doi: 10.3934/dcdsb.2021299
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

The cyclicity of a class of quadratic reversible centers defining elliptic curves

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

*Corresponding author: Changjian Liu

Received  August 2021 Revised  November 2021 Early access December 2021

Fund Project: The work was supported by the NSF of China (No.11771315)

In this paper, the cyclicity of period annulus of an one-parameter family quadratic reversible system under quadratic perturbations is studied which is equivalent to the number of zeros of any nontrivial linear combination of three Abelian integrals. By the criteria established in [28] and the asymptotic expansions of Abelian integrals, we obtain that the cyclicity is two when the parameter in $ (-\infty,-2)\cup[-\frac{8}{5},+\infty) $. Moreover, we develop new criteria which combined with the asymptotic expansions of Abelian integrals show that the cyclicity is three when the parameter belongs to $ (-2,-\frac{8}{5}) $.

Citation: Guilin Ji, Changjian Liu. The cyclicity of a class of quadratic reversible centers defining elliptic curves. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021299
References:
[1] A.A. AndronovA.A. Vitt and S.E. Khaikin, Theory of Oscillators, Pergamon Press, Oxford, New York, Toronto, 1966.   Google Scholar
[2]

G. ChenC. LiC. Liu and J. Llibre, The cyclicity of period annuli of some classes of reversible quadratic systems, Discrete Contin. Dyn. Syst., 16 (2006), 157-177.  doi: 10.3934/dcds.2006.16.157.  Google Scholar

[3]

L. ChenX. MaG. Zhang and C. Li, Cyclicity of several quadratic reversible systems with center of genus one, J. Appl. Anal. Comput., 1 (2011), 439-447.  doi: 10.11948/2011030.  Google Scholar

[4]

S. N. ChowC. Li and Y. Yi, The cyclicity of period annulus of degenerate quadratic Hamiltonian system with elliptic segment loop, Ergodic Theory Dynam. Systems, 22 (2002), 349-374.  doi: 10.1017/S0143385702000184.  Google Scholar

[5]

B. CollC. Li and R. Prohens, Quadratic perturbations of a class quadratic reversible systems with two centers, Discrete Contin. Dyn. Syst., 24 (2009), 699-729.  doi: 10.3934/dcds.2009.24.699.  Google Scholar

[6]

F. DumortierC. Li and Z. Zhang, Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops, J. Differ. Equations, 139 (1997), 146-193.  doi: 10.1006/jdeq.1997.3285.  Google Scholar

[7]

J.-P. FrançoiseL. Gavrilov and D. Xiao, Hilbert's 16th problem on a period annulus and Nash space of arcs, Math. Proc. Camb. Philos. Soc., 169 (2020), 377-409.  doi: 10.1017/S0305004119000239.  Google Scholar

[8]

S. GautierL. Gavrilov and I. D. Iliev, Perturbation of quadratic centers of genus one, Discrete Contin. Dyn. Syst., 25 (2009), 511-535.  doi: 10.3934/dcds.2009.25.511.  Google Scholar

[9]

M. GrauF. Mañosas and J. Villadelprat, A Chebyshev criterion for abelian integrals, Trans. Am. Math. Soc., 363 (2011), 109-129.  doi: 10.1090/S0002-9947-2010-05007-X.  Google Scholar

[10]

M. Han, Asymptotic expansions of Melnikov functions and limit cycle bifurcations, Int. J. Bifur. Chaos, 22 (2012), 1250296.  doi: 10.1142/S0218127412502963.  Google Scholar

[11]

M. HanJ. YangA. Tarta and Y. Gao, Limit cycles near homoclinic and heteroclinic loops, J. Dynam. Differ. Equ., 20 (2008), 923-944.  doi: 10.1007/s10884-008-9108-3.  Google Scholar

[12]

I. D. Iliev, The cyclicity of the period annulus of the quadratic Hamiltonian triangle, J. Differ. Equations, 128 (1996), 309-326.  doi: 10.1006/jdeq.1996.0097.  Google Scholar

[13]

I. D. Iliev, High-order Melnikov functions for degenerate cubic Hamiltonians, Adv. Differential Equations, 1 (1996), 689-708.   Google Scholar

[14]

I. D. Iliev, Perturbations of quadratic centers, Bull. Sci. Math., 122 (1998), 107-161.  doi: 10.1016/S0007-4497(98)80080-8.  Google Scholar

[15]

I. D. IlievC. Li and J. Yu, Bifurcation of limit cycles from quadratic non-Hamiltonian systems with two centers and two unbounded heteroclinic loops, Nonlinearity, 18 (2005), 305-330.  doi: 10.1088/0951-7715/18/1/016.  Google Scholar

[16]

I. D. IlievC. Li and J. Yu, Bifurcation of limit cycles in a reversible quadratic system with a center, a saddle and two nodes, Commun. Pure Appl. Anal., 9 (2010), 583-610.  doi: 10.3934/cpaa.2010.9.583.  Google Scholar

[17]

C. Li, Abelian integrals and limit cycles, Qual. Theory Dyn. Syst., 11 (2012), 111-128.  doi: 10.1007/s12346-011-0051-z.  Google Scholar

[18]

C. Li and J. Llibre, A unified study on the cyclicity of period annulus of the reversible quadratic Hamiltonian systems, J. Dynam. Differ. Equations, 16 (2004), 271-295.  doi: 10.1007/s10884-004-2781-y.  Google Scholar

[19]

C. Li and J. Llibre, The cyclicity of period annulus of a quadratic reversible Lotka-Volterra system, Nonlinearity, 22 (2009), 2971-2979.  doi: 10.1088/0951-7715/22/12/009.  Google Scholar

[20]

C. Li and Z. Zhang, A criterion for determining the monotonocity of the ratio of two Abelian integrals, J. Differ. Equations, 124 (1996), 407-424.  doi: 10.1006/jdeq.1996.0017.  Google Scholar

[21]

J. Li, Limit cycles bifurcated from a reversible quadratic center, Qual. Theory Dyn. Syst., 6 (2005), 205-215.  doi: 10.1007/BF02972673.  Google Scholar

[22]

H. LiangK. Wu and Y. Zhao, Quadratic perturbations of a class of quadratic reversible center of genus one, Sci. China Math., 56 (2013), 577-596.  doi: 10.1007/s11425-012-4488-6.  Google Scholar

[23]

H. Liang and Y. Zhao, Quadratic perturbations of a class of quadratic reversible systems with one center, Discrete Contin. Dyn. Syst., 27 (2010), 325-335.  doi: 10.3934/dcds.2010.27.325.  Google Scholar

[24]

H. Liang and Y. Zhao, Limit cycles bifurcated from a class of quadratic reversible center of genus one, J. Math. Anal. Appl., 391 (2012), 240-254.  doi: 10.1016/j.jmaa.2012.02.014.  Google Scholar

[25]

C. Liu, The cyclicity of period annuli of a class of quadratic reversible systems with two centers, J. Differ. Equations, 252 (2012), 5260-5273.  doi: 10.1016/j.jde.2012.02.005.  Google Scholar

[26]

C. Liu, Limit cycles bifurcated from some reversible quadratic centres with a non-algebraic first integral, Nonlinearity, 25 (2012), 1653-1660.  doi: 10.1088/0951-7715/25/6/1653.  Google Scholar

[27]

C. Liu and D. Xiao, The monotonicity of the ratio of two Abelian integrals, Trans. Amer. Math. Soc., 365 (2013), 5525-5544.  doi: 10.1090/S0002-9947-2013-05934-X.  Google Scholar

[28]

C. Liu and D. Xiao, The smallest upper bound on the number of zeros of Abelian integrals, J. Differ. Equations, 269 (2020), 3816-3852.  doi: 10.1016/j.jde.2020.03.016.  Google Scholar

[29]

F. Mañosas and J. Villadelprat, Bounding the number of zeros of certain Abelian integrals, J. Differ. Equations, 251 (2011), 1656-1669.  doi: 10.1016/j.jde.2011.05.026.  Google Scholar

[30]

L. Peng, Unfolding of a quadratic integrable system with a homoclinic loop, Acta Math. Sin. (Engl. Ser.), 18 (2002), 737-754.  doi: 10.1007/s10114-002-0196-4.  Google Scholar

[31]

Y. Shao and Y. Zhao, The cyclicity of a class of quadratic reversible system of genus one, Chaos Solitons Fractals, 44 (2011), 827-835.  doi: 10.1016/j.chaos.2011.06.015.  Google Scholar

[32]

Y. Shao and Y. Zhao, The cyclicity of the period annulus of a class of quadratic reverdible system, Commun. Pure Appl. Anal., 11 (2012), 1269-1283.  doi: 10.3934/cpaa.2012.11.1269.  Google Scholar

[33]

Y. Sun and C. Liu, The poincaré bifurcation of a SD oscillator, Discrete Contin. Dyn. Syst. B, 26 (2021), 1565-1577.  doi: 10.3934/dcdsb.2020173.  Google Scholar

[34]

J. Yu and C. Li, Bifurcation of a class of planar non-Hamiltonian integrable systems with one center and one homoclinic loop, J. Math. Anal. Appl., 269 (2002), 227-243.  doi: 10.1016/S0022-247X(02)00018-5.  Google Scholar

[35]

Y. ZhaoZ. Liang and G. Lu, The cyclicity of period annulus of the quadratic Hamiltonian systems with non-Morsean point, J. Differ. Equations, 162 (2000), 199-223.  doi: 10.1006/jdeq.1999.3704.  Google Scholar

[36]

Y. Zhao and H. Zhu, Bifurcation of limit cycles from a non-Hamiltonian quadratic integrable system with homoclinic loop, Infinite Dimensional Dynamical Systems, 64 (2013), 445-479.  doi: 10.1007/978-1-4614-4523-4.  Google Scholar

[37]

Y. Zhao and S. Zhu, Perturbations of the non-generic quadratic Hamiltonian vector fields with hyperbolic segment, Bull. Sci. Math., 125 (2001), 109-138.  doi: 10.1016/S0007-4497(00)01069-1.  Google Scholar

[38]

H. Żołądek, Quadratic systems with center and their perturbations, J. Differ. Equations, 109 (1994), 223-273.  doi: 10.1006/jdeq.1994.1049.  Google Scholar

show all references

References:
[1] A.A. AndronovA.A. Vitt and S.E. Khaikin, Theory of Oscillators, Pergamon Press, Oxford, New York, Toronto, 1966.   Google Scholar
[2]

G. ChenC. LiC. Liu and J. Llibre, The cyclicity of period annuli of some classes of reversible quadratic systems, Discrete Contin. Dyn. Syst., 16 (2006), 157-177.  doi: 10.3934/dcds.2006.16.157.  Google Scholar

[3]

L. ChenX. MaG. Zhang and C. Li, Cyclicity of several quadratic reversible systems with center of genus one, J. Appl. Anal. Comput., 1 (2011), 439-447.  doi: 10.11948/2011030.  Google Scholar

[4]

S. N. ChowC. Li and Y. Yi, The cyclicity of period annulus of degenerate quadratic Hamiltonian system with elliptic segment loop, Ergodic Theory Dynam. Systems, 22 (2002), 349-374.  doi: 10.1017/S0143385702000184.  Google Scholar

[5]

B. CollC. Li and R. Prohens, Quadratic perturbations of a class quadratic reversible systems with two centers, Discrete Contin. Dyn. Syst., 24 (2009), 699-729.  doi: 10.3934/dcds.2009.24.699.  Google Scholar

[6]

F. DumortierC. Li and Z. Zhang, Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops, J. Differ. Equations, 139 (1997), 146-193.  doi: 10.1006/jdeq.1997.3285.  Google Scholar

[7]

J.-P. FrançoiseL. Gavrilov and D. Xiao, Hilbert's 16th problem on a period annulus and Nash space of arcs, Math. Proc. Camb. Philos. Soc., 169 (2020), 377-409.  doi: 10.1017/S0305004119000239.  Google Scholar

[8]

S. GautierL. Gavrilov and I. D. Iliev, Perturbation of quadratic centers of genus one, Discrete Contin. Dyn. Syst., 25 (2009), 511-535.  doi: 10.3934/dcds.2009.25.511.  Google Scholar

[9]

M. GrauF. Mañosas and J. Villadelprat, A Chebyshev criterion for abelian integrals, Trans. Am. Math. Soc., 363 (2011), 109-129.  doi: 10.1090/S0002-9947-2010-05007-X.  Google Scholar

[10]

M. Han, Asymptotic expansions of Melnikov functions and limit cycle bifurcations, Int. J. Bifur. Chaos, 22 (2012), 1250296.  doi: 10.1142/S0218127412502963.  Google Scholar

[11]

M. HanJ. YangA. Tarta and Y. Gao, Limit cycles near homoclinic and heteroclinic loops, J. Dynam. Differ. Equ., 20 (2008), 923-944.  doi: 10.1007/s10884-008-9108-3.  Google Scholar

[12]

I. D. Iliev, The cyclicity of the period annulus of the quadratic Hamiltonian triangle, J. Differ. Equations, 128 (1996), 309-326.  doi: 10.1006/jdeq.1996.0097.  Google Scholar

[13]

I. D. Iliev, High-order Melnikov functions for degenerate cubic Hamiltonians, Adv. Differential Equations, 1 (1996), 689-708.   Google Scholar

[14]

I. D. Iliev, Perturbations of quadratic centers, Bull. Sci. Math., 122 (1998), 107-161.  doi: 10.1016/S0007-4497(98)80080-8.  Google Scholar

[15]

I. D. IlievC. Li and J. Yu, Bifurcation of limit cycles from quadratic non-Hamiltonian systems with two centers and two unbounded heteroclinic loops, Nonlinearity, 18 (2005), 305-330.  doi: 10.1088/0951-7715/18/1/016.  Google Scholar

[16]

I. D. IlievC. Li and J. Yu, Bifurcation of limit cycles in a reversible quadratic system with a center, a saddle and two nodes, Commun. Pure Appl. Anal., 9 (2010), 583-610.  doi: 10.3934/cpaa.2010.9.583.  Google Scholar

[17]

C. Li, Abelian integrals and limit cycles, Qual. Theory Dyn. Syst., 11 (2012), 111-128.  doi: 10.1007/s12346-011-0051-z.  Google Scholar

[18]

C. Li and J. Llibre, A unified study on the cyclicity of period annulus of the reversible quadratic Hamiltonian systems, J. Dynam. Differ. Equations, 16 (2004), 271-295.  doi: 10.1007/s10884-004-2781-y.  Google Scholar

[19]

C. Li and J. Llibre, The cyclicity of period annulus of a quadratic reversible Lotka-Volterra system, Nonlinearity, 22 (2009), 2971-2979.  doi: 10.1088/0951-7715/22/12/009.  Google Scholar

[20]

C. Li and Z. Zhang, A criterion for determining the monotonocity of the ratio of two Abelian integrals, J. Differ. Equations, 124 (1996), 407-424.  doi: 10.1006/jdeq.1996.0017.  Google Scholar

[21]

J. Li, Limit cycles bifurcated from a reversible quadratic center, Qual. Theory Dyn. Syst., 6 (2005), 205-215.  doi: 10.1007/BF02972673.  Google Scholar

[22]

H. LiangK. Wu and Y. Zhao, Quadratic perturbations of a class of quadratic reversible center of genus one, Sci. China Math., 56 (2013), 577-596.  doi: 10.1007/s11425-012-4488-6.  Google Scholar

[23]

H. Liang and Y. Zhao, Quadratic perturbations of a class of quadratic reversible systems with one center, Discrete Contin. Dyn. Syst., 27 (2010), 325-335.  doi: 10.3934/dcds.2010.27.325.  Google Scholar

[24]

H. Liang and Y. Zhao, Limit cycles bifurcated from a class of quadratic reversible center of genus one, J. Math. Anal. Appl., 391 (2012), 240-254.  doi: 10.1016/j.jmaa.2012.02.014.  Google Scholar

[25]

C. Liu, The cyclicity of period annuli of a class of quadratic reversible systems with two centers, J. Differ. Equations, 252 (2012), 5260-5273.  doi: 10.1016/j.jde.2012.02.005.  Google Scholar

[26]

C. Liu, Limit cycles bifurcated from some reversible quadratic centres with a non-algebraic first integral, Nonlinearity, 25 (2012), 1653-1660.  doi: 10.1088/0951-7715/25/6/1653.  Google Scholar

[27]

C. Liu and D. Xiao, The monotonicity of the ratio of two Abelian integrals, Trans. Amer. Math. Soc., 365 (2013), 5525-5544.  doi: 10.1090/S0002-9947-2013-05934-X.  Google Scholar

[28]

C. Liu and D. Xiao, The smallest upper bound on the number of zeros of Abelian integrals, J. Differ. Equations, 269 (2020), 3816-3852.  doi: 10.1016/j.jde.2020.03.016.  Google Scholar

[29]

F. Mañosas and J. Villadelprat, Bounding the number of zeros of certain Abelian integrals, J. Differ. Equations, 251 (2011), 1656-1669.  doi: 10.1016/j.jde.2011.05.026.  Google Scholar

[30]

L. Peng, Unfolding of a quadratic integrable system with a homoclinic loop, Acta Math. Sin. (Engl. Ser.), 18 (2002), 737-754.  doi: 10.1007/s10114-002-0196-4.  Google Scholar

[31]

Y. Shao and Y. Zhao, The cyclicity of a class of quadratic reversible system of genus one, Chaos Solitons Fractals, 44 (2011), 827-835.  doi: 10.1016/j.chaos.2011.06.015.  Google Scholar

[32]

Y. Shao and Y. Zhao, The cyclicity of the period annulus of a class of quadratic reverdible system, Commun. Pure Appl. Anal., 11 (2012), 1269-1283.  doi: 10.3934/cpaa.2012.11.1269.  Google Scholar

[33]

Y. Sun and C. Liu, The poincaré bifurcation of a SD oscillator, Discrete Contin. Dyn. Syst. B, 26 (2021), 1565-1577.  doi: 10.3934/dcdsb.2020173.  Google Scholar

[34]

J. Yu and C. Li, Bifurcation of a class of planar non-Hamiltonian integrable systems with one center and one homoclinic loop, J. Math. Anal. Appl., 269 (2002), 227-243.  doi: 10.1016/S0022-247X(02)00018-5.  Google Scholar

[35]

Y. ZhaoZ. Liang and G. Lu, The cyclicity of period annulus of the quadratic Hamiltonian systems with non-Morsean point, J. Differ. Equations, 162 (2000), 199-223.  doi: 10.1006/jdeq.1999.3704.  Google Scholar

[36]

Y. Zhao and H. Zhu, Bifurcation of limit cycles from a non-Hamiltonian quadratic integrable system with homoclinic loop, Infinite Dimensional Dynamical Systems, 64 (2013), 445-479.  doi: 10.1007/978-1-4614-4523-4.  Google Scholar

