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October  2021, 26(10): 5661-5679. doi: 10.3934/dcdsb.2020375

Parameter identification on Abelian integrals to achieve Chebyshev property

1. 

Department of Applied Mathematics, Guangxi University of Finance and Economics, Nanning 530003, China

2. 

Department of Applied Mathematics, Western University, London, Ontario, Canada N6A 5B7

* Corresponding author: pyu@uwo.ca

Received  May 2020 Revised  August 2020 Published  October 2021 Early access  December 2020

Fund Project: Pei Yu was supported by the Natural Sciences and Engineering Research Council of Canada (No. R2686A02). Xianbo Sun was partially supported by the Ontario Graduate Scholarship, NSFC(11861009), NSFGX(2018GXNSFAA138198) and Program for Innovative Team of GUFE(2018-2021)

Chebyshev criterion is a powerful tool in the study of limit cycle bifurcations in dynamical systems based on Abelian integrals, but it is difficult when the Abelian integrals involve parameters. In this paper, we consider the Abelian integrals on the periodic annuli of a Hamiltonian with one parameter, arising from the generalized Liénard system, and identify the parameter values such that the Abelian integrals have Chebyshev property. In particular, the bounds on the number of zeros of the Abelian integrals are derived for different parameter intervals. The main mathematical tools are transformations and polynomial boundary theory, which overcome the difficulties in symbolic computations and analysis, arising from large parametric-semi-algebraic systems.

Citation: Xianbo Sun, Zhanbo Chen, Pei Yu. Parameter identification on Abelian integrals to achieve Chebyshev property. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5661-5679. doi: 10.3934/dcdsb.2020375
References:
[1]

V. I. Arnold, Ten problems, in Theory of Singularities and its Applications, Amer. Math. Soc., Providence, RI 1990, 1–8.

[2]

B. Buchberger, Gröbner bases: An algorithmic method in polynomial ideal theory, in Multidimensional Systems Theory, Reidel, 1985.

[3]

J. CartwrightV. EguiluzE. Hernandez-Garcia and O. Piro, Dynamics of elastic excitable media, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 2197-2202.  doi: 10.1142/S0218127499001620.

[4]

F. ChenC. LiJ. Libre and Z. Zhang, A unified proof on the weak Hilbert 16th problem for $n = 2$, J. Differential Equations, 221 (2006), 309-342.  doi: 10.1016/j.jde.2005.01.009.

[5]

C. Chen and M. M. Maza, Algorithms for computing triangular decomposition of polynomial systems, J. Symbolic Comput., 47 (2012), 610-642.  doi: 10.1016/j.jsc.2011.12.023.

[6]

L. Cveticanin, Strong Nonlinear Oscillators. Analytical Solutions, 2$^nd$ edition, Springer, Cham, 2018. doi: 10.1007/978-3-319-58826-1.

[7]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysics J., 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.

[8]

M. GrauF. Ma$\widetilde{n}$osas and J. Villadelprat, A Chebyshev criterion for Abelian integrals, Trans. Amer. Math. Soc., 363 (2011), 109-129.  doi: 10.1090/S0002-9947-2010-05007-X.

[9]

M. Han, Asymptotic expansions of Melnikov functions and limit cycle bifurcations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250296, 30 pp. doi: 10.1142/S0218127412502963.

[10]

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

[11]

M. Han and P. Yu, Normal forms, Melnikov functions and bifurcations of limit cycles, Springer, London, 2012. doi: 10.1007/978-1-4471-2918-9.

[12]

M. HanH. Zang and J. Yang, Limit cycle bifurcations by perturbing a cuspidal loop in a Hamiltonian system, J. Differential Equations, 246 (2009), 129-163.  doi: 10.1016/j.jde.2008.06.039.

[13]

D. Hilbert, Mathematical problems, Bull. Amer. Math, Soc., 8 (1902), 437-479.  doi: 10.1090/S0002-9904-1902-00923-3.

[14]

R. Kazemi and H. R. Z. Zangeneh, Bifurcation of limit cycles in small perturbations of a hyper-elliptic Hamiltonian system with two nilpotent saddles, J. Appl. Anal. Comput., 2 (2012), 395-413.  doi: 10.11948/2012029.

[15]

C. Li and C. Rousseau, A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: The cusp of order 4, J. Differential Equations, 79 (1989), 132-167.  doi: 10.1016/0022-0396(89)90117-4.

[16]

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

[17]

A. Liénard, Etude des oscillations entretenues, Revue générale de l'électricité, 23 (1928), 901-912. 

[18]

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.

[19]

F. Ma$\widetilde{n}$osas and J. Villadelprat, Bounding the number of zeros of certain Abelian integrals, J. Differential Equations, 251 (2011), 1656-1669.  doi: 10.1016/j.jde.2011.05.026.

[20]

P. Mardešić, Chebyshev Systems and the Versal Unfolding of the Cusps of Order $n$, vol. 57, Hermann, Paris, 1998.

[21]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE., 50 (1962), 2061-2070.  doi: 10.1109/JRPROC.1962.288235.

[22]

S. Smale, Mathematical problems for the next century, Math. Intelligencer, 20 (1998), 7-15.  doi: 10.1007/BF03025291.

[23]

X. Sun, Perturbation of a period annulus bounded by a heteroclinic loop connecting two hyperbolic saddles, Qual. Theory Dyn. Syst., 16 (2017), 187-203.  doi: 10.1007/s12346-015-0186-4.

[24]

X. Sun, Bifurcation of limit cycles from a Liénard system with a heteroclinic loop connecting two nilpotent saddles, Nonlinear Dynam., 73 (2013), 869-880.  doi: 10.1007/s11071-013-0838-3.

[25]

X. SunM. Han and J. Yang, Bifurcation of limit cycles from a heteroclinic loop with a cusp, Nonlinear Anal., 74 (2011), 2948-2965.  doi: 10.1016/j.na.2011.01.013.

[26]

X. Sun and P. Yu, Exact bound on the number of zeros of Abelian integrals for two hyper-elliptic Hamiltonian systems of degree $4$, J. Differential Equations, 267 (2019), 7369-7384.  doi: 10.1016/j.jde.2019.07.023.

[27]

X. Sun and L. Zhao, Perturbations of a class of hyper-elliptic Hamiltonian systems of degree seven with nilpotent singular points, Appl. Math. Comput., 289 (2016), 194-203.  doi: 10.1016/j.amc.2016.04.018.

[28]

Y. Tsai, Numbers of relative equilibria in the planar four-vortex problem: Some special cases, J. Nonlinear Sci., 27 (2017), 775-815.  doi: 10.1007/s00332-016-9350-5.

[29]

B. Van der Pol, The nonlinear theory of electric oscillations, Proceedings of the Institute of Radio Engineers, 22 (1934), 1051–1086.

[30]

D. Wang, Decomposing polynomial systems into simple systems, J. Symbolic Comput., 25 (1998) 295–314. doi: 10.1006/jsco.1997.0177.

[31]

W. T. Wu, On the decision problem and the mechanization of theorem-proving in elementary geometry, Sci. Sinica., 21 (1978) 159–172.

[32]

L. Yang, X. Hou and B. Xia, A complete algorithm for automated discovering of a class of inequality-type theorems, Sci. China (Ser. F), 44 (2001) 33–49. doi: 10.1007/BF02713938.

[33]

S. Yang, B. Qin, G. Xia and Y. Xia, Perturbation of a period annulus bounded by a saddle-saddle cycle in a hyperelliptic Hamiltonian systems of degree 7, Qual. Theory Dyn. Syst., 19 (2020), Paper No. 33, 20 pp. doi: 10.1007/s12346-020-00348-7.

[34]

L. Yang and B. Xia, Real solution classification for parametric semi-algebraic systems, in Dolzmann, Algorithmic Algebra and Logic, 2005,281–289.

[35]

P. Yu and W. Zhang, Complex dynamics in a unified SIR and HIV disease model: A bifurcation theory approach, J. Nonlinear Sci., 29 (2019), 2447-2500.  doi: 10.1007/s00332-019-09550-7.

[36]

H. Zhu, S. Yang, X. Hu and W. Huang, Perturbation of a period annulus with a unique two-saddle cycle in higher order Hamiltonian, Complexity, (2019), 5813596. doi: 10.1155/2019/5813596.

show all references

References:
[1]

V. I. Arnold, Ten problems, in Theory of Singularities and its Applications, Amer. Math. Soc., Providence, RI 1990, 1–8.

[2]

B. Buchberger, Gröbner bases: An algorithmic method in polynomial ideal theory, in Multidimensional Systems Theory, Reidel, 1985.

[3]

J. CartwrightV. EguiluzE. Hernandez-Garcia and O. Piro, Dynamics of elastic excitable media, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 2197-2202.  doi: 10.1142/S0218127499001620.

[4]

F. ChenC. LiJ. Libre and Z. Zhang, A unified proof on the weak Hilbert 16th problem for $n = 2$, J. Differential Equations, 221 (2006), 309-342.  doi: 10.1016/j.jde.2005.01.009.

[5]

C. Chen and M. M. Maza, Algorithms for computing triangular decomposition of polynomial systems, J. Symbolic Comput., 47 (2012), 610-642.  doi: 10.1016/j.jsc.2011.12.023.

[6]

L. Cveticanin, Strong Nonlinear Oscillators. Analytical Solutions, 2$^nd$ edition, Springer, Cham, 2018. doi: 10.1007/978-3-319-58826-1.

[7]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysics J., 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.

[8]

M. GrauF. Ma$\widetilde{n}$osas and J. Villadelprat, A Chebyshev criterion for Abelian integrals, Trans. Amer. Math. Soc., 363 (2011), 109-129.  doi: 10.1090/S0002-9947-2010-05007-X.

[9]

M. Han, Asymptotic expansions of Melnikov functions and limit cycle bifurcations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 1250296, 30 pp. doi: 10.1142/S0218127412502963.

[10]

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

[11]

M. Han and P. Yu, Normal forms, Melnikov functions and bifurcations of limit cycles, Springer, London, 2012. doi: 10.1007/978-1-4471-2918-9.

[12]

M. HanH. Zang and J. Yang, Limit cycle bifurcations by perturbing a cuspidal loop in a Hamiltonian system, J. Differential Equations, 246 (2009), 129-163.  doi: 10.1016/j.jde.2008.06.039.

[13]

D. Hilbert, Mathematical problems, Bull. Amer. Math, Soc., 8 (1902), 437-479.  doi: 10.1090/S0002-9904-1902-00923-3.

[14]

R. Kazemi and H. R. Z. Zangeneh, Bifurcation of limit cycles in small perturbations of a hyper-elliptic Hamiltonian system with two nilpotent saddles, J. Appl. Anal. Comput., 2 (2012), 395-413.  doi: 10.11948/2012029.

[15]

C. Li and C. Rousseau, A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: The cusp of order 4, J. Differential Equations, 79 (1989), 132-167.  doi: 10.1016/0022-0396(89)90117-4.

[16]

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

[17]

A. Liénard, Etude des oscillations entretenues, Revue générale de l'électricité, 23 (1928), 901-912. 

[18]

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.

[19]

F. Ma$\widetilde{n}$osas and J. Villadelprat, Bounding the number of zeros of certain Abelian integrals, J. Differential Equations, 251 (2011), 1656-1669.  doi: 10.1016/j.jde.2011.05.026.

[20]

P. Mardešić, Chebyshev Systems and the Versal Unfolding of the Cusps of Order $n$, vol. 57, Hermann, Paris, 1998.

[21]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE., 50 (1962), 2061-2070.  doi: 10.1109/JRPROC.1962.288235.

[22]

S. Smale, Mathematical problems for the next century, Math. Intelligencer, 20 (1998), 7-15.  doi: 10.1007/BF03025291.

[23]

X. Sun, Perturbation of a period annulus bounded by a heteroclinic loop connecting two hyperbolic saddles, Qual. Theory Dyn. Syst., 16 (2017), 187-203.  doi: 10.1007/s12346-015-0186-4.

[24]

X. Sun, Bifurcation of limit cycles from a Liénard system with a heteroclinic loop connecting two nilpotent saddles, Nonlinear Dynam., 73 (2013), 869-880.  doi: 10.1007/s11071-013-0838-3.

[25]

X. SunM. Han and J. Yang, Bifurcation of limit cycles from a heteroclinic loop with a cusp, Nonlinear Anal., 74 (2011), 2948-2965.  doi: 10.1016/j.na.2011.01.013.

[26]

X. Sun and P. Yu, Exact bound on the number of zeros of Abelian integrals for two hyper-elliptic Hamiltonian systems of degree $4$, J. Differential Equations, 267 (2019), 7369-7384.  doi: 10.1016/j.jde.2019.07.023.

[27]

X. Sun and L. Zhao, Perturbations of a class of hyper-elliptic Hamiltonian systems of degree seven with nilpotent singular points, Appl. Math. Comput., 289 (2016), 194-203.  doi: 10.1016/j.amc.2016.04.018.

[28]

Y. Tsai, Numbers of relative equilibria in the planar four-vortex problem: Some special cases, J. Nonlinear Sci., 27 (2017), 775-815.  doi: 10.1007/s00332-016-9350-5.

[29]

B. Van der Pol, The nonlinear theory of electric oscillations, Proceedings of the Institute of Radio Engineers, 22 (1934), 1051–1086.

[30]

D. Wang, Decomposing polynomial systems into simple systems, J. Symbolic Comput., 25 (1998) 295–314. doi: 10.1006/jsco.1997.0177.

[31]

W. T. Wu, On the decision problem and the mechanization of theorem-proving in elementary geometry, Sci. Sinica., 21 (1978) 159–172.

[32]

L. Yang, X. Hou and B. Xia, A complete algorithm for automated discovering of a class of inequality-type theorems, Sci. China (Ser. F), 44 (2001) 33–49. doi: 10.1007/BF02713938.

[33]

S. Yang, B. Qin, G. Xia and Y. Xia, Perturbation of a period annulus bounded by a saddle-saddle cycle in a hyperelliptic Hamiltonian systems of degree 7, Qual. Theory Dyn. Syst., 19 (2020), Paper No. 33, 20 pp. doi: 10.1007/s12346-020-00348-7.

[34]

L. Yang and B. Xia, Real solution classification for parametric semi-algebraic systems, in Dolzmann, Algorithmic Algebra and Logic, 2005,281–289.

[35]

P. Yu and W. Zhang, Complex dynamics in a unified SIR and HIV disease model: A bifurcation theory approach, J. Nonlinear Sci., 29 (2019), 2447-2500.  doi: 10.1007/s00332-019-09550-7.

[36]

H. Zhu, S. Yang, X. Hu and W. Huang, Perturbation of a period annulus with a unique two-saddle cycle in higher order Hamiltonian, Complexity, (2019), 5813596. doi: 10.1155/2019/5813596.

Table 1.  Eight different topological phase eortraits of system (8)$ _{\epsilon = 0} $ with heteroclinic loops (red curves)
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