Application of interval iterations to the entrainment problem in respiratory physiology

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 367, Issue 1908, 2009, Pages 4717-4739

  Demongeot, Jacques ^a   Waku, Jules ^a  

^a LABORATOIRE TIMC IMAG   (France)

Author keywords

Interval iterations; Intervals limit cycles; Invariant domain; Population dynamics; Respiratory entrainment; Set valued map

Indexed keywords

ASYMPTOTIC ANALYSIS; NUMERICAL ANALYSIS; POPULATION DYNAMICS; RESPIRATORS;

ASYMPTOTIC BEHAVIOUR; DYNAMICAL BEHAVIOURS; INTERVAL ITERATIONS; INTERVAL LIMITS; LIMIT CYCLE; NEIGHBOURHOOD; NUMERICAL ASPECTS; NUMERICAL RESULTS; SET-VALUED MAP; VERHULST;

SET THEORY;

ANIMAL; ARTICLE; BIOLOGICAL MODEL; HUMAN; MATHEMATICAL COMPUTING; RABBIT; RESPIRATORY FUNCTION;

ANIMALS; HUMANS; MODELS, BIOLOGICAL; NUMERICAL ANALYSIS, COMPUTER-ASSISTED; RABBITS; RESPIRATORY PHYSIOLOGICAL PROCESSES;

EID: 73349104942     PISSN: 1364503X     EISSN: None     Source Type: Journal    
DOI: 10.1098/rsta.2009.0177     Document Type: Article

References (34)

1. Aubin, J. & Frankowska, H. 1990 Set-valued analysis. Berlin, Germany: Birkhauser.
2. Chan, K. S. & Tong, H. 1994 A note on noisy chaos. J. R. Stat. Soc. 56, 301-311.
3. Cheban, D. & Mammana, C. 2005 Relation between different types of global attractors of set-valued nonautonomous dynamical systems. Set-Valued Anal. 13, 291-321. (doi:10.1007/s11228-004-0046-x)
4. Chen, G., Huang, X. & Yang, X. 2005 Vector optimization. Set-valued and variational analysis. Berlin, Germay: Springer Verlag.
5. Collet, P. & Eckmann, J. P. 1980 Iterated maps on the interval as dynamical systems. Berlin, Germany: Birkhauser.
6. Cosnard, M. 1983 Contributions à l'étude du comportement itératif des transformations unidimensionnelles. Grenoble, France: The&grave'se Université J. Fourier.
7. De Farias, D. P. & van Roy, B. 2000 On the existence of fixed points for approximate value iteration and temporal-difference learning. J. Optimiz. Theory Appl. 105, 589-608. (doi:10.1023/A:1004641123405)
8. Demongeot, J. & Leitner, F. 1996 Compact set valued flows I: applications in medical imaging. C. R. Acad. Sci. II B 323, 747-754.
9. Demongeot, J. & Waku, J. 2005 Counter-examples about lower and upper bounded population growth. Math. Popul. Stud. 12, 199-209. (doi:10.1080/08898480500301785)
10. Demongeot, J., Jacob, C. & Cinquin, P. 1987 Periodicity and chaos in biological systems: new tools for the study of attractors. Life Sci. Ser. Plenum 138, 255-266.
11. Demongeot, J., Kulesa, P. & Murray, J. D. 1997 Compact set valued flows II: applications in biological modelling. C. R. Acad. Sci. Paris, Série II b 324, 107-115. (doi:10.1016/s1251-8069(99)80014-1)
12. Desheng, L. & Kloeden, P. E. 2005 Equi-attraction and continuous dependence of strong attractors of set-valued dynamical systems on parameters. Set-Valued Anal. 13, 405-416. (doi:10.1007/s11228-005-2971-8)
13. Dobbs, N. 2007 Visible measures of maximal entropy in dimension one. Bull. Lond. Math. Soc. 39, 366-376. (doi:10.1112/blms/bdm005)
14. Graczyk, J., Sands, D. & Swiatek, G. 2004 Metric attractors for smooth unimodal maps. Ann. Math. 159, 725-740. (doi:10.4007/annals.2004.159.725)
15. Holmgren, R. A. 1996 A first course in discrete dynamical systems. Berlin, Germany: Springer Verlag.
16. Homburg, A. J. & Young, T. 2006 Hard bifurcations in dynamical systems with bounded random perturbations. Regul. Chaotic Dyn. 11, 247-258. (doi:10.1070/RD2006v011n02ABEH000348)
17. Hoppensteadt, F. C. 1993 Analysis and simulation ofchaotic systems. Berlin, Germany: Springer Verlag.
18. Kamran, T. 2007 Multi-valued f-weakly Picard mappings. Nonlinear Anal. 67, 2289-2296. (doi:10.1016/j.na.2006.09.010)
19. Kleptsyn, V. A. & Nalskii, M. B. 2004 Contraction of orbits in random dynamical systems on the circle. Funct. Anal. Appl. 38, 36-54. (doi:10.1007/s10688-005-0005-9)
20. Kloeden, P. E. & Valero, J. 2005 Attractors of weakly asymptotically compact set-valued dynamical systems. Set-Valued Anal. 13, 381-404. (doi:10.1007/s11228-004-0047-9)
21. Lange, L. 1986 Continued fraction applications to zero location. Lect. Notes Math. 1199, 220-262. Levy, P. 1939 L'addition des variables aléatoires définies sur une circonfé rence. Bull. Soc. Math. Fr. 67, 1-41.
22. May, R. M. 1976 Simple mathematical models with very complicated dynamics. Nature 261, 459-467. (doi:10.1038/261459a0)
23. May, R. M. & Oster, G. F. 1976 Bifurcations and dynamic complexity in simple ecological models. Am. Nat. 110, 573-599. (doi:10.1086/283092)
24. Murray, J. D. 2002 Mathematical biology, 3rd edn. New York, NY: Springer-Verlag.
25. Nadler, S. B. 1969 Multi-valued contraction mappings. Pac. J. Math. 30, 475-488.
26. Pham Dinh, T., Demongeot, J., Baconnier, P. & Benchetrit, G. 1983 Simulation of a biological oscillator: the respiratory rhythm. J. Theor. Biol. 103, 113-132. (doi:10.1016/0022-5193(83)90202-3)
27. Przytycki, F. 1987 Chaos after bifurcation of a Morse-Smale diffeomorphism through a one-cycle saddle-node and iterations of multivalued mappings of an interval and a circle. Bull. Braz. Math. Soc. 18, 29-79. (doi:10.1007/BF02584831)
28. Reich, S. & Zaslavski, A. J. 2002 Generic existence of fixed points for set-valued mappings. Set-Valued Anal. 10, 287-296. (doi:10.1023/A:1020602030873)
29. Schaffer, W. M., Eller S. & Kot, M. M. 1986 Effects of noise on some dynamical models in ecology. J. Math. Biol. 24, 479-523. (doi:10.1007/BF00275681)
30. Smajdor, A. 1985 Iterations of multi-valued functions. Katowice, Poland: Prace Naukowe Uniwersytetu Slaskiego.
31. Stenseth, N. C., Falck, W., Bjornstad, O. N. & Krebs, C. J. 1997 Population regulation in Snowshoe Hare and Canadian Lynx: asymmetric food web configurations between Hare and Lynx. Proc. Natl Acad. Sci. USA 94, 5147-5152. (doi:10.1073/pnas.94.10.5147)
32. Tong, H. 1983 Threshold models in non-linear time series analysis. New York, NY: Springer-Verlag.
33. Wlodarczyk, K., Plebaniak, R. & Obczynski, C. 2007 Endpoints of set-valued dynamical systems of asymptotic contractions of Meir-Keeler type and strict contractions in uniform spaces. Nonlinear Anal. 67, 1668-1679. (doi:10.1016/j.na.2006.07.039)
34. Wu, W. B. & Shao, X. 2004 Limit theorems for iterated random functions. J. Appl. Probab. 41, 425-436. (doi:10.1239/jap/ 1082999076)