October  2010, 14(3): 977-1000. doi: 10.3934/dcdsb.2010.14.977

On the high order approximation of the centre manifold for ODEs

1. 

Universitat de Barcelona, Gran Via de les Corts Catalanes 585, Barcelona, 08007, Spain, Spain

Received  September 2009 Revised  April 2010 Published  July 2010

Many times in dynamical systems one wants to understand the bounded motion around an equilibrium point. From a numerical point of view, we can take arbitrary initial conditions close to the equilibrium points, integrate the trajectories and plot them to have a rough idea of motion. If the dimension of the phase space is high, we can take suitable Poincaré sections and/or projections to visualise the dynamics. Of course, if the linear behaviour around the equilibrium point has an unstable direction, this procedure is useless as the trajectories will escape quickly. We need to get rid, in some way, of the instability of the system.
   Here we focus on equilibrium points whose linear dynamics is a cross product of one hyperbolic directions and several elliptic ones. We will compute a high order approximation of the centre manifold around the equilibrium point and use it to describe the behaviour of the system in an extended neighbourhood of this point. Our approach is based on the graph transform method. To derive an efficient algorithm we use recurrent expressions for the expansion of the non - linear terms on the equations of motion.
   Although this method does not require the system to be Hamiltonian, we have taken a Hamiltonian system as an example. We have compared its efficiency with a more classical approach for this type of systems, the Lie series method. It turns out that in this example the graph transform method is more efficient than the Lie series method. Finally, we have used this high order approximation of the centre manifold to describe the bounded motion of the system around and unstable equilibrium point.
Citation: Ariadna Farrés, Àngel Jorba. On the high order approximation of the centre manifold for ODEs. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 977-1000. doi: 10.3934/dcdsb.2010.14.977
[1]

Henk Broer, Aaron Hagen, Gert Vegter. Numerical approximation of normally hyperbolic invariant manifolds. Conference Publications, 2003, 2003 (Special) : 133-140. doi: 10.3934/proc.2003.2003.133

[2]

Maciej J. Capiński, Piotr Zgliczyński. Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 641-670. doi: 10.3934/dcds.2011.30.641

[3]

Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1089-1112. doi: 10.3934/dcds.2013.33.1089

[4]

Inmaculada Baldomá, Ernest Fontich, Pau Martín. Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4159-4190. doi: 10.3934/dcds.2017177

[5]

Rovella Alvaro, Vilamajó Francesc, Romero Neptalí. Invariant manifolds for delay endomorphisms. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 35-50. doi: 10.3934/dcds.2001.7.35

[6]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233

[7]

Zhihong Xia. Hyperbolic invariant sets with positive measures. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 811-818. doi: 10.3934/dcds.2006.15.811

[8]

Brent Everitt, John Ratcliffe and Steven Tschantz. The smallest hyperbolic 6-manifolds. Electronic Research Announcements, 2005, 11: 40-46.

[9]

José F. Alves, Davide Azevedo. Statistical properties of diffeomorphisms with weak invariant manifolds. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 1-41. doi: 10.3934/dcds.2016.36.1

[10]

George Osipenko. Indestructibility of invariant locally non-unique manifolds. Discrete & Continuous Dynamical Systems, 1996, 2 (2) : 203-219. doi: 10.3934/dcds.1996.2.203

[11]

Christopher K. R. T. Jones, Siu-Kei Tin. Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 967-1023. doi: 10.3934/dcdss.2009.2.967

[12]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579

[13]

Arturo Echeverría-Enríquez, Alberto Ibort, Miguel C. Muñoz-Lecanda, Narciso Román-Roy. Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds. Journal of Geometric Mechanics, 2012, 4 (4) : 397-419. doi: 10.3934/jgm.2012.4.397

[14]

Roberto Castelli. Efficient representation of invariant manifolds of periodic orbits in the CRTBP. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 563-586. doi: 10.3934/dcdsb.2018197

[15]

Pablo Aguirre, Bernd Krauskopf, Hinke M. Osinga. Global invariant manifolds near a Shilnikov homoclinic bifurcation. Journal of Computational Dynamics, 2014, 1 (1) : 1-38. doi: 10.3934/jcd.2014.1.1

[16]

Clara Cufí-Cabré, Ernest Fontich. Differentiable invariant manifolds of nilpotent parabolic points. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4667-4704. doi: 10.3934/dcds.2021053

[17]

Alexey Gorshkov. Stable invariant manifolds with application to control problems. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021040

[18]

Fuzhong Cong, Yong Li. Invariant hyperbolic tori for Hamiltonian systems with degeneracy. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 371-382. doi: 10.3934/dcds.1997.3.371

[19]

Francois Ledrappier and Omri Sarig. Invariant measures for the horocycle flow on periodic hyperbolic surfaces. Electronic Research Announcements, 2005, 11: 89-94.

[20]

Lennard F. Bakker, Pedro Martins Rodrigues. A profinite group invariant for hyperbolic toral automorphisms. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 1965-1976. doi: 10.3934/dcds.2012.32.1965

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (63)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]