[37]

Y. Zhao and S. Zhu, Perturbations of the non-generic quadratic Hamiltonian vector fields with hyperbolic segment, Bull. Sci. Math., 125 (2001), 109-138.  doi: 10.1016/S0007-4497(00)01069-1.  Google Scholar

[38]

H. Żołądek, Quadratic systems with center and their perturbations, J. Differ. Equations, 109 (1994), 223-273.  doi: 10.1006/jdeq.1994.1049.  Google Scholar

Figure 1.  The phase portrait of system (2.1)
Figure 2.  The real constant $ \alpha_2 $ and $ \frac{W[\bar{F}_1,\bar{F}_3](x)}{W[\bar{F}_1,\bar{F}_2](x)} $ have at most two intersections
Figure 3.  The phase portrait of system (1.5) with $ b\in(-\infty,-2)\cup(-2,0]\cup[2,+\infty) $
Figure 4.  The range of $ (u,b). $
Figure 5.  The value of $ N_3(u,b) $
[1]

Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047

[2]

Iliya D. Iliev, Chengzhi Li, Jiang Yu. Bifurcations of limit cycles in a reversible quadratic system with a center, a saddle and two nodes. Communications on Pure & Applied Analysis, 2010, 9 (3) : 583-610. doi: 10.3934/cpaa.2010.9.583

[3]

Sze-Bi Hsu, Junping Shi. Relaxation oscillation profile of limit cycle in predator-prey system. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 893-911. doi: 10.3934/dcdsb.2009.11.893

[4]

Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114

[5]

Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439

[6]

Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447

[7]

Magdalena Caubergh, Freddy Dumortier, Robert Roussarie. Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle. Communications on Pure & Applied Analysis, 2007, 6 (1) : 1-21. doi: 10.3934/cpaa.2007.6.1

[8]

Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2417-2425. doi: 10.3934/dcdsb.2017123

[9]

Stijn Luca, Freddy Dumortier, Magdalena Caubergh, Robert Roussarie. Detecting alien limit cycles near a Hamiltonian 2-saddle cycle. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1081-1108. doi: 10.3934/dcds.2009.25.1081

[10]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257

[11]

Shengqing Hu, Bin Liu. Degenerate lower dimensional invariant tori in reversible system. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3735-3763. doi: 10.3934/dcds.2018162

[12]

Linping Peng, Yazhi Lei. The cyclicity of the period annulus of a quadratic reversible system with a hemicycle. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 873-890. doi: 10.3934/dcds.2011.30.873

[13]

Yi Shao, Yulin Zhao. The cyclicity of the period annulus of a class of quadratic reversible system. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1269-1283. doi: 10.3934/cpaa.2012.11.1269

[14]

Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855

[15]

Changlu Liu, Shuangli Qiao. Symmetry and monotonicity for a system of integral equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1925-1932. doi: 10.3934/cpaa.2009.8.1925

[16]

Yingshu Lü, Chunqin Zhou. Symmetry for an integral system with general nonlinearity. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1533-1543. doi: 10.3934/dcds.2018121

[17]

Wenxiong Chen, Congming Li. Regularity of solutions for a system of integral equations. Communications on Pure & Applied Analysis, 2005, 4 (1) : 1-8. doi: 10.3934/cpaa.2005.4.1

[18]

Linping Peng, Zhaosheng Feng, Changjian Liu. Quadratic perturbations of a quadratic reversible Lotka-Volterra system with two centers. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4807-4826. doi: 10.3934/dcds.2014.34.4807

[19]

Fang Wu, Lihong Huang, Jiafu Wang. Bifurcation of the critical crossing cycle in a planar piecewise smooth system with two zones. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021264

[20]

Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (63)
  • HTML views (44)
  • Cited by (0)

Other articles
by authors

[Back to Top]