American Institute of Mathematical Sciences

Advanced
2011, 2011(Special): 475-484. doi: 10.3934/proc.2011.2011.475

Comparing the efficiency of numerical techniques for the integration of variational equations

Enrico Gerlach 1, and  Charlampos Skokos 2,

1. 

Lohrmann Observatory, Technical University Dresden, D-01062 Dredsen, Germany
2. 

Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, D-01187 Dresden, Germany

Received  July 2010 Revised  April 2011 Published  October 2011

We present a comparison of different numerical techniques for the integration of variational equations. The methods presented can be applied to any autonomous Hamiltonian system whose kinetic energy is quadratic in the generalized momenta, and whose potential is a function of the generalized positions. We apply the various techniques to the well-known H´enon-Heiles system, and use the Smaller Alignment Index (SALI) method of chaos detection to evaluate the percentage of its chaotic orbits. The accuracy and the speed of the integration schemes in evaluating this percentage are used to investigate the numerical efficiency of the various techniques.
Keywords: SALI method, Variational equations, H´enon-Heiles system, chaos.
Mathematics Subject Classification: Primary: 37M25, 65P10; Secondary: 70H07, 37J99, 65P20.
Citation: Enrico Gerlach, Charlampos Skokos. Comparing the efficiency of numerical techniques for the integration of variational equations. Conference Publications, 2011, 2011 (Special) : 475-484. doi: 10.3934/proc.2011.2011.475
[1]

Xianwei Chen, Zhujun Jing, Xiangling Fu. Chaos control in a pendulum system with excitations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 373-383. doi: 10.3934/dcdsb.2015.20.373
[2]

Iuliana Oprea, Gerhard Dangelmayr. A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 273-296. doi: 10.3934/dcdsb.2018095
[3]

Ting Yang. Homoclinic orbits and chaos in the generalized Lorenz system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1097-1108. doi: 10.3934/dcdsb.2019210
[4]

Zhujun Jing, K.Y. Chan, Dashun Xu, Hongjun Cao. Bifurcations of periodic solutions and chaos in Josephson system. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 573-592. doi: 10.3934/dcds.2001.7.573
[5]

Alberto Bressan, Yilun Jiang. The vanishing viscosity limit for a system of H-J equations related to a debt management problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 793-824. doi: 10.3934/dcdss.2018050
[6]

Hongxia Yin. An iterative method for general variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 201-209. doi: 10.3934/jimo.2005.1.201
[7]

Piotr Gwiazda, Karolina Kropielnicka, Anna Marciniak-Czochra. The Escalator Boxcar Train method for a system of age-structured equations. Networks & Heterogeneous Media, 2016, 11 (1) : 123-143. doi: 10.3934/nhm.2016.11.123
[8]

Tahereh Salimi Siahkolaei, Davod Khojasteh Salkuyeh. A preconditioned SSOR iteration method for solving complex symmetric system of linear equations. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 483-492. doi: 10.3934/naco.2019033
[9]

Alfonso Ruiz-Herrera. Chaos in delay differential equations with applications in population dynamics. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1633-1644. doi: 10.3934/dcds.2013.33.1633
[10]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827
[11]

Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079
[12]

Xianwei Chen, Xiangling Fu, Zhujun Jing. Chaos control in a special pendulum system for ultra-subharmonic resonance. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020144
[13]

Yoshifumi Aimoto, Takayasu Matsuo, Yuto Miyatake. A local discontinuous Galerkin method based on variational structure. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 817-832. doi: 10.3934/dcdss.2015.8.817
[14]

Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713
[15]

Luca Lorenzi. Optimal Hölder regularity for nonautonomous Kolmogorov equations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 169-191. doi: 10.3934/dcdss.2011.4.169
[16]

Peter Giesl, Martin Rasmussen. A note on almost periodic variational equations. Communications on Pure & Applied Analysis, 2011, 10 (3) : 983-994. doi: 10.3934/cpaa.2011.10.983
[17]

Luigi Ambrosio. Variational models for incompressible Euler equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 1-10. doi: 10.3934/dcdsb.2009.11.1
[18]

Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437
[19]

Nassif Ghoussoub. A variational principle for nonlinear transport equations. Communications on Pure & Applied Analysis, 2005, 4 (4) : 735-742. doi: 10.3934/cpaa.2005.4.735
[20]

Salahuddin. System of generalized mixed nonlinear ordered variational inclusions. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 445-460. doi: 10.3934/naco.2019026

 Impact Factor: 

Tools

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences