Dauby, Pierre ; Université de Liège - ULiège > Département d'astrophys., géophysique et océanographie (AGO) > Thermodynamique des phénomènes irréversibles
Language :
English
Title :
Bifurcation analysis of a periodically forced relaxation oscillator: Differential model versus phase-resetting map
Publication date :
2009
Journal title :
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
ISSN :
1539-3755
eISSN :
1550-2376
Publisher :
American Physical Society, College Park, United States - Maryland
J. Guckenheimer, K. Hoffman, and W. Weckesser, SIAM J. Appl. Dyn. Syst. 2, 1 (2003). 10.1137/S1111111102404738
R. FitzHugh, Biophys. J. 1, 445 (1961).
K. F. Bonhoeffer, J. Gen. Physiol. 32, 69 (1948). 10.1085/jgp.32.1.69
L. Glass and M. C. Mackey, From Clocks to Chaos (Princeton University Press, Princeton, NJ, 1988).
S. Scott, Ph.D. thesis, State University of New York, Buffalo, New York, 1979.
M. R. Guevara, L. Glass, M. C. Mackey, and A. Shrier, IEEE Trans. Syst. Man Cybern. 13, 790 (1983).
T. Nomura, S. Sato, S. Doi, J. P. Segundo, and M. D. Stiber, Biol. Cybern. 69, 429 (1993).
T. Nomura, S. Sato, S. Doi, J. P. Segundo, and M. D. Stiber, Biol. Cybern. 72, 55 (1994). 10.1007/BF00206238
K. Yoshino, T. Nomura, K. Pakdaman, and S. Sato, Phys. Rev. E 59, 956 (1999). 10.1103/PhysRevE.59.956
A. Rabinovitch and M. Friedman, Phys. Lett. A 355, 319 (2006). 10.1016/j.physleta.2006.02.059
E. A. Jackson, Perspectives of Nonlinear Dynamics (Cambridge University Press, Cambridge, U.K., 1990), Vol. 1.
R. Mettin, U. Parlitz, and W. Lauterborn, Int. J. Bifurcation Chaos Appl. Sci. Eng. 3, 1529 (1993). 10.1142/S0218127493001203
E. J. Ding, Phys. Scr. 38, 9 (1988). 10.1088/0031-8949/38/1/001
C. Hayashi, H. Shibayama, and Y. Nishikawa, IRE Trans. Circuit Theory 7, 413 (1960).
C. Hayashi, Nonlinear Oscillations in Physical Systems (McGraw-Hill, New York, 1964).
K. Tomita and T. Kai, J. Stat. Phys. 21, 65 (1979). 10.1007/BF01011482
Y. Ueda and N. Akamatsu, IEEE Trans. Circuits Syst. CAS-28, 217 (1981). 10.1109/TCS.1981.1084975
M. R. Guevara and L. Glass, J. Math. Biol. 14, 1 (1982). 10.1007/BF02154750
D. L. Gonzalez and O. Piro, Phys. Rev. Lett. 50, 870 (1983). 10.1103/PhysRevLett.50.870
H. Kawakami, IEEE Trans. Circuits Syst. CAS-31, 248 (1984). 10.1109/TCS.1984.1085495
D. G. Aronson, R. P. McGehee, I. G. Kevrekidis, and R. Aris, Phys. Rev. A 33, 2190 (1986). 10.1103/PhysRevA.33.2190
I. G. Kevrekidis, L. D. Schmidt, and R. Aris, Chem. Eng. Sci. 41, 1263 (1986). 10.1016/0009-2509(86)87099-3
I. G. Kevrekidis, R. Aris, and L. D. Schmidt, Chem. Eng. Sci. 41, 1549 (1986). 10.1016/0009-2509(86)85237-X
I. G. Kevrekidis, R. Aris, and L. D. Schmidt, Physica D 23, 391 (1986). 10.1016/0167-2789(86)90145-4
E. J. Ding, Phys. Rev. A 35, 2669 (1987). 10.1103/PhysRevA.35.2669
E. J. Ding, Phys. Rev. A 36, 1488 (1987). 10.1103/PhysRevA.36.1488
I. G. Kevrekidis, AIChE J. 33, 1850 (1987). 10.1002/aic.690331112
I. Schreiber, M. Dolník, P. Choc, and M. Marek, Phys. Lett. A 128, 66 (1988). 10.1016/0375-9601(88)91045-6
M. Dolník, J. Finkeová, I. Schreiber, and M. Marek, J. Phys. Chem. 93, 2764 (1989). 10.1021/j100344a015
W. Vance and J. Ross, J. Chem. Phys. 91, 7654 (1989). 10.1063/1.457235
B. B. Peckham, Nonlinearity 3, 261 (1990). 10.1088/0951-7715/3/2/002
C. Knudsen, J. Sturis, and J. S. Thomsen, Phys. Rev. A 44, 3503 (1991). 10.1103/PhysRevA.44.3503
L. Glass and J. Sun, Phys. Rev. E 50, 5077 (1994). 10.1103/PhysRevE.50. 5077
N. Inaba, R. Fujimoto, H. Kawakami, and T. Yoshinaga, Electron. Commun. Jpn., Part 2: Electron. 83, 35 (2000).
M. Sekikawa, N. Inaba, T. Yoshinaga, and H. Kawakami, Electron. Commun Jpn Part 3: Fundam. Electron. Sci. 87, 30 (2004). 10.1002/ecjc.10126
F. Schilder and B. B. Peckham, J. Comput. Phys. 220, 932 (2007). 10.1016/j.jcp.2006.05.041
M. Feingold, D. L. Gonzalez, O. Piro, and H. Viturro, Phys. Rev. A 37, 4060 (1988). 10.1103/PhysRevA.37.4060
J. C. Alexander, E. S. Doedel, and H. G. Othmer, SIAM J. Appl. Math. 50, 1373 (1990). 10.1137/0150082
S. Sato and S. Doi, Math. Biosci. 112, 243 (1992). 10.1016/0025-5564(92) 90026-S
A. Rabinovitch, R. Thieberger, and M. Friedman, Phys. Rev. E 50, 1572 (1994). 10.1103/PhysRevE.50.1572
S. Doi and S. Sato, Math. Biosci. 125, 229 (1995). 10.1016/0025-5564(94) 00035-X
D. T. Kaplan, J. R. Clay, T. Manning, L. Glass, M. R. Guevara, and A. Shrier, Phys. Rev. Lett. 76, 4074 (1996). 10.1103/PhysRevLett.76.4074
M. Xie, H. G. Othmer, and M. Watanabe, Physica D 98, 75 (1996). 10.1016/0167-2789(96)00092-9
H. G. Othmer and M. Xie, J. Math. Biol. 39, 139 (1999). 10.1007/s002850050166
S. Coombes and A. H. Osbaldestin, Phys. Rev. E 62, 4057 (2000). 10.1103/PhysRevE.62.4057
E. N. Cytrynbaum, J. Theor. Biol. 229, 69 (2004). 10.1016/j.jtbi.2004.03. 004
H. Croisier and P. C. Dauby, J. Theor. Biol. 246, 430 (2007). 10.1016/j.jtbi.2007.01.017
L. Glass and M. C. Mackey, J. Math. Biol. 7, 339 (1979). 10.1007/BF00275153
L. Glass and R. Perez, Phys. Rev. Lett. 48, 1772 (1982). 10.1103/PhysRevLett.48.1772
L. Glass and J. Bélair, in Nonlinear Oscillations in Biology and Chemistry, edited by, H. Othmer, Lectures Notes in Biomathematics Vol. 66, (Springer-Verlag, Berlin, 1986), pp. 232-243.
D. H. Perkel, J. H. Schulman, T. H. Bullock, G. P. Moore, and J. P. Segundo, Science 145, 61 (1964). 10.1126/science.145.3627.61
M. R. Guevara, L. Glass, and A. Shrier, Science 214, 1350 (1981). 10.1126/science.7313693
L. Glass, M. R. Guevara, J. Belair, and A. Shrier, Phys. Rev. A 29, 1348 (1984). 10.1103/PhysRevA.29.1348
A. Campbell, A. Gonzalez, D. L. Gonzalez, O. Piro, and H. A. Larrondo, Physica A 155, 565 (1989). 10.1016/0378-4371(89)90006-X
J. M. Anumonwo, M. Delmar, A. Vinet, D. C. Michaels, and J. Jalife, Circ. Res. 68, 1138 (1991).
M. R. Guevara, A. Shrier, and L. Glass, in Cardiac Electrophysiology: From Cell to Bedside, 1st ed., edited by, D. P. Zipes, and, J. Jalife, (Saunders, Philadelphia, 1990), pp. 192-201.
G. A. Cecchi, D. L. Gonzalez, M. O. Magnasco, G. B. Mindlin, O. Piro, and A. J. Santillan, Chaos 3, 51 (1993). 10.1063/1.165978
A. T. Winfree, The Geometry of Biological Time, 2nd ed. (Springer, New York, 2000).
L. Glass, Nature (London) 410, 277 (2001). 10.1038/35065745
J. Blair and L. Glass, Physica D 16, 143 (1985). 10.1016/0167-2789(85) 90055-7
E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede, and X. Wang, AUTO97: Continuation and bifurcation software for ordinary differential equations, 1998, http://cmvl.cs.concordia.ca/publications/ auto97.ps.gz.
B. B. Peckham and I. G. Kevrekidis, Nonlinearity 15, 405 (2002). 10.1088/0951-7715/15/2/310
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer, New-York, 1990).
N. Fenichel, Indiana Univ. Math. J. 21, 193 (1971). 10.1512/iumj.1971.21. 21017
S. Wiggins, Normally Hyperbolic Invariant Manifolds in Dynamical Systems (Springer-Verlag, New York, 1994).
H. Poincaré, Oeuvres I (Gauthier-Villar, Paris, 1928).
G. R. Hall, SIAM J. Math. Anal. 15, 1075 (1984). 10.1137/0515083
V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations (Springer, New York, 1983).
R. P. McGehee and B. B. Peckham, Int. J. Bifurcation Chaos Appl. Sci. Eng. 6, 315 (1996). 10.1142/S0218127496000072
J. Guckenheimer, K. Hoffman, and W. Weckesser, Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, 2669 (2000). 10.1142/S0218127400001742
M. R. Guevara, A. Shrier, and Å. Glass, Am. J. Physiol. 251, H1298 (1986).
M. R. Guevara and H. J. Jongsma, Am. J. Physiol. 258, H734 (1990).
D. G. Tsalikakis, H. G. Zhang, D. I. Fotiadis, G. P. Kremmydas, and L. K. Michalis, Comput. Biol. Med. 37, 8 (2007). 10.1016/j.compbiomed.2005.09.011
M. R. Guevara and A. Shrier, Biophys. J. 52, 165 (1987).
M. Kawato, J. Math. Biol. 12, 13 (1981). 10.1007/BF00275200
See EPAPS Document No. E-PLEEE8-78-102812. For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html.
B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz, and O. Junge, Int. J. Bifurcation Chaos Appl. Sci. Eng. 15, 763 (2005). 10.1142/S0218127405012533
L. Glass and A. T. Winfree, Am. J. Physiol. 246, R251 (1984).
D. L. Gonzalez and O. Piro, Phys. Lett. 101, 455 (1984). 10.1016/0375-9601(84)90417-1
J. P. Keener and L. Glass, J. Math. Biol. 21, 175 (1984). 10.1007/BF00277669
T. Krogh-Madsen, L. Glass, E. J. Doedel, and M. R. Guevara, J. Theor. Biol. 230, 499 (2004). 10.1016/j.jtbi.2004.03.027
B. van der Pol and J. van der Mark, Nature (London) 120, 363 (1927). 10.1038/120363a0
R. Perez and L. Glass, Phys. Lett. 90A, 441 (1982).
H. Gonzalez, H. Arce, and M. R. Guevara, Phys. Rev. E 78, 036217 (2008). 10.1103/PhysRevE.78.036217
P. Rehmus, W. Vance, and J. Ross, J. Chem. Phys. 80, 3373 (1984). 10.1063/1.447091
M. Rivera, P. Parmananda, and M. Eiswirth, Phys. Rev. E 65, 025201 (R) (2002). 10.1103/PhysRevE.65.025201
B. van der Pol, Philos. Mag. 3, 65 (1927).
T. Kostova, R. Ravindran, and M. Schonbek, Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, 913 (2004). 10.1142/S0218127404009685