September  2018, 38(9): 4657-4674. doi: 10.3934/dcds.2018204

Lyapunov stability for regular equations and applications to the Liebau phenomenon

1. 

School of Mathematics and Physics, Changzhou University, Changzhou 213164, China

2. 

Department of Mathematics, Nanjing University, Nanjing 210093, China

3. 

Departamento de Matemáticas, Universidade de Vigo, 32004, Pabellón 3, Campus de Ourense, Spain

4. 

Department of Functional Analysis, Faculty of Mathematics and Natural Sciences, University of Rzeszów, Pigonia 1, 35-959 Rzeszów, Poland

* Corresponding author: J. A. Cid

Received  December 2017 Revised  March 2018 Published  June 2018

Fund Project: F. Wang was sponsored by Qing Lan Project of Jiangsu Province, and was supported by the National Natural Science Foundation of China (Grant No. 11501055 and No. 11401166), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 15KJB110001), China Postdoctoral Science Foundation funded project (Grant No. 2017M610315), Hainan Natural Science Foundation (Grant No.117005), Jiangsu Planned Projects for Postdoctoral Research Funds. J. A. Cid was partially supported by Ministerio de Educación y Ciencia, Spain, and FEDER, Project MTM2017-85054-C2-1-P. M. Zima was partially supported by the Centre for Innovation and Transfer of Natural Science and Engineering Knowledge of University of Rzeszów.

We study the existence and stability of periodic solutions of two kinds of regular equations by means of classical topological techniques like the Kolmogorov-Arnold-Moser (KAM) theory, the Moser twist theorem, the averaging method and the method of upper and lower solutions in the reversed order. As an application, we present some results on the existence and stability of $ T$-periodic solutions of a Liebau-type equation.

Citation: Feng Wang, José Ángel Cid, Mirosława Zima. Lyapunov stability for regular equations and applications to the Liebau phenomenon. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4657-4674. doi: 10.3934/dcds.2018204
References:
[1]

J. Chu and M. Li, Twist periodic solutions of second order singular differential equations, J. Math. Anal. Appl., 355 (2009), 830-838.  doi: 10.1016/j.jmaa.2009.02.033.

[2]

J. ChuP. J. Torres and F. Wang, Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem, Discrete Contin. Dyn. Syst., 35 (2015), 1921-1932.  doi: 10.3934/dcds.2015.35.1921.

[3]

J. ChuP. J. Torres and F. Wang, Twist periodic solutions for differential equations with a combined attractive-repulsive singularity, J. Math. Anal. Appl., 437 (2016), 1070-1083.  doi: 10.1016/j.jmaa.2016.01.057.

[4]

J. Chu and M. Zhang, Rotation numbers and Lyapunov stability of elliptic periodic solutions, Discrete Contin. Dyn. Syst., 21 (2008), 1071-1094.  doi: 10.3934/dcds.2008.21.1071.

[5]

J. A. CidG. Propst and M. Tvrdý, On the pumping effect in a pipe/tank flow configuration with friction, Phys. D, 273/274 (2014), 28-33.  doi: 10.1016/j.physd.2014.01.010.

[6]

J. A. CidG. InfanteM. Tvrdý and M. Zima, A topological approach to periodic oscillations related to the Liebau phenomenon, J. Math. Anal. Appl., 423 (2015), 1546-1556.  doi: 10.1016/j.jmaa.2014.10.054.

[7]

J. A. CidG. InfanteM. Tvrdý and M. Zima, New results for the Liebau phenomenon via fixed point index, Nonlinear Anal. Real World Appl., 35 (2017), 457-469.  doi: 10.1016/j.nonrwa.2016.11.009.

[8]

C. De Coster and P. Habets, Two-point Boundary Value Problems: Lower and Upper Solutions, Mathematics in Science and Engineering, 205, Elsevier B. V., Amsterdam, 2006.

[9]

E. N. Dancer and R. Ortega, The index of Lyapunov stable fixed points in two dimensions, J. Dynam. Differential Equations, 6 (1994), 631-637.  doi: 10.1007/BF02218851.

[10]

G. Feltrin and F. Zanolin, Multiple positive solutions for a superlinear problem: A topological approach, J. Differential Equations, 259 (2015), 925-963.  doi: 10.1016/j.jde.2015.02.032.

[11]

J. K. Hale, Ordinary Differential Equations, 2nd edition, Robert E. Krieger Publishing Co., Inc., Huntington, New York, 1980.

[12]

J. LeiX. LiP. Yan and M. Zhang, Twist character of the least amplitude periodic solution of the forced pendulum, SIAM J. Math. Anal., 35 (2003), 844-867.  doi: 10.1137/S003614100241037X.

[13]

F. F. Liao, Periodic solutions of Liebau-type differential equations, Appl. Math. Lett., 69 (2017), 8-14.  doi: 10.1016/j.aml.2017.02.001.

[14]

G. Liebau, Über ein ventilloses Pumpprinzip, Naturwissenschaften, 41 (1954), 327. 

[15]

R. Ortega, Periodic solution of a Newtonian equation: Stability by the third approximation, J. Differential Equations, 128 (1996), 491-518.  doi: 10.1006/jdeq.1996.0103.

[16]

R. Ortega and G. Verzini, A variational method for the existence of bounded solutions of a sublinear forced oscillator, Proc. London Math. Soc., 88 (2004), 775-795.  doi: 10.1112/S0024611503014515.

[17]

G. Propst, Pumping effects in models of periodically forced flow configurations, Phys. D, 217 (2006), 193-201.  doi: 10.1016/j.physd.2006.04.007.

[18]

C. Siegel and J. Moser, Lectures on Celestial Mechanics, Reprint of the 1971 translation, Classics in Mathematics, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-87284-6.

[19]

P. J. Torres, Mathematical Models with Singularities. A Zoo of Singular Creatures, Atlantis Briefs in Differential Equations, 1, Atlantis Press, Paris, 2015. doi: 10.2991/978-94-6239-106-2.

[20]

P. J. Torres and M. Zhang, Twist periodic solutions of repulsive singular equations, Nonlinear Anal., 56 (2004), 591-599.  doi: 10.1016/j.na.2003.10.005.

[21]

M. Zhang, The best bound on the rotations in the stability of periodic solutions of a Newtonian equation, J. London Math. Soc., 67 (2003), 137-148.  doi: 10.1112/S0024610702003939.

show all references

References:
[1]

J. Chu and M. Li, Twist periodic solutions of second order singular differential equations, J. Math. Anal. Appl., 355 (2009), 830-838.  doi: 10.1016/j.jmaa.2009.02.033.

[2]

J. ChuP. J. Torres and F. Wang, Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem, Discrete Contin. Dyn. Syst., 35 (2015), 1921-1932.  doi: 10.3934/dcds.2015.35.1921.

[3]

J. ChuP. J. Torres and F. Wang, Twist periodic solutions for differential equations with a combined attractive-repulsive singularity, J. Math. Anal. Appl., 437 (2016), 1070-1083.  doi: 10.1016/j.jmaa.2016.01.057.

[4]

J. Chu and M. Zhang, Rotation numbers and Lyapunov stability of elliptic periodic solutions, Discrete Contin. Dyn. Syst., 21 (2008), 1071-1094.  doi: 10.3934/dcds.2008.21.1071.

[5]

J. A. CidG. Propst and M. Tvrdý, On the pumping effect in a pipe/tank flow configuration with friction, Phys. D, 273/274 (2014), 28-33.  doi: 10.1016/j.physd.2014.01.010.

[6]

J. A. CidG. InfanteM. Tvrdý and M. Zima, A topological approach to periodic oscillations related to the Liebau phenomenon, J. Math. Anal. Appl., 423 (2015), 1546-1556.  doi: 10.1016/j.jmaa.2014.10.054.

[7]

J. A. CidG. InfanteM. Tvrdý and M. Zima, New results for the Liebau phenomenon via fixed point index, Nonlinear Anal. Real World Appl., 35 (2017), 457-469.  doi: 10.1016/j.nonrwa.2016.11.009.

[8]

C. De Coster and P. Habets, Two-point Boundary Value Problems: Lower and Upper Solutions, Mathematics in Science and Engineering, 205, Elsevier B. V., Amsterdam, 2006.

[9]

E. N. Dancer and R. Ortega, The index of Lyapunov stable fixed points in two dimensions, J. Dynam. Differential Equations, 6 (1994), 631-637.  doi: 10.1007/BF02218851.

[10]

G. Feltrin and F. Zanolin, Multiple positive solutions for a superlinear problem: A topological approach, J. Differential Equations, 259 (2015), 925-963.  doi: 10.1016/j.jde.2015.02.032.

[11]

J. K. Hale, Ordinary Differential Equations, 2nd edition, Robert E. Krieger Publishing Co., Inc., Huntington, New York, 1980.

[12]

J. LeiX. LiP. Yan and M. Zhang, Twist character of the least amplitude periodic solution of the forced pendulum, SIAM J. Math. Anal., 35 (2003), 844-867.  doi: 10.1137/S003614100241037X.

[13]

F. F. Liao, Periodic solutions of Liebau-type differential equations, Appl. Math. Lett., 69 (2017), 8-14.  doi: 10.1016/j.aml.2017.02.001.

[14]

G. Liebau, Über ein ventilloses Pumpprinzip, Naturwissenschaften, 41 (1954), 327. 

[15]

R. Ortega, Periodic solution of a Newtonian equation: Stability by the third approximation, J. Differential Equations, 128 (1996), 491-518.  doi: 10.1006/jdeq.1996.0103.

[16]

R. Ortega and G. Verzini, A variational method for the existence of bounded solutions of a sublinear forced oscillator, Proc. London Math. Soc., 88 (2004), 775-795.  doi: 10.1112/S0024611503014515.

[17]

G. Propst, Pumping effects in models of periodically forced flow configurations, Phys. D, 217 (2006), 193-201.  doi: 10.1016/j.physd.2006.04.007.

[18]

C. Siegel and J. Moser, Lectures on Celestial Mechanics, Reprint of the 1971 translation, Classics in Mathematics, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-87284-6.

[19]

P. J. Torres, Mathematical Models with Singularities. A Zoo of Singular Creatures, Atlantis Briefs in Differential Equations, 1, Atlantis Press, Paris, 2015. doi: 10.2991/978-94-6239-106-2.

[20]

P. J. Torres and M. Zhang, Twist periodic solutions of repulsive singular equations, Nonlinear Anal., 56 (2004), 591-599.  doi: 10.1016/j.na.2003.10.005.

[21]

M. Zhang, The best bound on the rotations in the stability of periodic solutions of a Newtonian equation, J. London Math. Soc., 67 (2003), 137-148.  doi: 10.1112/S0024610702003939.

Figure 1.  $2\pi$-periodic solution of equation (39) with $b = 1.55$ and $c = 0.4$
Figure 2.  $2\pi$-periodic positive solution of equation (40) with $b = 3/2$ and $c = 0.133333$
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