# American Institute of Mathematical Sciences

September  2011, 16(2): 423-443. doi: 10.3934/dcdsb.2011.16.423

## Lyapunov stability for conservative systems with lower degrees of freedom

 1 Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China 2 Zhou Pei-Yuan Center for Applied Mathematics, MOE Key Laboratory of Bioinformatics, Tsinghua University, Beijing 100084 3 Department of Mathematical Sciences, Tsinghua University, Beijing 100084

Received  February 2010 Revised  October 2010 Published  June 2011

It is a central theme to study the Lyapunov stability of periodic solutions of nonlinear differential equations or systems. For dissipative systems, the Lyapunov direct method is an important tool to study the stability. However, this method is not applicable to conservative systems such as Lagrangian equations and Hamiltonian systems. In the last decade, a method that is now known as the 'third order approximation' has been developed by Ortega, and has been applied to particular types of conservative systems including time periodic scalar Lagrangian equations (Ortega, J. Differential Equations, 128(1996), 491-518). This method is based on Moser's twist theorem, a prototype of the KAM theory. Latter, the twist coefficients were re-explained by Zhang in 2003 through the unique positive periodic solutions of the Ermakov-Pinney equation that is associated to the first order approximation (Zhang, J. London Math. Soc., 67(2003), 137-148). After that, Zhang and his collaborators have obtained some important twist criteria and applied the results to some interesting examples of time periodic scalar Lagrangian equations and planar Hamiltonian systems. In this survey, we will introduce the fundamental ideas in these works and will review recent progresses in this field, including applications to examples such as swing, the (relativistic) pendulum and singular equations. Some unsolved problems will be imposed for future study.
Citation: Jifeng Chu, Jinzhi Lei, Meirong Zhang. Lyapunov stability for conservative systems with lower degrees of freedom. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 423-443. doi: 10.3934/dcdsb.2011.16.423
##### References:
 [1] J. Chu, "Stability of Periodic Solutions of Lagrange Equations and Planar Hamiltonian Systems,", PhD Thesis, (2008).   Google Scholar [2] J. Chu, J. Lei and M. Zhang, The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator,, J. Differential Equations, 247 (2009), 530.  doi: 10.1016/j.jde.2008.11.013.  Google Scholar [3] J. Chu and M. Li, Twist periodic solutions of second order singular differential equations,, J. Math. Anal. Appl., 355 (2009), 830.  doi: 10.1016/j.jmaa.2009.02.033.  Google Scholar [4] J. Chu and T. Xia, The Lyapunov stability for the linear and nonlinear damped oscillator with time-periodic parameters,, Abstr. Appl. Anal., 2010 (2010).   Google Scholar [5] J. Chu and M. Zhang, Rotation numbers and Lyapunov stability of elliptic periodic solutions,, Discrete Contin. Dyn. Syst. A, 21 (2008), 1071.  doi: 10.3934/dcds.2008.21.1071.  Google Scholar [6] H. Feng and M. Zhang, Optimal estimates on rotation number of almost periodic systems,, Z. Angew. Math. Phys., 57 (2006), 183.  doi: 10.1007/s00033-005-0020-y.  Google Scholar [7] A. Fonda and A. J. Ureña, Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force,, Discrete Contin. Dyn. Syst. A, 29 (2011), 169.  doi: 10.3934/dcds.2011.29.169.  Google Scholar [8] J. K. Hale, "Ordinary Differential Equations," 2nd Edition,, Robert E. Krieger Publishing Co., (1980).   Google Scholar [9] M. G. Krein, On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability,, AMS Translations, 1 (1955), 163.   Google Scholar [10] J. Lei, X. Li, P. Yan and M. Zhang, Twist character of the least amplitude periodic solution of the forced pendulum,, SIAM J. Math. Anal., 35 (2003), 844.  doi: 10.1137/S003614100241037X.  Google Scholar [11] J. Lei and P. J. Torres, $L^1$ criteria for stability of periodic solutions of a newtonian equation,, Math. Proc. Cambridge Philos. Soc., 140 (2006), 359.  doi: 10.1017/S0305004105008959.  Google Scholar [12] J. Lei, P. J. Torres and M. Zhang, Twist character of the fourth order resonant periodic solution,, J. Dynam. Differential Equations, 17 (2005), 21.  doi: 10.1007/s10884-005-2937-4.  Google Scholar [13] J. Lei and M. Zhang, Twist property of periodic motion of an atom near a charged wire,, Lett. Math. Phys., 60 (2002), 9.  doi: 10.1023/A:1015797310039.  Google Scholar [14] B. Liu, The stability of the equilibrium of a conservative system,, J. Math. Anal. Appl., 202 (1996), 133.  doi: 10.1006/jmaa.1996.0307.  Google Scholar [15] B. Liu, The stability of the equilibrium of reversible system,, Trans. Amer. Math. Soc., 351 (1999), 515.  doi: 10.1090/S0002-9947-99-01965-0.  Google Scholar [16] B. Liu, The stability of the equilibrium of planar Hamiltonian and reversible system,, J. Dynam. Differential Equations, 18 (2006), 975.  doi: 10.1007/s10884-006-9027-0.  Google Scholar [17] B. Liu, The stability of equilibrium of quasi-periodic planar Hamiltonian and reversible systems,, Sci. China Math., 53 (2010), 125.  doi: 10.1007/s11425-009-0117-4.  Google Scholar [18] Q. Liu, D. Qian and Z. Wang, The stability of the equilibrium of the damped oscillator with damping changing sign,, Nonlinear Anal., 73 (2010), 2071.  doi: 10.1016/j.na.2010.05.035.  Google Scholar [19] J. Llibre and R. Ortega, On the families of periodic orbits of the Sitnikov problem,, SIAM J. Appl. Dynam. Syst., 7 (2008), 561.  doi: 10.1137/070695253.  Google Scholar [20] W. Magnus and S. Winkler, "Hill's Equation,", Dover, (1979).   Google Scholar [21] D. Núñez, The method of lower and upper solutions and the stability of periodic oscillations,, Nonlinear Anal., 51 (2002), 1207.  doi: 10.1016/S0362-546X(01)00888-4.  Google Scholar [22] D. Núñez and R. Ortega, Parabolic fixed points and stability criteria for nonlinear Hill's equation,, Z. Angew. Math. Phys., 51 (2000), 890.   Google Scholar [23] D. Núñez and P. J. Torres, KAM dynamics and stabilization of a particle sliding over a periodically driven curve,, Appl. Math. Lett., 20 (2007), 610.  doi: 10.1016/j.aml.2006.05.023.  Google Scholar [24] D. Núñez and P. J. Torres, Stabilization by vertical vibrations,, Math. Meth. Appl. Sci., 32 (2009), 1118.  doi: 10.1002/mma.1082.  Google Scholar [25] D. Núñez and P. J. Torres, On the motion of an oscillator with a periodically time-varying mass,, Nonlinear Anal. Real World Appl., 10 (2009), 1976.  doi: 10.1016/j.nonrwa.2008.03.003.  Google Scholar [26] R. Ortega, The twist coefficient of periodic solutions of a time-dependent Newton's equation,, J. Dynam. Differential Equations, 4 (1992), 651.  doi: 10.1007/BF01048263.  Google Scholar [27] R. Ortega, The stability of equilibrium of a nonlinear Hill's equation,, SIAM J. Math. Anal., 25 (1994), 1393.  doi: 10.1137/S003614109223920X.  Google Scholar [28] R. Ortega, Periodic solution of a Newtonian equation: Stability by the third approximation,, J. Differential Equations, 128 (1996), 491.  doi: 10.1006/jdeq.1996.0103.  Google Scholar [29] R. Ortega, The stability of the equilibrium: a search for the right approximation,, in, (2005), 215.   Google Scholar [30] R. Ortega and M. Zhang, Optimal bounds for bifurcation values of a superlinear periodic problem,, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 119.  doi: 10.1017/S0308210500003796.  Google Scholar [31] C. Siegel and J. Moser, "Lectures on Celestial Mechanics,", Springer-Verlag, (1971).   Google Scholar [32] C. Simo, Stability of degenerate fixed points of analytic area preserving mappings,, Astérisque, 98-99 (1982), 98.   Google Scholar [33] P. J. Torres, Twist solutions of a Hill's equations with singular term,, Adv. Nonlinear Stud., 2 (2002), 279.   Google Scholar [34] P. J. Torres, Existence and stability of periodic solutions for second order semilinear differential equations with a singular nonlinearity,, Proc. Royal Soc. Edinburgh Sect. A, 137 (2007), 195.   Google Scholar [35] P. J. Torres and M. Zhang, Twist periodic solutions of repulsive singular equations,, Nonlinear Anal., 56 (2004), 591.  doi: 10.1016/j.na.2003.10.005.  Google Scholar [36] X. Wang, Stability criteria for linear periodic Hamiltonian systems,, J. Math. Anal. Appl., 367 (2010), 329.  doi: 10.1016/j.jmaa.2010.01.027.  Google Scholar [37] J. R. Ward, Asymptotic conditions for periodic solutions of ordinary differential equations,, Proc. Amer. Math. Soc., 81 (1981), 415.  doi: 10.1090/S0002-9939-1981-0597653-2.  Google Scholar [38] 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.  doi: 10.1112/S0024610702003939.  Google Scholar [39] M. Zhang, Periodic solutions of equations of Emarkov-Pinney type,, Adv. Nonlinear Stud., 6 (2006), 57.   Google Scholar [40] M. Zhang, Sobolev inequalities and ellipticity of planar linear Hamiltonian systems,, Adv. Nonlinear Stud., 8 (2008), 633.   Google Scholar [41] M. Zhang, J. Chu and X. Li, Lyapunov stability of periodic solutions of the quadratic Newtonian equation,, Math. Nachr., 282 (2009), 1354.  doi: 10.1002/mana.200610799.  Google Scholar [42] M. Zhang and W. Li, A Lyapunov-type stability criterion using $L^\alpha$ norms,, Proc. Amer. Math. Soc., 130 (2002), 3325.  doi: 10.1090/S0002-9939-02-06462-6.  Google Scholar

show all references

##### References:
 [1] J. Chu, "Stability of Periodic Solutions of Lagrange Equations and Planar Hamiltonian Systems,", PhD Thesis, (2008).   Google Scholar [2] J. Chu, J. Lei and M. Zhang, The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator,, J. Differential Equations, 247 (2009), 530.  doi: 10.1016/j.jde.2008.11.013.  Google Scholar [3] J. Chu and M. Li, Twist periodic solutions of second order singular differential equations,, J. Math. Anal. Appl., 355 (2009), 830.  doi: 10.1016/j.jmaa.2009.02.033.  Google Scholar [4] J. Chu and T. Xia, The Lyapunov stability for the linear and nonlinear damped oscillator with time-periodic parameters,, Abstr. Appl. Anal., 2010 (2010).   Google Scholar [5] J. Chu and M. Zhang, Rotation numbers and Lyapunov stability of elliptic periodic solutions,, Discrete Contin. Dyn. Syst. A, 21 (2008), 1071.  doi: 10.3934/dcds.2008.21.1071.  Google Scholar [6] H. Feng and M. Zhang, Optimal estimates on rotation number of almost periodic systems,, Z. Angew. Math. Phys., 57 (2006), 183.  doi: 10.1007/s00033-005-0020-y.  Google Scholar [7] A. Fonda and A. J. Ureña, Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force,, Discrete Contin. Dyn. Syst. A, 29 (2011), 169.  doi: 10.3934/dcds.2011.29.169.  Google Scholar [8] J. K. Hale, "Ordinary Differential Equations," 2nd Edition,, Robert E. Krieger Publishing Co., (1980).   Google Scholar [9] M. G. Krein, On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability,, AMS Translations, 1 (1955), 163.   Google Scholar [10] J. Lei, X. Li, P. Yan and M. Zhang, Twist character of the least amplitude periodic solution of the forced pendulum,, SIAM J. Math. Anal., 35 (2003), 844.  doi: 10.1137/S003614100241037X.  Google Scholar [11] J. Lei and P. J. Torres, $L^1$ criteria for stability of periodic solutions of a newtonian equation,, Math. Proc. Cambridge Philos. Soc., 140 (2006), 359.  doi: 10.1017/S0305004105008959.  Google Scholar [12] J. Lei, P. J. Torres and M. Zhang, Twist character of the fourth order resonant periodic solution,, J. Dynam. Differential Equations, 17 (2005), 21.  doi: 10.1007/s10884-005-2937-4.  Google Scholar [13] J. Lei and M. Zhang, Twist property of periodic motion of an atom near a charged wire,, Lett. Math. Phys., 60 (2002), 9.  doi: 10.1023/A:1015797310039.  Google Scholar [14] B. Liu, The stability of the equilibrium of a conservative system,, J. Math. Anal. Appl., 202 (1996), 133.  doi: 10.1006/jmaa.1996.0307.  Google Scholar [15] B. Liu, The stability of the equilibrium of reversible system,, Trans. Amer. Math. Soc., 351 (1999), 515.  doi: 10.1090/S0002-9947-99-01965-0.  Google Scholar [16] B. Liu, The stability of the equilibrium of planar Hamiltonian and reversible system,, J. Dynam. Differential Equations, 18 (2006), 975.  doi: 10.1007/s10884-006-9027-0.  Google Scholar [17] B. Liu, The stability of equilibrium of quasi-periodic planar Hamiltonian and reversible systems,, Sci. China Math., 53 (2010), 125.  doi: 10.1007/s11425-009-0117-4.  Google Scholar [18] Q. Liu, D. Qian and Z. Wang, The stability of the equilibrium of the damped oscillator with damping changing sign,, Nonlinear Anal., 73 (2010), 2071.  doi: 10.1016/j.na.2010.05.035.  Google Scholar [19] J. Llibre and R. Ortega, On the families of periodic orbits of the Sitnikov problem,, SIAM J. Appl. Dynam. Syst., 7 (2008), 561.  doi: 10.1137/070695253.  Google Scholar [20] W. Magnus and S. Winkler, "Hill's Equation,", Dover, (1979).   Google Scholar [21] D. Núñez, The method of lower and upper solutions and the stability of periodic oscillations,, Nonlinear Anal., 51 (2002), 1207.  doi: 10.1016/S0362-546X(01)00888-4.  Google Scholar [22] D. Núñez and R. Ortega, Parabolic fixed points and stability criteria for nonlinear Hill's equation,, Z. Angew. Math. Phys., 51 (2000), 890.   Google Scholar [23] D. Núñez and P. J. Torres, KAM dynamics and stabilization of a particle sliding over a periodically driven curve,, Appl. Math. Lett., 20 (2007), 610.  doi: 10.1016/j.aml.2006.05.023.  Google Scholar [24] D. Núñez and P. J. Torres, Stabilization by vertical vibrations,, Math. Meth. Appl. Sci., 32 (2009), 1118.  doi: 10.1002/mma.1082.  Google Scholar [25] D. Núñez and P. J. Torres, On the motion of an oscillator with a periodically time-varying mass,, Nonlinear Anal. Real World Appl., 10 (2009), 1976.  doi: 10.1016/j.nonrwa.2008.03.003.  Google Scholar [26] R. Ortega, The twist coefficient of periodic solutions of a time-dependent Newton's equation,, J. Dynam. Differential Equations, 4 (1992), 651.  doi: 10.1007/BF01048263.  Google Scholar [27] R. Ortega, The stability of equilibrium of a nonlinear Hill's equation,, SIAM J. Math. Anal., 25 (1994), 1393.  doi: 10.1137/S003614109223920X.  Google Scholar [28] R. Ortega, Periodic solution of a Newtonian equation: Stability by the third approximation,, J. Differential Equations, 128 (1996), 491.  doi: 10.1006/jdeq.1996.0103.  Google Scholar [29] R. Ortega, The stability of the equilibrium: a search for the right approximation,, in, (2005), 215.   Google Scholar [30] R. Ortega and M. Zhang, Optimal bounds for bifurcation values of a superlinear periodic problem,, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 119.  doi: 10.1017/S0308210500003796.  Google Scholar [31] C. Siegel and J. Moser, "Lectures on Celestial Mechanics,", Springer-Verlag, (1971).   Google Scholar [32] C. Simo, Stability of degenerate fixed points of analytic area preserving mappings,, Astérisque, 98-99 (1982), 98.   Google Scholar [33] P. J. Torres, Twist solutions of a Hill's equations with singular term,, Adv. Nonlinear Stud., 2 (2002), 279.   Google Scholar [34] P. J. Torres, Existence and stability of periodic solutions for second order semilinear differential equations with a singular nonlinearity,, Proc. Royal Soc. Edinburgh Sect. A, 137 (2007), 195.   Google Scholar [35] P. J. Torres and M. Zhang, Twist periodic solutions of repulsive singular equations,, Nonlinear Anal., 56 (2004), 591.  doi: 10.1016/j.na.2003.10.005.  Google Scholar [36] X. Wang, Stability criteria for linear periodic Hamiltonian systems,, J. Math. Anal. Appl., 367 (2010), 329.  doi: 10.1016/j.jmaa.2010.01.027.  Google Scholar [37] J. R. Ward, Asymptotic conditions for periodic solutions of ordinary differential equations,, Proc. Amer. Math. Soc., 81 (1981), 415.  doi: 10.1090/S0002-9939-1981-0597653-2.  Google Scholar [38] 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.  doi: 10.1112/S0024610702003939.  Google Scholar [39] M. Zhang, Periodic solutions of equations of Emarkov-Pinney type,, Adv. Nonlinear Stud., 6 (2006), 57.   Google Scholar [40] M. Zhang, Sobolev inequalities and ellipticity of planar linear Hamiltonian systems,, Adv. Nonlinear Stud., 8 (2008), 633.   Google Scholar [41] M. Zhang, J. Chu and X. Li, Lyapunov stability of periodic solutions of the quadratic Newtonian equation,, Math. Nachr., 282 (2009), 1354.  doi: 10.1002/mana.200610799.  Google Scholar [42] M. Zhang and W. Li, A Lyapunov-type stability criterion using $L^\alpha$ norms,, Proc. Amer. Math. Soc., 130 (2002), 3325.  doi: 10.1090/S0002-9939-02-06462-6.  Google Scholar
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