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Relativistic pendulum and invariant curves
1. | Dipartimento di Matematica - Università di Torino, Via Carlo Alberto, 10, 10123 Torino, Italy |
References:
[1] |
J. M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance,, Nonlinearity, 9 (1996), 1099.
doi: 10.1088/0951-7715/9/5/003. |
[2] |
C. Bereanu, P. Jebelean and J. Mawhin, Periodic solutions of pendulum-like perturbations of singular and bounded $\phi$-Laplacians,, J. Dynam. Differential Equations, 22 (2010), 463.
doi: 10.1007/s10884-010-9172-3. |
[3] |
C. Bereanu and P. J. Torres, Existence of at least two periodic solutions of the forced relativistic pendulum,, Proc. Amer. Math. Soc., 140 (2012), 2713.
doi: 10.1090/S0002-9939-2011-11101-8. |
[4] |
H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum,, Differential Integral Equations, 23 (2010), 801.
|
[5] |
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. |
[6] |
A. Fonda and R. Toader, Periodic solutions of pendulum-like Hamiltonian systems in the plane,, Adv. Nonlinear Stud., 12 (2012), 395.
|
[7] |
M. Herman, Sur Les Courbes Invariantes Par Les Difféomorphismes de L'anneau, Vol. 1, vol. 103 of Astérisque,, Société Mathématique de France, (1983). Google Scholar |
[8] |
M. Herman, Sur Les Courbes Invariantes Par Les Difféomorphismes de L'anneau, Vol. 2,, Astérisque, (). Google Scholar |
[9] |
M. Kunze and R. Ortega, Twist mappings with non-periodic angles,, in Stability and bifurcation theory for non-autonomous differential equations, (2013), 265.
doi: 10.1007/978-3-642-32906-7_5. |
[10] |
M. Levi, KAM theory for particles in periodic potentials,, Ergodic Theory Dynam. Systems, 10 (1990), 777.
doi: 10.1017/S0143385700005897. |
[11] |
S. Marò, Coexistence of bounded and unbounded motions in a bouncing ball model,, Nonlinearity, 26 (2013), 1439.
doi: 10.1088/0951-7715/26/5/1439. |
[12] |
S. Marò, Periodic solutions of a forced relativistic pendulum via twist dynamics,, Topol. Methods Nonlinear Anal., 42 (2013), 51.
|
[13] |
J. Mather, Existence of quasiperiodic orbits for twist homeomorphisms of the annulus,, Topology, 21 (1982), 457.
doi: 10.1016/0040-9383(82)90023-4. |
[14] |
J. Mather, Variational construction of orbits of twist diffeomorphisms,, J. Amer. Math. Soc., 4 (1991), 207.
doi: 10.1090/S0894-0347-1991-1080112-5. |
[15] |
J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962 (1962), 1.
|
[16] |
J. Moser, Selected Chapters in the Calculus of Variations,, Lectures in Mathematics ETH Zürich, (2003).
doi: 10.1007/978-3-0348-8057-2. |
[17] |
R. Ortega, Asymmetric oscillators and twist mappings,, J. London Math. Soc. (2), 53 (1996), 325.
doi: 10.1112/jlms/53.2.325. |
[18] |
R. Ortega, Invariant curves of mappings with averaged small twist,, Adv. Nonlinear Stud., 1 (2001), 14.
|
[19] |
R. Ortega, Twist mappings, invariant curves and periodic differential equations,, in Nonlinear analysis and its applications to differential equations (Lisbon, (2001), 85.
|
[20] |
P. Torres, Periodic oscillations of the relativistic pendulum with friction,, Phys. Lett. A, 372 (2008), 6386.
doi: 10.1016/j.physleta.2008.08.060. |
[21] |
J. You, Invariant tori and Lagrange stability of pendulum-type equations,, J. Differential Equations, 85 (1990), 54.
doi: 10.1016/0022-0396(90)90088-7. |
show all references
References:
[1] |
J. M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance,, Nonlinearity, 9 (1996), 1099.
doi: 10.1088/0951-7715/9/5/003. |
[2] |
C. Bereanu, P. Jebelean and J. Mawhin, Periodic solutions of pendulum-like perturbations of singular and bounded $\phi$-Laplacians,, J. Dynam. Differential Equations, 22 (2010), 463.
doi: 10.1007/s10884-010-9172-3. |
[3] |
C. Bereanu and P. J. Torres, Existence of at least two periodic solutions of the forced relativistic pendulum,, Proc. Amer. Math. Soc., 140 (2012), 2713.
doi: 10.1090/S0002-9939-2011-11101-8. |
[4] |
H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum,, Differential Integral Equations, 23 (2010), 801.
|
[5] |
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. |
[6] |
A. Fonda and R. Toader, Periodic solutions of pendulum-like Hamiltonian systems in the plane,, Adv. Nonlinear Stud., 12 (2012), 395.
|
[7] |
M. Herman, Sur Les Courbes Invariantes Par Les Difféomorphismes de L'anneau, Vol. 1, vol. 103 of Astérisque,, Société Mathématique de France, (1983). Google Scholar |
[8] |
M. Herman, Sur Les Courbes Invariantes Par Les Difféomorphismes de L'anneau, Vol. 2,, Astérisque, (). Google Scholar |
[9] |
M. Kunze and R. Ortega, Twist mappings with non-periodic angles,, in Stability and bifurcation theory for non-autonomous differential equations, (2013), 265.
doi: 10.1007/978-3-642-32906-7_5. |
[10] |
M. Levi, KAM theory for particles in periodic potentials,, Ergodic Theory Dynam. Systems, 10 (1990), 777.
doi: 10.1017/S0143385700005897. |
[11] |
S. Marò, Coexistence of bounded and unbounded motions in a bouncing ball model,, Nonlinearity, 26 (2013), 1439.
doi: 10.1088/0951-7715/26/5/1439. |
[12] |
S. Marò, Periodic solutions of a forced relativistic pendulum via twist dynamics,, Topol. Methods Nonlinear Anal., 42 (2013), 51.
|
[13] |
J. Mather, Existence of quasiperiodic orbits for twist homeomorphisms of the annulus,, Topology, 21 (1982), 457.
doi: 10.1016/0040-9383(82)90023-4. |
[14] |
J. Mather, Variational construction of orbits of twist diffeomorphisms,, J. Amer. Math. Soc., 4 (1991), 207.
doi: 10.1090/S0894-0347-1991-1080112-5. |
[15] |
J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962 (1962), 1.
|
[16] |
J. Moser, Selected Chapters in the Calculus of Variations,, Lectures in Mathematics ETH Zürich, (2003).
doi: 10.1007/978-3-0348-8057-2. |
[17] |
R. Ortega, Asymmetric oscillators and twist mappings,, J. London Math. Soc. (2), 53 (1996), 325.
doi: 10.1112/jlms/53.2.325. |
[18] |
R. Ortega, Invariant curves of mappings with averaged small twist,, Adv. Nonlinear Stud., 1 (2001), 14.
|
[19] |
R. Ortega, Twist mappings, invariant curves and periodic differential equations,, in Nonlinear analysis and its applications to differential equations (Lisbon, (2001), 85.
|
[20] |
P. Torres, Periodic oscillations of the relativistic pendulum with friction,, Phys. Lett. A, 372 (2008), 6386.
doi: 10.1016/j.physleta.2008.08.060. |
[21] |
J. You, Invariant tori and Lagrange stability of pendulum-type equations,, J. Differential Equations, 85 (1990), 54.
doi: 10.1016/0022-0396(90)90088-7. |
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