American Institute of Mathematical Sciences

Advanced
November  2007, 18(4): 793-807. doi: 10.3934/dcds.2007.18.793

Existence, uniqueness, and stability of periodic solutions of an equation of duffing type

Hongbin Chen 1, and  Yi Li 2,

1. 

Department of Mathematics, Xi'an Jiaotong University, Xi'an, China
2. 

Department of Mathematics, The University of Iowa, Iowa City, IA 52242, United States

Received  January 2006 Revised  March 2007 Published  May 2007

We consider a second-order equation of Duffing type. Bounds for the derivative of the restoring force are given which ensure the existence and uniqueness of a periodic solution. Furthermore, the unique periodic solution is asymptotically stable with sharp rate of exponential decay. In particular, for a restoring term independent of the variable $t$, a necessary and sufficient condition is obtained which guarantees the existence and uniqueness of a periodic solution that is stable.
Keywords: stability., topological degree, Periodic solution.
Mathematics Subject Classification: 34C25, 34C1.
Citation: Hongbin Chen, Yi Li. Existence, uniqueness, and stability of periodic solutions of an equation of duffing type. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 793-807. doi: 10.3934/dcds.2007.18.793
[1]

Mi-Young Kim. Uniqueness and stability of positive periodic numerical solution of an epidemic model. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 365-375. doi: 10.3934/dcdsb.2007.7.365
[2]

Marc Henrard. Homoclinic and multibump solutions for perturbed second order systems using topological degree. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 765-782. doi: 10.3934/dcds.1999.5.765
[3]

Anna Capietto, Walter Dambrosio. A topological degree approach to sublinear systems of second order differential equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 861-874. doi: 10.3934/dcds.2000.6.861
[4]

Jian Lu, Huaiyu Jian. Topological degree method for the rotationally symmetric $L_p$-Minkowski problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 971-980. doi: 10.3934/dcds.2016.36.971
[5]

Matthieu Arfeux, Jan Kiwi. Topological cubic polynomials with one periodic ramification point. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1799-1811. doi: 10.3934/dcds.2020094
[6]

Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115
[7]

Keonhee Lee, Ngoc-Thach Nguyen, Yinong Yang. Topological stability and spectral decomposition for homeomorphisms on noncompact spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2487-2503. doi: 10.3934/dcds.2018103
[8]

Alexanger Arbieto, Carlos Arnoldo Morales Rojas. Topological stability from Gromov-Hausdorff viewpoint. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3531-3544. doi: 10.3934/dcds.2017151
[9]

Anthony W. Baker, Michael Dellnitz, Oliver Junge. Topological method for rigorously computing periodic orbits using Fourier modes. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 901-920. doi: 10.3934/dcds.2005.13.901
[10]

João Ferreira Alves, Michal Málek. Zeta functions and topological entropy of periodic nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 465-482. doi: 10.3934/dcds.2013.33.465
[11]

Guowei Yu. Periodic solutions of the planar N-center problem with topological constraints. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5131-5162. doi: 10.3934/dcds.2016023
[12]

Paweł Pilarczyk. Topological-numerical approach to the existence of periodic trajectories in ODE's. Conference Publications, 2003, 2003 (Special) : 701-708. doi: 10.3934/proc.2003.2003.701
[13]

Daniel Wilczak, Piotr Zgliczyński. Topological method for symmetric periodic orbits for maps with a reversing symmetry. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 629-652. doi: 10.3934/dcds.2007.17.629
[14]

Wolfgang Krieger, Kengo Matsumoto. Markov-Dyck shifts, neutral periodic points and topological conjugacy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 1-18. doi: 10.3934/dcds.2019001
[15]

Zhiyuan Geng, Wei Wang, Pingwen Zhang, Zhifei Zhang. Stability of half-degree point defect profiles for 2-D nematic liquid crystal. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6227-6242. doi: 10.3934/dcds.2017269
[16]

Yaping Wu, Xiuxia Xing, Qixiao Ye. Stability of travelling waves with algebraic decay for $n$-degree Fisher-type equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 47-66. doi: 10.3934/dcds.2006.16.47
[17]

Yaping Wu, Xiuxia Xing. Stability of traveling waves with critical speeds for $P$-degree Fisher-type equations. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1123-1139. doi: 10.3934/dcds.2008.20.1123
[18]

Yujun Zhu. Topological quasi-stability of partially hyperbolic diffeomorphisms under random perturbations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 869-882. doi: 10.3934/dcds.2014.34.869
[19]

Qiuxia Liu, Peidong Liu. Topological stability of hyperbolic sets of flows under random perturbations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 117-127. doi: 10.3934/dcdsb.2010.13.117
[20]

Noriaki Kawaguchi. Topological stability and shadowing of zero-dimensional dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2743-2761. doi: 10.3934/dcds.2019115

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences