January  2011, 29(1): 169-192. doi: 10.3934/dcds.2011.29.169

Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force

1. 

Dipartimento di Matematica e Geoscienze, Università di Trieste, P. le Europa 1, I-34127 Trieste, Italy

2. 

Departamento de Matematica Aplicada, Facultad de Ciencias, Universidad de Granada, E-18071, Granada, Spain

Received  November 2009 Revised  February 2010 Published  September 2010

We consider planar systems driven by a central force which depends periodically on time. If the force is sublinear and attractive, then there is a connected set of subharmonic and quasi-periodic solutions rotating around the origin at different speeds; moreover, this connected set stretches from zero to infinity. The result still holds allowing the force to be attractive only in average provided that an uniformity condition is satisfied and there are no periodic oscillations with zero angular momentum. We provide examples showing that these assumptions cannot be skipped.
Citation: Alessandro Fonda, Antonio J. Ureña. Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 169-192. doi: 10.3934/dcds.2011.29.169
References:
[1]

A. Ambrosetti and V. Coti Zelati, "Periodic Solutions of Singular Lagrangian Systems, Progress in Nonlinear Differential Equations and their Applications," Birkhäuser Boston, Inc., Boston, MA, 1993.

[2]

J.C. Alexander, A primer on connectivity, Fixed point theory (Sherbrooke, Que., 1980), pp. 455-483, Lecture Notes in Math. 886, Springer, Berlin-New York, 1981.

[3]

V. I. Arnold, "Mathematical Methods of Classical Mechanics," 2nd edition, Graduate Texts in Mathematics 60, Springer, New York, Heidelberg, 1978.

[4]

E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems, Ann. Mat. Pura Appl. (4), 131 (1982), 167-185. doi: doi:10.1007/BF01765151.

[5]

K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin Heidelberg, 1985,

[6]

A. Fonda and R. Toader, Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach, J. Differential Equations, 244 (2008), 3235-3264. doi: doi:10.1016/j.jde.2007.11.005.

[7]

P. Habets and L. Sanchez, Periodic solutions of dissipative dynamical systems with singular potentials, Differential Integral Equations, 3 (1990), 1139-1149.

[8]

J. Leray and J. Schauder, Topologie et équations fonctionnelles (French), Ann. Sci. École Norm. Sup. (3), 51 (1934), 45-78.

[9]

J. Mawhin, Leray-Schauder continuation theorems in the absence of a priori bounds, Topol. Methods Nonlinear Anal., 9 (1997), 179-200.

[10]

J. Mawhin, C. Rebelo and F. Zanolin, Continuation theorems for Ambrosetti-Prodi type periodic problems, Communications in Contemporary Mathematics, 2 (2000), 87-126.

[11]

W. Magnus and S. Winkler, "Hill's Equation," Dover Publ., Dover, 1979.

[12]

I. Newton, "Principes Mathématiques de la Philosophie Naturelle," 2nd section, Livre Premier, Paris, 1759.

show all references

References:
[1]

A. Ambrosetti and V. Coti Zelati, "Periodic Solutions of Singular Lagrangian Systems, Progress in Nonlinear Differential Equations and their Applications," Birkhäuser Boston, Inc., Boston, MA, 1993.

[2]

J.C. Alexander, A primer on connectivity, Fixed point theory (Sherbrooke, Que., 1980), pp. 455-483, Lecture Notes in Math. 886, Springer, Berlin-New York, 1981.

[3]

V. I. Arnold, "Mathematical Methods of Classical Mechanics," 2nd edition, Graduate Texts in Mathematics 60, Springer, New York, Heidelberg, 1978.

[4]

E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems, Ann. Mat. Pura Appl. (4), 131 (1982), 167-185. doi: doi:10.1007/BF01765151.

[5]

K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin Heidelberg, 1985,

[6]

A. Fonda and R. Toader, Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach, J. Differential Equations, 244 (2008), 3235-3264. doi: doi:10.1016/j.jde.2007.11.005.

[7]

P. Habets and L. Sanchez, Periodic solutions of dissipative dynamical systems with singular potentials, Differential Integral Equations, 3 (1990), 1139-1149.

[8]

J. Leray and J. Schauder, Topologie et équations fonctionnelles (French), Ann. Sci. École Norm. Sup. (3), 51 (1934), 45-78.

[9]

J. Mawhin, Leray-Schauder continuation theorems in the absence of a priori bounds, Topol. Methods Nonlinear Anal., 9 (1997), 179-200.

[10]

J. Mawhin, C. Rebelo and F. Zanolin, Continuation theorems for Ambrosetti-Prodi type periodic problems, Communications in Contemporary Mathematics, 2 (2000), 87-126.

[11]

W. Magnus and S. Winkler, "Hill's Equation," Dover Publ., Dover, 1979.

[12]

I. Newton, "Principes Mathématiques de la Philosophie Naturelle," 2nd section, Livre Premier, Paris, 1759.

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