American Institute of Mathematical Sciences

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

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

Yilei Tang 1,2,

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.

Keywords: Quasi-homogeneous polynomial systems, global phase portrait, bifurcation of piecewise system, Melnikov function.
Mathematics Subject Classification: Primary: 37G05; Secondary: 37G10, 34C23, 34C20.
Citation: Yilei Tang. Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems. Discrete and Continuous Dynamical Systems, 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.

show all references

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.

Figure 1.  Existence of closed orbits for system $(I)$
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Figure 2.  The global phase portraits of system $(I)$
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Figure 3.  The global phase portraits of system $(III)$
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Figure 4.  The closed orbit of system $(I)$ and its perturbation
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Table 1.  Parameter conditions of Figure 2
Figure 2Parameter 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 $\tilde{a}_1>0$
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Figure 2Parameter 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 $\tilde{a}_1>0$
Table 2.  Parameter conditions of Figure 3
Figure 3Parameter 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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Figure 3Parameter 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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