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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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 $c = 0.133333$
[1]

Yanmin Niu, Xiong Li. An application of Moser's twist theorem to superlinear impulsive differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 431-445. doi: 10.3934/dcds.2019017

[2]

Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413

[3]

Pedro Teixeira. Dacorogna-Moser theorem on the Jacobian determinant equation with control of support. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4071-4089. doi: 10.3934/dcds.2017173

[4]

Sigurdur Freyr Hafstein. A constructive converse Lyapunov theorem on exponential stability. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 657-678. doi: 10.3934/dcds.2004.10.657

[5]

Viktor L. Ginzburg and Basak Z. Gurel. The Generalized Weinstein--Moser Theorem. Electronic Research Announcements, 2007, 14: 20-29. doi: 10.3934/era.2007.14.20

[6]

Florian Wagener. A parametrised version of Moser's modifying terms theorem. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 719-768. doi: 10.3934/dcdss.2010.3.719

[7]

Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57

[8]

Antonio Fernández, Pedro L. García. Regular discretizations in optimal control theory. Journal of Geometric Mechanics, 2013, 5 (4) : 415-432. doi: 10.3934/jgm.2013.5.415

[9]

Salvador Addas-Zanata. Stability for the vertical rotation interval of twist mappings. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 631-642. doi: 10.3934/dcds.2006.14.631

[10]

Huiping Jin. Boundedness in a class of duffing equations with oscillating potentials via the twist theorem. Communications on Pure & Applied Analysis, 2011, 10 (1) : 179-192. doi: 10.3934/cpaa.2011.10.179

[11]

Daniel Núñez, Pedro J. Torres. Periodic solutions of twist type of an earth satellite equation. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 303-306. doi: 10.3934/dcds.2001.7.303

[12]

Dongfeng Yan. KAM Tori for generalized Benjamin-Ono equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 941-957. doi: 10.3934/cpaa.2015.14.941

[13]

Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069

[14]

Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080

[15]

Linfeng Mei, Wei Dong, Changhe Guo. Concentration phenomenon in a nonlocal equation modeling phytoplankton growth. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 587-597. doi: 10.3934/dcdsb.2015.20.587

[16]

Olivier Bonnefon, Jérôme Coville, Guillaume Legendre. Concentration phenomenon in some non-local equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 763-781. doi: 10.3934/dcdsb.2017037

[17]

Alexander J. Zaslavski. Stability of a turnpike phenomenon for a class of optimal control systems in metric spaces. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 245-260. doi: 10.3934/naco.2011.1.245

[18]

Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2141-2160. doi: 10.3934/dcds.2017092

[19]

Hongzi Cong, Lufang Mi, Yunfeng Shi, Yuan Wu. On the existence of full dimensional KAM torus for nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6599-6630. doi: 10.3934/dcds.2019287

[20]

Luis Barreira, César Silva. Lyapunov exponents for continuous transformations and dimension theory. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 469-490. doi: 10.3934/dcds.2005.13.469

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (72)
  • HTML views (83)
  • Cited by (0)

Other articles
by authors

[Back to Top]