American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Spatial propagation for a parabolic system with multiple species competing for single resource
  • DCDS-B Home
  • This Issue
  • Next Article
    Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time
April  2019, 24(4): 1769-1784. doi: 10.3934/dcdsb.2018236

Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center

Jaume Llibre 1, and  Yilei Tang 2,

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
2. 

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

Received  November 2017 Revised  February 2018 Published  August 2018

We apply the averaging theory of high order for computing the limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. These discontinuous piecewise differential systems are formed by two either quadratic, or cubic polynomial differential systems separated by a straight line.

We compute the maximum number of limit cycles of these discontinuous piecewise polynomial perturbations of the linear center, which can be obtained by using the averaging theory of order $ n $ for $ n = 1, 2, 3, 4, 5 $. Of course these limit cycles bifurcate from the periodic orbits of the linear center. As it was expected, using the averaging theory of the same order, the results show that the discontinuous quadratic and cubic polynomial perturbations of the linear center have more limit cycles than the ones found for continuous and discontinuous linear perturbations.

Moreover we provide sufficient and necessary conditions for the existence of a center or a focus at infinity if the discontinuous piecewise perturbations of the linear center are general quadratic polynomials or cubic quasi-homogenous polynomials.

Keywords: Periodic solution, limit cycle, discontinuous piecewise differential system, averaging theory.
Mathematics Subject Classification: 34C29, 34C25, 34C05.
Citation: Jaume Llibre, Yilei Tang. Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1769-1784. doi: 10.3934/dcdsb.2018236
References:
[1]

A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966.  Google Scholar

[2]

V. I. Arnold, Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields, Funct. Anal. Appl., 11 (1977), 85-92.   Google Scholar

[3]

V. I. Arnold, Ten problems, Adv. Soviet Math., 1 (1990), 1-8.   Google Scholar

[4]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems. Theory and Applications, Appl. Math. Sci. Series, 163, Springer-Verlag London, Ltd., London, 2008.  Google Scholar

[5]

I. S. Berezin and N. P. Zhidkov, Computing Methods, Reading, Mass. -London, 1965.  Google Scholar

[6]

D. C. Braga and L. F. Mello, Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane, Nonlinear Dynam., 73 (2013), 1283-1288.  doi: 10.1007/s11071-013-0862-3.  Google Scholar

[7]

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.  Google Scholar

[8]

X. Chen, J. Llibre and W. Zhang, Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3953-3965.  doi: 10.3934/dcdsb.2017203.  Google Scholar

[9]

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

[10]

B. Coll, Qualitative Study of Some Vector Fields in the Plane (Ph. Thesis in Catalan), Universitat Autònoma de Barcelona, 1987. Google Scholar

[11]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006.  Google Scholar

[12]

E. Freire, E. Ponce, F. Rodrigo and F. Torres, Bifurcation sets of continuous piecewise linear systems with two zones, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 2073-2097.  doi: 10.1142/S0218127498001728.  Google Scholar

[13]

B. García, J. Llibre and J. S. Pérez del Río, Planar quasihomogeneous polynomial differential systems and their integrability, J. Differential Equation, 255 (2013), 3185-3204.  doi: 10.1016/j.jde.2013.07.032.  Google Scholar

[14]

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.  Google Scholar

[15]

S. Huan and X. Yang, The number of limit cycles in general planar piecewise linear systems, Discrete Contin. Dyn. Syst., 32 (2012), 2147-2164.  doi: 10.3934/dcds.2012.32.2147.  Google Scholar

[16]

J. Itikawa, J. Llibre and D. D. Novaes, A new result on averaging theory for a class of discontinuous planar differential systems with applications, Revista Matemática Iberoamericana, 33 (2017), 1247-1265.  doi: 10.4171/RMI/970.  Google Scholar

[17]

A. G. Khovansky, Real analytic manifolds with finiteness properties and complex Abelian integrals, Funct. Anal. Appl., 18 (1984), 40-50.   Google Scholar

[18]

I. D. Iliev, The number of limit cycles due to polynomial perturbations of the harmonic oscillator, Math. Proc. Cambridge Philos. Soc., 127 (1999), 317-322.  doi: 10.1017/S0305004199003795.  Google Scholar

[19]

J. Itikawa and J. Llibre, Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers, J. of Computational and Applied Mathematic, 277 (2015), 171-191.  doi: 10.1016/j.cam.2014.09.007.  Google Scholar

[20]

J. Itikawa and J. Llibre, Phase portraits of uniform isochronous quartic centers, J. of Computational and Applied Mathematics, 287 (2015), 98-114.  doi: 10.1016/j.cam.2015.02.046.  Google Scholar

[21]

L. Li, Three crossing limit cycles in planar piecewise linear systems with saddle-focus type, Electron. J. Qual. Theory Differ. Equ., 2014 (2014), 14pp.  Google Scholar

[22]

J. Llibre, A. C. Mereu and D. D. Novaes, Averaging theory for discontinuous piecewise differential systems, J. Differential Equation, 258 (2015), 4007-4032.  doi: 10.1016/j.jde.2015.01.022.  Google Scholar

[23]

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

[24]

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.   Google Scholar

[25]

J. Llibre, E. Ponce and C. Valls, Uniqueness and non-uniqueness of limit cycles of piecewise linear differential systems with three zones and no symmetry, J. Nonlinear Science, 25 (2015), 861-887.  doi: 10.1007/s00332-015-9244-y.  Google Scholar

[26]

R. Lum and L. O. Chua, Global properties of continuous piecewise-linear vector fields. Part Ⅰ: Simplest case in $ \mathbb{R} ^2 $, Internat. J. Circuit Theory Appl., 19 (1991), 251-307.   Google Scholar

[27]

R. Lum and L. O. Chua, Global properties of continuous piecewise-linear vector fields. Part Ⅱ: simplest symmetric in $ \mathbb R^2 $, Internat. J. Circuit Theory Appl., 20 (1992), 9-46.  doi: 10.1002/cta.4490200103.  Google Scholar

[28]

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

[29]

G. Sansone and R. Conti, Non-Linear Differential Equations, 2$ ^{nd} $ edition, Pergamon Press, New York, 1964.  Google Scholar

[30]

D. Schlomiuk and N. Vulpe, Geometry of quadratic differential systems in the neighborhood of infinity, J. Differential Equations, 215 (2005), 357-400.  doi: 10.1016/j.jde.2004.11.001.  Google Scholar

[31]

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

[32]

A. N. Varchenko, An estimate of the number of zeros of an Abelian integral depending on a parameter and limiting cycles, Funct. Anal. Appl., 18 (1984), 14-25.   Google Scholar

[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.  doi: 10.3934/dcds.2016.36.2803.  Google Scholar

[34]

Y. Q. Ye, Theory of Limit Cycles, Trans. Math. Monographs 66, Amer. Math. Soc., Providence, RI, 1986.  Google Scholar

[35]

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.  Google Scholar

show all references

References:
[1]

A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966.  Google Scholar

[2]

V. I. Arnold, Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields, Funct. Anal. Appl., 11 (1977), 85-92.   Google Scholar

[3]

V. I. Arnold, Ten problems, Adv. Soviet Math., 1 (1990), 1-8.   Google Scholar

[4]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems. Theory and Applications, Appl. Math. Sci. Series, 163, Springer-Verlag London, Ltd., London, 2008.  Google Scholar

[5]

I. S. Berezin and N. P. Zhidkov, Computing Methods, Reading, Mass. -London, 1965.  Google Scholar

[6]

D. C. Braga and L. F. Mello, Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane, Nonlinear Dynam., 73 (2013), 1283-1288.  doi: 10.1007/s11071-013-0862-3.  Google Scholar

[7]

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.  Google Scholar

[8]

X. Chen, J. Llibre and W. Zhang, Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3953-3965.  doi: 10.3934/dcdsb.2017203.  Google Scholar

[9]

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

[10]

B. Coll, Qualitative Study of Some Vector Fields in the Plane (Ph. Thesis in Catalan), Universitat Autònoma de Barcelona, 1987. Google Scholar

[11]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006.  Google Scholar

[12]

E. Freire, E. Ponce, F. Rodrigo and F. Torres, Bifurcation sets of continuous piecewise linear systems with two zones, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), 2073-2097.  doi: 10.1142/S0218127498001728.  Google Scholar

[13]

B. García, J. Llibre and J. S. Pérez del Río, Planar quasihomogeneous polynomial differential systems and their integrability, J. Differential Equation, 255 (2013), 3185-3204.  doi: 10.1016/j.jde.2013.07.032.  Google Scholar

[14]

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.  Google Scholar

[15]

S. Huan and X. Yang, The number of limit cycles in general planar piecewise linear systems, Discrete Contin. Dyn. Syst., 32 (2012), 2147-2164.  doi: 10.3934/dcds.2012.32.2147.  Google Scholar

[16]

J. Itikawa, J. Llibre and D. D. Novaes, A new result on averaging theory for a class of discontinuous planar differential systems with applications, Revista Matemática Iberoamericana, 33 (2017), 1247-1265.  doi: 10.4171/RMI/970.  Google Scholar

[17]

A. G. Khovansky, Real analytic manifolds with finiteness properties and complex Abelian integrals, Funct. Anal. Appl., 18 (1984), 40-50.   Google Scholar

[18]

I. D. Iliev, The number of limit cycles due to polynomial perturbations of the harmonic oscillator, Math. Proc. Cambridge Philos. Soc., 127 (1999), 317-322.  doi: 10.1017/S0305004199003795.  Google Scholar

[19]

J. Itikawa and J. Llibre, Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers, J. of Computational and Applied Mathematic, 277 (2015), 171-191.  doi: 10.1016/j.cam.2014.09.007.  Google Scholar

[20]

J. Itikawa and J. Llibre, Phase portraits of uniform isochronous quartic centers, J. of Computational and Applied Mathematics, 287 (2015), 98-114.  doi: 10.1016/j.cam.2015.02.046.  Google Scholar

[21]

L. Li, Three crossing limit cycles in planar piecewise linear systems with saddle-focus type, Electron. J. Qual. Theory Differ. Equ., 2014 (2014), 14pp.  Google Scholar

[22]

J. Llibre, A. C. Mereu and D. D. Novaes, Averaging theory for discontinuous piecewise differential systems, J. Differential Equation, 258 (2015), 4007-4032.  doi: 10.1016/j.jde.2015.01.022.  Google Scholar

[23]

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

[24]

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.   Google Scholar

[25]

J. Llibre, E. Ponce and C. Valls, Uniqueness and non-uniqueness of limit cycles of piecewise linear differential systems with three zones and no symmetry, J. Nonlinear Science, 25 (2015), 861-887.  doi: 10.1007/s00332-015-9244-y.  Google Scholar

[26]

R. Lum and L. O. Chua, Global properties of continuous piecewise-linear vector fields. Part Ⅰ: Simplest case in $ \mathbb{R} ^2 $, Internat. J. Circuit Theory Appl., 19 (1991), 251-307.   Google Scholar

[27]

R. Lum and L. O. Chua, Global properties of continuous piecewise-linear vector fields. Part Ⅱ: simplest symmetric in $ \mathbb R^2 $, Internat. J. Circuit Theory Appl., 20 (1992), 9-46.  doi: 10.1002/cta.4490200103.  Google Scholar

[28]

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

[29]

G. Sansone and R. Conti, Non-Linear Differential Equations, 2$ ^{nd} $ edition, Pergamon Press, New York, 1964.  Google Scholar

[30]

D. Schlomiuk and N. Vulpe, Geometry of quadratic differential systems in the neighborhood of infinity, J. Differential Equations, 215 (2005), 357-400.  doi: 10.1016/j.jde.2004.11.001.  Google Scholar

[31]

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

[32]

A. N. Varchenko, An estimate of the number of zeros of an Abelian integral depending on a parameter and limiting cycles, Funct. Anal. Appl., 18 (1984), 14-25.   Google Scholar

[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.  doi: 10.3934/dcds.2016.36.2803.  Google Scholar

[34]

Y. Q. Ye, Theory of Limit Cycles, Trans. Math. Monographs 66, Amer. Math. Soc., Providence, RI, 1986.  Google Scholar

[35]

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.  Google Scholar

Figure 1.  Existence of closed orbits for system (29)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Existence of global center for system (29)
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Maximum number of limit cycles bifurcating from the periodic orbits of the linear center using averaging theory of order $n$
Order $n$ $L_1(n)$ $L_2(n)$ $L_3(n)$ $L_2^I(n)$ $L_3^I(n)$
112301
213512
325813
4361124
5381325
Table Options

Download as excel

Order $n$ $L_1(n)$ $L_2(n)$ $L_3(n)$ $L_2^I(n)$ $L_3^I(n)$
112301
213512
325813
4361124
5381325
[1]

Shanshan Liu, Maoan Han. Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3115-3124. doi: 10.3934/dcdss.2020133
[2]

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
[3]

Jaume Llibre, Lucyjane de A. S. Menezes. On the limit cycles of a class of discontinuous piecewise linear differential systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1835-1858. doi: 10.3934/dcdsb.2020005
[4]

Shimin Li, Jaume Llibre. On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5885-5901. doi: 10.3934/dcdsb.2019111
[5]

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
[6]

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
[7]

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
[8]

Hebai Chen, Jaume Llibre, Yilei Tang. Centers of discontinuous piecewise smooth quasi–homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6495-6509. doi: 10.3934/dcdsb.2019150
[9]

Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020368
[10]

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
[11]

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
[12]

Luca Dieci, Timo Eirola, Cinzia Elia. Periodic orbits of planar discontinuous system under discretization. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2743-2762. doi: 10.3934/dcdsb.2018103
[13]

Xingwu Chen, Jaume Llibre, Weinian Zhang. Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3953-3965. doi: 10.3934/dcdsb.2017203
[14]

Jackson Itikawa, Jaume Llibre, Ana Cristina Mereu, Regilene Oliveira. Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3259-3272. doi: 10.3934/dcdsb.2017136
[15]

Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369
[16]

Jaume Llibre, Clàudia Valls. Hopf bifurcation for some analytic differential systems in $\R^3$ via averaging theory. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 779-790. doi: 10.3934/dcds.2011.30.779
[17]

Wenjie Li, Lihong Huang, Jinchen Ji. Globally exponentially stable periodic solution in a general delayed predator-prey model under discontinuous prey control strategy. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2639-2664. doi: 10.3934/dcdsb.2020026
[18]

Thi-Bich-Ngoc Mac. Existence of solution for a system of repulsion and alignment: Comparison between theory and simulation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3013-3027. doi: 10.3934/dcdsb.2015.20.3013
[19]

Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121
[20]

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

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (176)
  • HTML views (573)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences