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Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model
Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems
1. | Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China |
2. | Department de Matemátiques, Universitat Autónoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain |
3. | Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China |
It is well known that the cyclicity of a Hopf bifurcation in continuous quadratic polynomial differential systems in $\mathbb{R}^2$ is $$. In contrast here we consider discontinuous differential systems in $\mathbb{R}^2$ defined in two half-planes separated by a straight line. In one half plane we have a general linear center at the origin of $\mathbb{R}^2$, and in the other a general quadratic polynomial differential system having a focus or a center at the origin of $\mathbb{R}^2$. Using averaging theory, we prove that the cyclicity of a Hopf bifurcation for such discontinuous differential systems is at least 5. Our computations show that only one of the averaged functions of fifth order can produce 5 limit cycles and there are no more limit cycles up to sixth order averaged function.
References:
[1] |
A. Andronov, A. Vitt and S. Khaikin,
Theory of Oscillations, Pergamon Press, Oxford, 1966. |
[2] |
N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients
from an equilibrium position of focus or center type, Math. Sbornik, 30 (1952), 181-196 (in
Russian); Transl. Amer. Math. Soc. , 1954 (1954), 19pp. |
[3] |
M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk,
Piecewise-Smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008. |
[4] |
N. N. Bogoliubov,
On Some Statistical Methods in Mathematical Physics, Akademiya Nauk Ukrainsko$\check{\rm i}$, 1945. |
[5] |
N. N. Bogoliubov and N. Krylov,
The Application of Methods of Nonlinear Mechanics in the Theory of Stationary Oscillations, Ukrainian Academy of Science, 1934. |
[6] |
X. Chen and Z. Du,
Limit cycles bifurcate from centers of discontinuous quadratic systems, Comput. Math. Appl., 59 (2010), 3836-3848.
doi: 10.1016/j.camwa.2010.04.019. |
[7] |
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. |
[8] |
X. Chen and W. Zhang,
Normal forms of planar switching systems, Disc. Cont. Dyn. Syst., 36 (2016), 6715-6736.
doi: 10.3934/dcds.2016092. |
[9] |
C. Christopher and C. Li,
Limit Cycles of Differential Equations, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2007. |
[10] |
B. Coll, A. Gasull and R. Prohens,
Degenerate Hopf bifurcations in discontinuous planar systems, J. Math. Anal. Appl., 253 (2001), 671-690.
doi: 10.1006/jmaa.2000.7188. |
[11] |
B. Coll, A. Gasull and R. Prohens,
Bifurcation of limit cycles from two families of centers, Dyn. Contin. Disc. Impu. Syst., 12 (2005), 275-287.
|
[12] |
W. A. Coppel,
A survey on quadratic systems, J. Diff. Equa., 2 (1966), 293-304.
doi: 10.1016/0022-0396(66)90070-2. |
[13] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, New York, 2006.
![]() ![]() |
[14] |
A. F. Filippov,
Differential Equations with Discontinuous Righthand Sides, Kluwer Academic, Amsterdam, 1988.
doi: 10.1007/978-94-015-7793-9. |
[15] |
A. Gasull and J. Torregrosa,
Center-focus problem for discontinuous planar differential equations, Int. J. Bifurc. Chaos, 13 (2003), 1755-1765.
doi: 10.1142/S0218127403007618. |
[16] |
J. Giné, M. Grau and J. Llibre,
Averaging theory at any order for computing periodic orbits, Physica D, 250 (2013), 58-65.
doi: 10.1016/j.physd.2013.01.015. |
[17] |
M. Han and W. Zhang,
On Hopf bifurcation in non-smooth planar systems, J. Diff. Equa., 248 (2010), 2399-2416.
doi: 10.1016/j.jde.2009.10.002. |
[18] |
Yu. Ilyashenko,
Centennial history of Hilbert's $16$th problem, Bull. (New Series) Amer. Math. Soc., 39 (2002), 301-354.
doi: 10.1090/S0273-0979-02-00946-1. |
[19] |
W. Kapteyn,
On the midpoints of integral curves of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland, 19 (1911), 1446-1457.
|
[20] |
W. Kapteyn,
New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland, 20 (1912), 1354-1365.
|
[21] |
M. Kunze,
Non-Smooth Dynamical Systems Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0103843. |
[22] |
J. Li,
Hilbert's $16$th problem and bifurcations of planar polynomial vector fields, Int. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 47-106.
doi: 10.1142/S0218127403006352. |
[23] |
J. Llibre, D. D. Novaes and M. A. Teixeira,
Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearity, 27 (2014), 563-583.
