{31}>2$ (17) $a_{31}>0$, $b_3>0$, $a_{31}\le 2b_3$, $a_{32}>0$ and $0<\tilde{a}_{31}\le 2$ (18) $a_{31}>0$, $b_3>0$, $a_{31}\le 2b_3$, $a_{32}>0$ and $\tilde{a}_{31}<0$ (19) $a_{31}>0$, $b_3>0$, $a_{31}> 2b_3$, $a_{32}<0$ and $\tilde{a}_{31}>2$ (20) $a_{31}>0$, $b_3>0$, $a_{31}> 2b_3$, $a_{32}<0$ and $0<\tilde{a}_{31}\le 2$ (21) $a_{31}>0$, $b_3>0$, $a_{31}> 2b_3$, $a_{32}<0$ and $\tilde{a}_{31}<0$ (22) $a_{31}>0$, $b_3>0$, $a_{31}> 2b_3$, $a_{32}>0$ and $\tilde{a}_{31}>2$ (23) $a_{31}>0$, $b_3>0$, $a_{31}> 2b_3$, $a_{32}>0$ and $0<\tilde{a}_{31}\le 2$ (24) $a_{31}>0$, $b_3>0$, $a_{31}> 2b_3$, $a_{32}>0$ and $\tilde{a}_{31}<0$ (25) $a_{31}>0$, $b_3<0$, $a_{32}<0$ and $\tilde{a}_{31}>2$ (26) $a_{31}>0$, $b_3<0$, $a_{32}<0$ and $0<\tilde{a}_{31}\le 2$ (27) $a_{31}>0$, $b_3<0$, $a_{32}<0$ and $\tilde{a}_{31}<0$ (28) $a_{31}>0$, $b_3<0$, $a_{32}>0$ and $\tilde{a}_{31}>2$ (29) $a_{31}>0$, $b_3<0$, $a_{32}>0$ and $0<\tilde{a}_{31}\le 2$ (30) $a_{31}>0$, $b_3<0$, $a_{32}>0$ and $\tilde{a}_{31}<0$ (31) $a_{31}<0$, $b_3>0$, $a_{32}<0$ and $\tilde{a}_{31}>2$ (32) $a_{31}<0$, $b_3>0$, $a_{32}<0$ and $0<\tilde{a}_{31}\le 2$ (33) $a_{31}<0$, $b_3>0$, $a_{32}<0$ and $\tilde{a}_{31}<0$ (34) $a_{31}<0$, $b_3>0$, $a_{32}>0$ and $\tilde{a}_{31}>2$ (35) $a_{31}<0$, $b_3>0$, $a_{32}>0$ and $0<\tilde{a}_{31}\le 2$ (36) $a_{31}<0$, $b_3>0$, $a_{32}>0$ and $\tilde{a}_{31}<0$
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