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Stability in measure for uncertain heat equations
Cyclicity of $ (1,3) $-switching FF type equilibria
1. | Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China |
2. | Department de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain |
Hilbert's 16th Problem suggests a concern to the cyclicity of planar polynomial differential systems, but it is known that a key step to the answer is finding the cyclicity of center-focus equilibria of polynomial differential systems (even of order 2 or 3). Correspondingly, the same question for polynomial discontinuous differential systems is also interesting. Recently, it was proved that the cyclicity of $ (1, 2) $-switching FF type equilibria is at least 5. In this paper we prove that the cyclicity of $ (1, 3) $-switching FF type equilibria with homogeneous cubic nonlinearities is at least 3.
References:
[1] |
N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Math. Sbornik, 30 (1952), 181–196 (in Russian); Transl. Amer. Math. Soc., 100 (1954), 19pp. |
[2] |
M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008. |
[3] |
N. N. Bogoliubov, On Some Statistical Methods in Mathematical Physics, Akademiya Nauk Ukrainsko$ \mathop {\text{i}}\limits^ \vee $, 1945. |
[4] |
N. N. Bogoliubov and N. Krylov, The Application of Methods of Nonlinear Mechanics in the Theory of Stationary Oscillations, Ukrainian Academy of Science, 1934. |
[5] |
X. Cen, J. Llibre and M. Zhang,
Periodic solutions and their stability of some higher-order positively homogenous differential equations, Chaos, Solitons & Fractals, 106 (2018), 285-288.
doi: 10.1016/j.chaos.2017.11.032. |
[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, J. Llibre and W. Zhang,
Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems, Discrete and Continuous Dynamical Systems-Series B, 22 (2017), 3953-3965.
doi: 10.3934/dcdsb.2017203. |
[8] |
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. |
[9] |
X. Chen and W. Zhang,
Normal forms of planar switching systems, Disc. Cont. Dyn. Syst., 36 (2016), 6715-6736.
doi: 10.3934/dcds.2016092. |
[10] |
C. Christopher and C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2007. |
[11] |
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. |
[12] |
A. F. Filippov, Differential Equation with Discontinuous Right-Hand Sides, Kluwer Academic, Amsterdam, 1988.
doi: 10.1007/978-94-015-7793-9. |
[13] |
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. |
[14] |
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. |
[15] |
J. K. Hale and H. Hoçak, Dynamics and Bifurcations, Springer-Verlag, Berlin, 1991.
doi: 10.1007/978-1-4612-4426-4. |
[16] |
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. |
[17] |
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. |
[18] |
M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0103843. |
[19] |
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. |
[20] |
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. |
[21] |
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. |
[22] |
J. Llibre, D. D. Novaes and M. A. Teixeira,
Maximum number of limit cycles for certain piecewise linear dynamical systems, Nonlinear Dynamics, 82 (2015), 1159-1175.
doi: 10.1007/s11071-015-2223-x. |
[23] |
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. |
[24] |
P. Patou,
Sur le mouvement d'un système soumis à des forces à courte période, Bull. Soc. Math. France, 56 (1928), 98-139.
|
[25] |
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. |
[26] |
H. Żoldek,
Eleven small limit cycles in a cubic vector field, Nonlinearity, 8 (1995), 843-860.
doi: 10.1088/0951-7715/8/5/011. |
show all references
References:
[1] |
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., 100 (1954), 19pp. |
[2] |
M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008. |
[3] |
N. N. Bogoliubov, On Some Statistical Methods in Mathematical Physics, Akademiya Nauk Ukrainsko$ \mathop {\text{i}}\limits^ \vee $, 1945. |
[4] |
N. N. Bogoliubov and N. Krylov, The Application of Methods of Nonlinear Mechanics in the Theory of Stationary Oscillations, Ukrainian Academy of Science, 1934. |
[5] |
X. Cen, J. Llibre and M. Zhang,
Periodic solutions and their stability of some higher-order positively homogenous differential equations, Chaos, Solitons & Fractals, 106 (2018), 285-288.
doi: 10.1016/j.chaos.2017.11.032. |
[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, J. Llibre and W. Zhang,
Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems, Discrete and Continuous Dynamical Systems-Series B, 22 (2017), 3953-3965.
doi: 10.3934/dcdsb.2017203. |
[8] |
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. |
[9] |
X. Chen and W. Zhang,
Normal forms of planar switching systems, Disc. Cont. Dyn. Syst., 36 (2016), 6715-6736.
doi: 10.3934/dcds.2016092. |
[10] |
C. Christopher and C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2007. |
[11] |
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. |
[12] |
A. F. Filippov, Differential Equation with Discontinuous Right-Hand Sides, Kluwer Academic, Amsterdam, 1988.
doi: 10.1007/978-94-015-7793-9. |
[13] |
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. |
[14] |
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. |
[15] |
J. K. Hale and H. Hoçak, Dynamics and Bifurcations, Springer-Verlag, Berlin, 1991.
doi: 10.1007/978-1-4612-4426-4. |
[16] |
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. |
[17] |
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. |
[18] |
M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0103843. |
[19] |
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. |
[20] |
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. |
[21] |
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. |
[22] |
J. Llibre, D. D. Novaes and M. A. Teixeira,
Maximum number of limit cycles for certain piecewise linear dynamical systems, Nonlinear Dynamics, 82 (2015), 1159-1175.
doi: 10.1007/s11071-015-2223-x. |
[23] |
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. |
[24] |
P. Patou,
Sur le mouvement d'un système soumis à des forces à courte période, Bull. Soc. Math. France, 56 (1928), 98-139.
|
[25] |
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. |
[26] |
H. Żoldek,
Eleven small limit cycles in a cubic vector field, Nonlinearity, 8 (1995), 843-860.
doi: 10.1088/0951-7715/8/5/011. |
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