doi: 10.1088/0951-7715/27/3/563. |
[24] |
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. |
[25] |
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. |
[26] |
P. Patou,
Sur le mouvement d'un systéme soumis à des forces à courte période, Bull. Soc. Math. France, 56 (1928), 98-139.
|
[27] |
D. J. W. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, World Scientific
Series on Nonlinear Science A, vol 69, World Scientific, Singapore, 2010 |
show all references
References:
[1] |
A. Andronov, A. Vitt and S. Khaikin,
Theory of Oscillations, Pergamon Press, Oxford, 1966. |
[2] |
N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients
from an equilibrium position of focus or center type, Math. Sbornik, 30 (1952), 181-196 (in
Russian); Transl. Amer. Math. Soc. , 1954 (1954), 19pp. |
[3] |
M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk,
Piecewise-Smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008. |
[4] |
N. N. Bogoliubov,
On Some Statistical Methods in Mathematical Physics, Akademiya Nauk Ukrainsko$\check{\rm i}$, 1945. |
[5] |
N. N. Bogoliubov and N. Krylov,
The Application of Methods of Nonlinear Mechanics in the Theory of Stationary Oscillations, Ukrainian Academy of Science, 1934. |
[6] |
X. Chen and Z. Du,
Limit cycles bifurcate from centers of discontinuous quadratic systems, Comput. Math. Appl., 59 (2010), 3836-3848.
doi: 10.1016/j.camwa.2010.04.019. |
[7] |
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. |
[8] |
X. Chen and W. Zhang,
Normal forms of planar switching systems, Disc. Cont. Dyn. Syst., 36 (2016), 6715-6736.
doi: 10.3934/dcds.2016092. |
[9] |
C. Christopher and C. Li,
Limit Cycles of Differential Equations, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2007. |
[10] |
B. Coll, A. Gasull and R. Prohens,
Degenerate Hopf bifurcations in discontinuous planar systems, J. Math. Anal. Appl., 253 (2001), 671-690.
doi: 10.1006/jmaa.2000.7188. |
[11] |
B. Coll, A. Gasull and R. Prohens,
Bifurcation of limit cycles from two families of centers, Dyn. Contin. Disc. Impu. Syst., 12 (2005), 275-287.
|
[12] |
W. A. Coppel,
A survey on quadratic systems, J. Diff. Equa., 2 (1966), 293-304.
doi: 10.1016/0022-0396(66)90070-2. |
[13] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, New York, 2006.
![]() ![]() |
[14] |
A. F. Filippov,
Differential Equations with Discontinuous Righthand Sides, Kluwer Academic, Amsterdam, 1988.
doi: 10.1007/978-94-015-7793-9. |
[15] |
A. Gasull and J. Torregrosa,
Center-focus problem for discontinuous planar differential equations, Int. J. Bifurc. Chaos, 13 (2003), 1755-1765.
doi: 10.1142/S0218127403007618. |
[16] |
J. Giné, M. Grau and J. Llibre,
Averaging theory at any order for computing periodic orbits, Physica D, 250 (2013), 58-65.
doi: 10.1016/j.physd.2013.01.015. |
[17] |
M. Han and W. Zhang,
On Hopf bifurcation in non-smooth planar systems, J. Diff. Equa., 248 (2010), 2399-2416.
doi: 10.1016/j.jde.2009.10.002. |
[18] |
Yu. Ilyashenko,
Centennial history of Hilbert's $16$th problem, Bull. (New Series) Amer. Math. Soc., 39 (2002), 301-354.
doi: 10.1090/S0273-0979-02-00946-1. |
[19] |
W. Kapteyn,
On the midpoints of integral curves of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland, 19 (1911), 1446-1457.
|
[20] |
W. Kapteyn,
New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland, 20 (1912), 1354-1365.
|
[21] |
M. Kunze,
Non-Smooth Dynamical Systems Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0103843. |
[22] |
J. Li,
Hilbert's $16$th problem and bifurcations of planar polynomial vector fields, Int. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 47-106.
doi: 10.1142/S0218127403006352. |
[23] |
J. Llibre, D. D. Novaes and M. A. Teixeira,
Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearity, 27 (2014), 563-583.
doi: 10.1088/0951-7715/27/3/563. |
[24] |
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. |
[25] |
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. |
[26] |
P. Patou,
Sur le mouvement d'un systéme soumis à des forces à courte période, Bull. Soc. Math. France, 56 (1928), 98-139.
|
[27] |
D. J. W. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, World Scientific
Series on Nonlinear Science A, vol 69, World Scientific, Singapore, 2010 |
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