Dynamical Systems
Liénard systems and potential-Hamiltonian decomposition II – algorithm
Comptes Rendus. Mathématique, Volume 344 (2007) no. 3, pp. 191-194.

We show here how to approach with an increasing precision the limit-cycles of Liénard systems, bifurcating from a stable stationary state, by contour lines of Hamiltonian systems derived from a potential-Hamiltonian decomposition of the Liénard flow. We evoke the case (non polynomial) of pure potential systems (n-switches) and pure Hamiltonian systems (2D Lotka–Volterra), and we show that, with the proposed approximation, we can deal with the case of mixed systems (van der Pol or FitzHugh-Nagumo) frequently used for modelling oscillatory systems in biology. We suggest finally that the proposed algorithm, generic for PH-decomposition, can be used for estimating the isochronal fibration in some specific cases near the pure potential or Hamiltonian systems. In a following Note, we will give applications in biology of the potential-Hamiltonian decomposition.

Nous montrons ici comment approcher, avec une précision croissante, les cycles limites des systèmes de Liénard par les courbes de niveau du système hamiltonien obtenu à partir d'une décomposition potentielle-hamiltonienne. Nous présentons des cas non polynomiaux de systèmes purement potentiels (type n-switch) et purement hamiltoniens (type Lotka–Volterra 2D), puis nous donnons des exemples de décomposition potentielle-hamiltonienne pour des systèmes de type van der Pol (mixtes) qui sont d'usage fréquent en modélisation des systèmes biologiques oscillants. Nous suggérons enfin que l'algorithme proposé, générique pour la décomposition potentielle-hamiltonienne, soit utilisé pour estimer la fibration isochrone, dans le cas de systèmes voisins des systèmes purs (potentiels ou hamiltoniens). Dans une Note future, nous décrirons des applications biologiques précises de la décomposition potentielle-hamiltonienne.

Published online:
DOI: 10.1016/j.crma.2006.10.013
Demongeot, Jacques 1, 2; Glade, Nicolas 2; Forest, Loic 2

1 Institut Universitaire de France, France
2 TIMC-IMAG UMR CNRS 5525, Faculty of Medicine, University J. Fourier, Grenoble, 38700 La Tronche, France
     author = {Demongeot, Jacques and Glade, Nicolas and Forest, Loic},
     title = {Li\'enard systems and {potential-Hamiltonian} decomposition {II} {\textendash} algorithm},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {191--194},
     publisher = {Elsevier},
     volume = {344},
     number = {3},
     year = {2007},
     doi = {10.1016/j.crma.2006.10.013},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2006.10.013/}
AU  - Demongeot, Jacques
AU  - Glade, Nicolas
AU  - Forest, Loic
TI  - Liénard systems and potential-Hamiltonian decomposition II – algorithm
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 191
EP  - 194
VL  - 344
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2006.10.013/
DO  - 10.1016/j.crma.2006.10.013
LA  - en
ID  - CRMATH_2007__344_3_191_0
ER  - 
%0 Journal Article
%A Demongeot, Jacques
%A Glade, Nicolas
%A Forest, Loic
%T Liénard systems and potential-Hamiltonian decomposition II – algorithm
%J Comptes Rendus. Mathématique
%D 2007
%P 191-194
%V 344
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2006.10.013/
%R 10.1016/j.crma.2006.10.013
%G en
%F CRMATH_2007__344_3_191_0
Demongeot, Jacques; Glade, Nicolas; Forest, Loic. Liénard systems and potential-Hamiltonian decomposition II – algorithm. Comptes Rendus. Mathématique, Volume 344 (2007) no. 3, pp. 191-194. doi : 10.1016/j.crma.2006.10.013. http://www.numdam.org/articles/10.1016/j.crma.2006.10.013/

[1] Aracena, J.; Ben Lamine, S.; Mermet, M.A.; Cohen, O.; Demongeot, J. Mathematical modelling in genetic networks: relationships between the genetic expression and both chromosomic breakage and positive circuits, IEEE Trans. Systems Man Cyber., Volume 33 (2003), pp. 825-834

[2] Aracena, J.; Demongeot, J.; Goles, E. Mathematical modelling in genetic networks, IEEE Trans. Neural Networks, Volume 15 (2004), pp. 77-83

[3] Aracena, J.; Demongeot, J.; Goles, E. Fixed points and maximal independent sets on AND-OR networks, Discr. Appl. Math., Volume 138 (2004), pp. 277-288

[4] Aracena, J.; Demongeot, J.; Goles, E. On limit-cycles of monotone functions with symmetric connection graphs, Theoret. Comp. Sci., Volume 322 (2004), pp. 237-244

[5] Ben Abdallah, N.; Dolbeault, J. Relative Entropies For Kinetic Equations In Bounded Domains (irreversibility, stationary solutions, uniqueness), C. R. Acad. Sci. Paris, Sér. I Math., Volume 330 (2000), pp. 867-872

[6] Cinquin, O.; Demongeot, J. Positive and negative feedback: striking a balance between necessary antagonists, J. Theoret. Biol., Volume 216 (2002), pp. 229-241

[7] Cinquin, O.; Demongeot, J. Positive and negative feedback: mending the ways of sloppy systems, C. R. Acad. Sci. Biologies, Volume 325 (2002), pp. 1085-1095

[8] Demongeot, J. Modeling the genetic expression: positive interaction loops and genetic regulatory systems, Mathématiques et Biologie, Société Mathématique de France, Paris, 2002, pp. 67-94

[9] J. Demongeot, N. Glade, L. Forest, Liénard systems and potential-Hamiltonian decomposition I – methodology, C. R. Acad. Sci. Paris, Ser. I, | DOI

[10] Françoise, J.P. Oscillations en biologie. Analyse qualitative et modèles, Springer, Berlin, 2005

[11] Giacomini, H.; Neukirch, S. Algebraic approximations to bifurcation curves of limit-cycles for the Liénard equation, Phys. Lett. A, Volume 244 (1998), pp. 53-58

[12] Hilbert, D. Sur les problèmes futurs des mathématiques : les 23 problèmes, Jacques Gabay, Paris, 1990

[13] Kopell, N.; Ermentrout, G.B. On chain of coupled oscillators forced at one end, SIAM J. Appl. Math., Volume 51 (1991), pp. 1397-1417

[14] Thom, R. Esquisse d'une sémiophysique, Interéditions, Paris, 1988

[15] Timoteo Carletti, T.; Villari, G. A note on existence and uniqueness of limit-cycles for Liénard systems, J. Math. Anal. Appl., Volume 307 (2005), pp. 763-773

[16] Tonnelier, A.; Meignen, S.; Bosch, H.; Demongeot, J. Synchronization and desynchronization of neural oscillators: comparison of two models, Neural Networks, Volume 12 (1999), pp. 1213-1228

[17] Winfree, A.T. An integrated view of the resetting of a circadian clock, J. Theor. Biol., Volume 28 (1970), pp. 327-374

[18] Winfree, A.T. The Geometry of the Biological Time, Springer-Verlag, Berlin, 1980

Cited by Sources: