Rotating periodic solutions of second order dissipative dynamical systems

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February 2016, 36(2): 643-652. doi: 10.3934/dcds.2016.36.643

Rotating periodic solutions of second order dissipative dynamical systems

Xiaojun Chang ^1, and Yong Li ^2,

1.	School of Mathematics and Statistics, & Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, 130024, China
2.	College of Mathematics, Jilin University, Changchun, 130012, China

Received June 2014 Published August 2015

Abstract
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This paper is devoted to the following second order dissipative dynamical system \begin{equation*} u''+cu'+ \nabla g(u)+h(u)=e(t) ~\mbox{in}~\mathbb{R}^n. \end{equation*} When $g(u)=g(|u|)$, $\nabla g$ is a coercive function and $h$ is bounded, we use the coincidence degree theory to obtain some existence results of rotating periodic solutions, i.e., $u(t+T)=Qu(t)$, $\forall t\in \mathbb{R}$, with $T>0$ and $Q$ an orthogonal matrix, for $g$ to be nonsingular and singular at zero respectively. Specially, when some strong force type assumption is supposed on $g$, we obtain some new existence results of non-collision solutions for singular systems.

Keywords: coincidence degree theory., second order dissipative dynamical systems, Rotating periodic solutions.

Mathematics Subject Classification: Primary: 34B15, 34C25; Secondary: 47H1.

Citation: Xiaojun Chang, Yong Li. Rotating periodic solutions of second order dissipative dynamical systems. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 643-652. doi: 10.3934/dcds.2016.36.643

References:

[1]	A. Ambrosetti and V. Coti Zelati, Periodic Solutions of Singular Lagrangian Systems, Progress in Nonlinear Differential Equations and Their Applications, 10, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0319-3.
[2]	K. C. Chang, Methods in Nonlinear Analysis, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.
[3]	J. F. Chu, P. J. Torres and M. R. Zhang, Periodic solutions of second order non-autonomous singular dynamical systems, J. Differential Equations, 239 (2007), 196-212. doi: 10.1016/j.jde.2007.05.007.
[4]	I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Results in Mathematics and Related Areas, 19, Springer, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.
[5]	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: 10.1016/j.jde.2007.11.005.
[6]	A. Fonda and J. A. Ureña, Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force, Discrete Contin. Dyn. Syst., 29 (2011), 169-192. doi: 10.3934/dcds.2011.29.169.
[7]	D. Franco and P. J. Torres, Periodic solutions of singular systems without the strong force condition, Proc. Amer. Math. Soc., 136 (2008), 1229-1236. doi: 10.1090/S0002-9939-07-09226-X.
[8]	D. Franco and J. R. L. Webb, Collisionless orbits of singular and non singular dynamical systems, Discrete Contin. Dyn. Syst., 15 (2006), 747-757. doi: 10.3934/dcds.2006.15.747.
[9]	W. B. Gordon, Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc., 204 (1975), 113-135. doi: 10.1090/S0002-9947-1975-0377983-1.
[10]	P. Habets and L. Sanchez, Periodic solutions of dissipative dynamical systems with singular potentials, Differential Integral Equations, 3 (1990), 1139-1149.
[11]	Y. M. Long, Index Theory for Symplectic Paths with Applications, Progress in Mathematics, 207, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8175-3.
[12]	J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics, 40, American Mathematical Society, Providence, RI, 1979.
[13]	J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.
[14]	P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65, American Mathematical Society, Providence, RI, 1986.
[15]	P. J. Torres, Non-collision periodic solutions of forced dynamical systems with weak singularities, Discrete Contin. Dyn. Syst., 11 (2004), 693-698. doi: 10.3934/dcds.2004.11.693.
[16]	P. J. Torres, A. J. Ureña and M. Zamora, Periodic and quasi-periodic motions of a relativistic particle under a central force field, Bull. Lond. Math. Soc., 45 (2013), 140-152. doi: 10.1112/blms/bds076.
[17]	J. R. Ward, Periodic solutions of first order systems, Discrete Contin. Dyn. Syst., 33 (2013), 381-389. doi: 10.3934/dcds.2013.33.381.
[18]	M. R. Zhang, Periodic solutions of damped differential systems with repulsive singular forces, Proc. Amer. Math. Soc., 127 (1999), 401-407. doi: 10.1090/S0002-9939-99-05120-5.

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References:

[1]	A. Ambrosetti and V. Coti Zelati, Periodic Solutions of Singular Lagrangian Systems, Progress in Nonlinear Differential Equations and Their Applications, 10, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0319-3.
[2]	K. C. Chang, Methods in Nonlinear Analysis, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.
[3]	J. F. Chu, P. J. Torres and M. R. Zhang, Periodic solutions of second order non-autonomous singular dynamical systems, J. Differential Equations, 239 (2007), 196-212. doi: 10.1016/j.jde.2007.05.007.
[4]	I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Results in Mathematics and Related Areas, 19, Springer, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.
[5]	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: 10.1016/j.jde.2007.11.005.
[6]	A. Fonda and J. A. Ureña, Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force, Discrete Contin. Dyn. Syst., 29 (2011), 169-192. doi: 10.3934/dcds.2011.29.169.
[7]	D. Franco and P. J. Torres, Periodic solutions of singular systems without the strong force condition, Proc. Amer. Math. Soc., 136 (2008), 1229-1236. doi: 10.1090/S0002-9939-07-09226-X.
[8]	D. Franco and J. R. L. Webb, Collisionless orbits of singular and non singular dynamical systems, Discrete Contin. Dyn. Syst., 15 (2006), 747-757. doi: 10.3934/dcds.2006.15.747.
[9]	W. B. Gordon, Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc., 204 (1975), 113-135. doi: 10.1090/S0002-9947-1975-0377983-1.
[10]	P. Habets and L. Sanchez, Periodic solutions of dissipative dynamical systems with singular potentials, Differential Integral Equations, 3 (1990), 1139-1149.
[11]	Y. M. Long, Index Theory for Symplectic Paths with Applications, Progress in Mathematics, 207, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8175-3.
[12]	J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics, 40, American Mathematical Society, Providence, RI, 1979.
[13]	J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.
[14]	P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65, American Mathematical Society, Providence, RI, 1986.
[15]	P. J. Torres, Non-collision periodic solutions of forced dynamical systems with weak singularities, Discrete Contin. Dyn. Syst., 11 (2004), 693-698. doi: 10.3934/dcds.2004.11.693.
[16]	P. J. Torres, A. J. Ureña and M. Zamora, Periodic and quasi-periodic motions of a relativistic particle under a central force field, Bull. Lond. Math. Soc., 45 (2013), 140-152. doi: 10.1112/blms/bds076.
[17]	J. R. Ward, Periodic solutions of first order systems, Discrete Contin. Dyn. Syst., 33 (2013), 381-389. doi: 10.3934/dcds.2013.33.381.
[18]	M. R. Zhang, Periodic solutions of damped differential systems with repulsive singular forces, Proc. Amer. Math. Soc., 127 (1999), 401-407. doi: 10.1090/S0002-9939-99-05120-5.

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