Liénard systems and potential-Hamiltonian decomposition. II: Algorithm. (English. French summary) Zbl 1111.37010
The authors propose a method for choosing a decomposition of a system of ordinary differential equations on the plane that models a biological phenomenon into the sum of a gradient and a Hamiltonian vector field. The division of parameters appearing in separate parts then elucidates the roles that they play in the underlying biological system.
For part I see: ibid., No. 2, 121-126 (2007; Zbl 1111.37011). For part III see: ibid., No. 4, 253–258 (2007; Zbl 1111.37012).
For part I see: ibid., No. 2, 121-126 (2007; Zbl 1111.37011). For part III see: ibid., No. 4, 253–258 (2007; Zbl 1111.37012).
Reviewer: Douglas S. Shafer (Charlotte)
References:
| [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., 33, 825-834 (2003) |
| [2] | Aracena, J.; Demongeot, J.; Goles, E., Mathematical modelling in genetic networks, IEEE Trans. Neural Networks, 15, 77-83 (2004) |
| [3] | Aracena, J.; Demongeot, J.; Goles, E., Fixed points and maximal independent sets on AND-OR networks, Discr. Appl. Math., 138, 277-288 (2004) · Zbl 1076.68047 |
| [4] | Aracena, J.; Demongeot, J.; Goles, E., On limit-cycles of monotone functions with symmetric connection graphs, Theoret. Comp. Sci., 322, 237-244 (2004) · Zbl 1068.68097 |
| [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., 330, 867-872 (2000) · Zbl 0960.35100 |
| [6] | Cinquin, O.; Demongeot, J., Positive and negative feedback: striking a balance between necessary antagonists, J. Theoret. Biol., 216, 229-241 (2002) |
| [7] | Cinquin, O.; Demongeot, J., Positive and negative feedback: mending the ways of sloppy systems, C. R. Acad. Sci. Biologies, 325, 1085-1095 (2002) |
| [8] | Demongeot, J., Modeling the genetic expression: positive interaction loops and genetic regulatory systems, (Mathématiques et Biologie (2002), Société Mathématique de France: Société Mathématique de France Paris), 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:10.1016/j.crma.2006.10.016; J. Demongeot, N. Glade, L. Forest, Liénard systems and potential-Hamiltonian decomposition I - methodology, C. R. Acad. Sci. Paris, Ser. I, DOI:10.1016/j.crma.2006.10.016 · Zbl 1111.37011 |
| [10] | Françoise, J. P., Oscillations en biologie. Analyse qualitative et modèles (2005), Springer: Springer Berlin · Zbl 1080.37103 |
| [11] | Giacomini, H.; Neukirch, S., Algebraic approximations to bifurcation curves of limit-cycles for the Liénard equation, Phys. Lett. A, 244, 53-58 (1998) · Zbl 0946.34035 |
| [12] | Hilbert, D., Sur les problèmes futurs des mathématiques : les 23 problèmes (1990), Jacques Gabay: Jacques Gabay Paris · JFM 32.0084.06 |
| [13] | Kopell, N.; Ermentrout, G. B., On chain of coupled oscillators forced at one end, SIAM J. Appl. Math., 51, 1397-1417 (1991) · Zbl 0748.34020 |
| [14] | Thom, R., Esquisse d’une sémiophysique (1988), Interéditions: Interéditions Paris |
| [15] | Timoteo Carletti, T.; Villari, G., A note on existence and uniqueness of limit-cycles for Liénard systems, J. Math. Anal. Appl., 307, 763-773 (2005) · Zbl 1081.34028 |
| [16] | Tonnelier, A.; Meignen, S.; Bosch, H.; Demongeot, J., Synchronization and desynchronization of neural oscillators: comparison of two models, Neural Networks, 12, 1213-1228 (1999) |
| [17] | Winfree, A. T., An integrated view of the resetting of a circadian clock, J. Theor. Biol., 28, 327-374 (1970) |
| [18] | Winfree, A. T., The Geometry of the Biological Time (1980), Springer-Verlag: Springer-Verlag Berlin · Zbl 0464.92001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
