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

Rotating periodic solutions of second order dissipative dynamical systems

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

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.
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.

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, 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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