2005, 13(4): 901-920. doi: 10.3934/dcds.2005.13.901

Topological method for rigorously computing periodic orbits using Fourier modes

1. 

Institute for Mathematics, University of Paderborn, D-33095 Paderborn, Germany, Germany, Germany

Received  December 2004 Revised  July 2005 Published  August 2005

We present a technique for the rigorous computation of periodic orbits in certain ordinary differential equations. The method combines set oriented numerical techniques for the computation of invariant sets in dynamical systems with topological index arguments. It not only allows for the proof of existence of periodic orbits but also for a precise (and rigorous) approximation of these. As an example we compute a periodic orbit for a differential equation introduced in [2].
Citation: 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
[1]

A. Aschwanden, A. Schulze-Halberg, D. Stoffer. Stable periodic solutions for delay equations with positive feedback - a computer-assisted proof. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 721-736. doi: 10.3934/dcds.2006.14.721

[2]

Piotr Zgliczyński. Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof. Journal of Computational Dynamics, 2015, 2 (1) : 95-142. doi: 10.3934/jcd.2015.2.95

[3]

Dietmar Szolnoki. Set oriented methods for computing reachable sets and control sets. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 361-382. doi: 10.3934/dcdsb.2003.3.361

[4]

Anete S. Cavalcanti. An existence proof of a symmetric periodic orbit in the octahedral six-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1903-1922. doi: 10.3934/dcds.2017080

[5]

Tatiane C. Batista, Juliano S. Gonschorowski, Fábio A. Tal. Density of the set of endomorphisms with a maximizing measure supported on a periodic orbit. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3315-3326. doi: 10.3934/dcds.2015.35.3315

[6]

Thomas Wanner. Computer-assisted equilibrium validation for the diblock copolymer model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1075-1107. doi: 10.3934/dcds.2017045

[7]

Katja Polotzek, Kathrin Padberg-Gehle, Tobias Jäger. Set-oriented numerical computation of rotation sets. Journal of Computational Dynamics, 2017, 4 (1&2) : 119-141. doi: 10.3934/jcd.2017004

[8]

Andrea Venturelli. A Variational proof of the existence of Von Schubart's orbit. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 699-717. doi: 10.3934/dcdsb.2008.10.699

[9]

Ugo Locatelli, Letizia Stefanelli. Quasi-periodic motions in a special class of dynamical equations with dissipative effects: A pair of detection methods. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1155-1187. doi: 10.3934/dcdsb.2015.20.1155

[10]

Jong-Shenq Guo, Chang-Hong Wu. Front propagation for a two-dimensional periodic monostable lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 197-223. doi: 10.3934/dcds.2010.26.197

[11]

Shi-Liang Wu, Cheng-Hsiung Hsu. Entire solutions with merging fronts to a bistable periodic lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2329-2346. doi: 10.3934/dcds.2016.36.2329

[12]

W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351

[13]

S. Aubry, G. Kopidakis, V. Kadelburg. Variational proof for hard Discrete breathers in some classes of Hamiltonian dynamical systems. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 271-298. doi: 10.3934/dcdsb.2001.1.271

[14]

Gary Froyland, Oliver Junge, Kathrin Padberg-Gehle. Preface: Special issue on the occasion of the 4th International Workshop on Set-Oriented Numerics (SON 13, Dresden, 2013). Journal of Computational Dynamics, 2015, 2 (1) : i-ii. doi: 10.3934/jcd.2015.2.1i

[15]

Mostafa Fazly, Mahmoud Hesaaraki. Periodic solutions for a semi-ratio-dependent predator-prey dynamical system with a class of functional responses on time scales. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 267-279. doi: 10.3934/dcdsb.2008.9.267

[16]

Peter Giesl, James McMichen. Determination of the basin of attraction of a periodic orbit in two dimensions using meshless collocation. Journal of Computational Dynamics, 2016, 3 (2) : 191-210. doi: 10.3934/jcd.2016010

[17]

Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185

[18]

Matt Holzer. A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2069-2084. doi: 10.3934/dcds.2016.36.2069

[19]

P.K. Newton. The dipole dynamical system. Conference Publications, 2005, 2005 (Special) : 692-699. doi: 10.3934/proc.2005.2005.692

[20]

B. S. Goh, W. J. Leong, Z. Siri. Partial Newton methods for a system of equations. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 463-469. doi: 10.3934/naco.2013.3.463

2016 Impact Factor: 1.099

Metrics

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

[Back to Top]