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Preface
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 |
References:
[1] |
J. Chu, "Stability of Periodic Solutions of Lagrange Equations and Planar Hamiltonian Systems," PhD Thesis, Tsinghua University, Beijing, 2008. |
[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), 530-542.
doi: 10.1016/j.jde.2008.11.013. |
[3] |
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. |
[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), Art. ID 286040, 12 pp. |
[5] |
J. Chu and M. Zhang, Rotation numbers and Lyapunov stability of elliptic periodic solutions, Discrete Contin. Dyn. Syst. A, 21 (2008), 1071-1094.
doi: 10.3934/dcds.2008.21.1071. |
[6] |
H. Feng and M. Zhang, Optimal estimates on rotation number of almost periodic systems, Z. Angew. Math. Phys., 57 (2006), 183-204.
doi: 10.1007/s00033-005-0020-y. |
[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), 169-192.
doi: 10.3934/dcds.2011.29.169. |
[8] |
J. K. Hale, "Ordinary Differential Equations," 2nd Edition, Robert E. Krieger Publishing Co., New York, 1980. |
[9] |
M. G. Krein, On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability, AMS Translations, Ser. 2, 1 (1955), 163-187. |
[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), 844-867.
doi: 10.1137/S003614100241037X. |
[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), 359-368.
doi: 10.1017/S0305004105008959. |
[12] |
J. Lei, P. J. Torres and M. Zhang, Twist character of the fourth order resonant periodic solution, J. Dynam. Differential Equations, 17 (2005), 21-50.
doi: 10.1007/s10884-005-2937-4. |
[13] |
J. Lei and M. Zhang, Twist property of periodic motion of an atom near a charged wire, Lett. Math. Phys., 60 (2002), 9-17.
doi: 10.1023/A:1015797310039. |
[14] |
B. Liu, The stability of the equilibrium of a conservative system, J. Math. Anal. Appl., 202 (1996), 133-149.
doi: 10.1006/jmaa.1996.0307. |
[15] |
B. Liu, The stability of the equilibrium of reversible system, Trans. Amer. Math. Soc., 351 (1999), 515-531.
doi: 10.1090/S0002-9947-99-01965-0. |
[16] |
B. Liu, The stability of the equilibrium of planar Hamiltonian and reversible system, J. Dynam. Differential Equations, 18 (2006), 975-990.
doi: 10.1007/s10884-006-9027-0. |
[17] |
B. Liu, The stability of equilibrium of quasi-periodic planar Hamiltonian and reversible systems, Sci. China Math., 53 (2010), 125-136.
doi: 10.1007/s11425-009-0117-4. |
[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), 2071-2077.
doi: 10.1016/j.na.2010.05.035. |
[19] |
J. Llibre and R. Ortega, On the families of periodic orbits of the Sitnikov problem, SIAM J. Appl. Dynam. Syst., 7 (2008), 561-576.
doi: 10.1137/070695253. |
[20] |
W. Magnus and S. Winkler, "Hill's Equation," Dover, New York, 1979. |
[21] |
D. Núñez, The method of lower and upper solutions and the stability of periodic oscillations, Nonlinear Anal., 51 (2002), 1207-1222.
doi: 10.1016/S0362-546X(01)00888-4. |
[22] |
D. Núñez and R. Ortega, Parabolic fixed points and stability criteria for nonlinear Hill's equation, Z. Angew. Math. Phys., 51 (2000), 890-911. |
[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), 610-615.
doi: 10.1016/j.aml.2006.05.023. |
[24] |
D. Núñez and P. J. Torres, Stabilization by vertical vibrations, Math. Meth. Appl. Sci., 32 (2009), 1118-1128.
doi: 10.1002/mma.1082. |
[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), 1976-1983.
doi: 10.1016/j.nonrwa.2008.03.003. |
[26] |
R. Ortega, The twist coefficient of periodic solutions of a time-dependent Newton's equation, J. Dynam. Differential Equations, 4 (1992), 651-665.
doi: 10.1007/BF01048263. |
[27] |
R. Ortega, The stability of equilibrium of a nonlinear Hill's equation, SIAM J. Math. Anal., 25 (1994), 1393-1401.
doi: 10.1137/S003614109223920X. |
[28] |
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. |
[29] |
R. Ortega, The stability of the equilibrium: a search for the right approximation, in "Ten Mathematical Essays on Approximation in Analysis and Topology," pp. 215-234, Elsevier B. V., Amsterdam, 2005. |
[30] |
R. Ortega and M. Zhang, Optimal bounds for bifurcation values of a superlinear periodic problem, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 119-132.
doi: 10.1017/S0308210500003796. |
[31] |
C. Siegel and J. Moser, "Lectures on Celestial Mechanics," Springer-Verlag, Berlin, 1971. |
[32] |
C. Simo, Stability of degenerate fixed points of analytic area preserving mappings, Astérisque, 98-99 (1982), 184-194. |
[33] |
P. J. Torres, Twist solutions of a Hill's equations with singular term, Adv. Nonlinear Stud., 2 (2002), 279-287. |
[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), 195-201. |
[35] |
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. |
[36] |
X. Wang, Stability criteria for linear periodic Hamiltonian systems, J. Math. Anal. Appl., 367 (2010), 329-336.
doi: 10.1016/j.jmaa.2010.01.027. |
[37] |
J. R. Ward, Asymptotic conditions for periodic solutions of ordinary differential equations, Proc. Amer. Math. Soc., 81 (1981), 415-420.
doi: 10.1090/S0002-9939-1981-0597653-2. |
[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), 137-148.
doi: 10.1112/S0024610702003939. |
[39] |
M. Zhang, Periodic solutions of equations of Emarkov-Pinney type, Adv. Nonlinear Stud., 6 (2006), 57-67. |
[40] |
M. Zhang, Sobolev inequalities and ellipticity of planar linear Hamiltonian systems, Adv. Nonlinear Stud., 8 (2008), 633-654. |
[41] |
M. Zhang, J. Chu and X. Li, Lyapunov stability of periodic solutions of the quadratic Newtonian equation, Math. Nachr., 282 (2009), 1354-1366.
doi: 10.1002/mana.200610799. |
[42] |
M. Zhang and W. Li, A Lyapunov-type stability criterion using $L^\alpha$ norms, Proc. Amer. Math. Soc., 130 (2002), 3325-3333.
doi: 10.1090/S0002-9939-02-06462-6. |
show all references
References:
[1] |
J. Chu, "Stability of Periodic Solutions of Lagrange Equations and Planar Hamiltonian Systems," PhD Thesis, Tsinghua University, Beijing, 2008. |
[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), 530-542.
doi: 10.1016/j.jde.2008.11.013. |
[3] |
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. |
[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), Art. ID 286040, 12 pp. |
[5] |
J. Chu and M. Zhang, Rotation numbers and Lyapunov stability of elliptic periodic solutions, Discrete Contin. Dyn. Syst. A, 21 (2008), 1071-1094.
doi: 10.3934/dcds.2008.21.1071. |
[6] |
H. Feng and M. Zhang, Optimal estimates on rotation number of almost periodic systems, Z. Angew. Math. Phys., 57 (2006), 183-204.
doi: 10.1007/s00033-005-0020-y. |
[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), 169-192.
doi: 10.3934/dcds.2011.29.169. |
[8] |
J. K. Hale, "Ordinary Differential Equations," 2nd Edition, Robert E. Krieger Publishing Co., New York, 1980. |
[9] |
M. G. Krein, On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability, AMS Translations, Ser. 2, 1 (1955), 163-187. |
[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), 844-867.
doi: 10.1137/S003614100241037X. |
[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), 359-368.
doi: 10.1017/S0305004105008959. |
[12] |
J. Lei, P. J. Torres and M. Zhang, Twist character of the fourth order resonant periodic solution, J. Dynam. Differential Equations, 17 (2005), 21-50.
doi: 10.1007/s10884-005-2937-4. |
[13] |
J. Lei and M. Zhang, Twist property of periodic motion of an atom near a charged wire, Lett. Math. Phys., 60 (2002), 9-17.
doi: 10.1023/A:1015797310039. |
[14] |
B. Liu, The stability of the equilibrium of a conservative system, J. Math. Anal. Appl., 202 (1996), 133-149.
doi: 10.1006/jmaa.1996.0307. |
[15] |
B. Liu, The stability of the equilibrium of reversible system, Trans. Amer. Math. Soc., 351 (1999), 515-531.
doi: 10.1090/S0002-9947-99-01965-0. |
[16] |
B. Liu, The stability of the equilibrium of planar Hamiltonian and reversible system, J. Dynam. Differential Equations, 18 (2006), 975-990.
doi: 10.1007/s10884-006-9027-0. |
[17] |
B. Liu, The stability of equilibrium of quasi-periodic planar Hamiltonian and reversible systems, Sci. China Math., 53 (2010), 125-136.
doi: 10.1007/s11425-009-0117-4. |
[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), 2071-2077.
doi: 10.1016/j.na.2010.05.035. |
[19] |
J. Llibre and R. Ortega, On the families of periodic orbits of the Sitnikov problem, SIAM J. Appl. Dynam. Syst., 7 (2008), 561-576.
doi: 10.1137/070695253. |
[20] |
W. Magnus and S. Winkler, "Hill's Equation," Dover, New York, 1979. |
[21] |
D. Núñez, The method of lower and upper solutions and the stability of periodic oscillations, Nonlinear Anal., 51 (2002), 1207-1222.
doi: 10.1016/S0362-546X(01)00888-4. |
[22] |
D. Núñez and R. Ortega, Parabolic fixed points and stability criteria for nonlinear Hill's equation, Z. Angew. Math. Phys., 51 (2000), 890-911. |
[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), 610-615.
doi: 10.1016/j.aml.2006.05.023. |
[24] |
D. Núñez and P. J. Torres, Stabilization by vertical vibrations, Math. Meth. Appl. Sci., 32 (2009), 1118-1128.
doi: 10.1002/mma.1082. |
[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), 1976-1983.
doi: 10.1016/j.nonrwa.2008.03.003. |
[26] |
R. Ortega, The twist coefficient of periodic solutions of a time-dependent Newton's equation, J. Dynam. Differential Equations, 4 (1992), 651-665.
doi: 10.1007/BF01048263. |
[27] |
R. Ortega, The stability of equilibrium of a nonlinear Hill's equation, SIAM J. Math. Anal., 25 (1994), 1393-1401.
doi: 10.1137/S003614109223920X. |
[28] |
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. |
[29] |
R. Ortega, The stability of the equilibrium: a search for the right approximation, in "Ten Mathematical Essays on Approximation in Analysis and Topology," pp. 215-234, Elsevier B. V., Amsterdam, 2005. |
[30] |
R. Ortega and M. Zhang, Optimal bounds for bifurcation values of a superlinear periodic problem, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 119-132.
doi: 10.1017/S0308210500003796. |
[31] |
C. Siegel and J. Moser, "Lectures on Celestial Mechanics," Springer-Verlag, Berlin, 1971. |
[32] |
C. Simo, Stability of degenerate fixed points of analytic area preserving mappings, Astérisque, 98-99 (1982), 184-194. |
[33] |
P. J. Torres, Twist solutions of a Hill's equations with singular term, Adv. Nonlinear Stud., 2 (2002), 279-287. |
[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), 195-201. |
[35] |
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. |
[36] |
X. Wang, Stability criteria for linear periodic Hamiltonian systems, J. Math. Anal. Appl., 367 (2010), 329-336.
doi: 10.1016/j.jmaa.2010.01.027. |
[37] |
J. R. Ward, Asymptotic conditions for periodic solutions of ordinary differential equations, Proc. Amer. Math. Soc., 81 (1981), 415-420.
doi: 10.1090/S0002-9939-1981-0597653-2. |
[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), 137-148.
doi: 10.1112/S0024610702003939. |
[39] |
M. Zhang, Periodic solutions of equations of Emarkov-Pinney type, Adv. Nonlinear Stud., 6 (2006), 57-67. |
[40] |
M. Zhang, Sobolev inequalities and ellipticity of planar linear Hamiltonian systems, Adv. Nonlinear Stud., 8 (2008), 633-654. |
[41] |
M. Zhang, J. Chu and X. Li, Lyapunov stability of periodic solutions of the quadratic Newtonian equation, Math. Nachr., 282 (2009), 1354-1366.
doi: 10.1002/mana.200610799. |
[42] |
M. Zhang and W. Li, A Lyapunov-type stability criterion using $L^\alpha$ norms, Proc. Amer. Math. Soc., 130 (2002), 3325-3333.
doi: 10.1090/S0002-9939-02-06462-6. |
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