American Institute of Mathematical Sciences

Advanced
August  2006, 15(3): 747-757. doi: 10.3934/dcds.2006.15.747

Collisionless orbits of singular and nonsingular dynamical systems

Daniel Franco 1, and  J. R. L. Webb 2,

1. 

Departamento de Matemática Aplicada, ETSI Industriales, UNED, Apartado de Correos 60149, Madrid, 28080
2. 

Department of Mathematics, University of Glasgow, Glasgow G12 8QW

Received  October 2005 Revised  December 2005 Published  April 2006

In this paper we study the existence of periodic solutions for some nonlinear systems of differential equations. We assume the nonlinearity satisfies suitable properties in one direction and we consider both the singular and the nonsingular case. As an application we present the existence of collisionless orbits for singular Lagrangian systems where the singular potential can have an attractive or repulsive behaviour near the singularity and we do not need to consider so-called strong force conditions. Our method is based on fixed point index theory for completely continuous operators, involving a new type of cone. In contrast with previous work using this type of technique, the nonlinearity neither needs to be positive nor to have a constant sign behaviour. The results improve recent work even in the scalar case.
Keywords: weak singularity., fixed point index, Periodic solutions.
Mathematics Subject Classification: Primary: 34C25; Secondary: 34B16, 47N2.
Citation: Daniel Franco, J. R. L. Webb. Collisionless orbits of singular and nonsingular dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 747-757. doi: 10.3934/dcds.2006.15.747
[1]

Zhibo Cheng, Jingli Ren. Periodic and subharmonic solutions for duffing equation with a singularity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1557-1574. doi: 10.3934/dcds.2012.32.1557
[2]

Rafael Ortega. Stability and index of periodic solutions of a nonlinear telegraph equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 823-837. doi: 10.3934/cpaa.2005.4.823
[3]

Pedro J. Torres. Non-collision periodic solutions of forced dynamical systems with weak singularities. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 693-698. doi: 10.3934/dcds.2004.11.693
[4]

Günther Hörmann, Hisashi Okamoto. Weak periodic solutions and numerical case studies of the Fornberg-Whitham equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4455-4469. doi: 10.3934/dcds.2019182
[5]

Chan-Gyun Kim, Yong-Hoon Lee. A bifurcation result for two point boundary value problem with a strong singularity. Conference Publications, 2011, 2011 (Special) : 834-843. doi: 10.3934/proc.2011.2011.834
[6]

Xing Liu, Yijing Sun. Multiple positive solutions for Kirchhoff type problems with singularity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 721-733. doi: 10.3934/cpaa.2013.12.721
[7]

Congming Li, Jisun Lim. The singularity analysis of solutions to some integral equations. Communications on Pure & Applied Analysis, 2007, 6 (2) : 453-464. doi: 10.3934/cpaa.2007.6.453
[8]

Gaston N'Guerekata. On weak-almost periodic mild solutions of some linear abstract differential equations. Conference Publications, 2003, 2003 (Special) : 672-677. doi: 10.3934/proc.2003.2003.672
[9]

Veronica Felli, Elsa M. Marchini, Susanna Terracini. On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 91-119. doi: 10.3934/dcds.2008.21.91
[10]

Jerry L. Bona, Stéphane Vento, Fred B. Weissler. Singularity formation and blowup of complex-valued solutions of the modified KdV equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4811-4840. doi: 10.3934/dcds.2013.33.4811
[11]

Richard Miles, Thomas Ward. A directional uniformity of periodic point distribution and mixing. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1181-1189. doi: 10.3934/dcds.2011.30.1181
[12]

Kung-Ching Chang, Zhi-Qiang Wang, Tan Zhang. On a new index theory and non semi-trivial solutions for elliptic systems. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 809-826. doi: 10.3934/dcds.2010.28.809
[13]

Takahiro Hashimoto. Nonexistence of positive solutions of quasilinear elliptic equations with singularity on the boundary in strip-like domains. Conference Publications, 2005, 2005 (Special) : 376-385. doi: 10.3934/proc.2005.2005.376
[14]

Tiancheng Ouyang, Zhifu Xie. Regularization of simultaneous binary collisions and solutions with singularity in the collinear four-body problem. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 909-932. doi: 10.3934/dcds.2009.24.909
[15]

Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807
[16]

Carlos J. Garcia-Cervera, Xiao-Ping Wang. Spin-polarized transport: Existence of weak solutions. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 87-100. doi: 10.3934/dcdsb.2007.7.87
[17]

Wanwan Wang, Hongxia Zhang, Huyuan Chen. Remarks on weak solutions of fractional elliptic equations. Communications on Pure & Applied Analysis, 2016, 15 (2) : 335-340. doi: 10.3934/cpaa.2016.15.335
[18]

Elder Jesús Villamizar-Roa, Henry Lamos-Díaz, Gilberto Arenas-Díaz. Very weak solutions for the magnetohydrodynamic type equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 957-972. doi: 10.3934/dcdsb.2008.10.957
[19]

Yuming Chu, Yihang Hao, Xiangao Liu. Global weak solutions to a general liquid crystals system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2681-2710. doi: 10.3934/dcds.2013.33.2681
[20]

Changchun Liu, Jingxue Yin, Juan Zhou. Existence of weak solutions for a generalized thin film equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 465-480. doi: 10.3934/cpaa.2007.6.465

2017 Impact Factor: 1.179

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (30)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences