Attractors in coherent systems of differential equations

Journal of Differential Equations

Volumn 246, Issue 8, 2009, Pages 3058-3076

  Angeli, David ^a,b   Hirsch, Morris W ^c,d   Sontag, Eduardo D ^e  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

^b UNIVERSITY OF FLORENCE   (Italy)

^c UNIVERSITY OF CALIFORNIA   (United States)

^d UNIVERSITY OF WISCONSIN MADISON   (United States)

^e RUTGERS UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 61849090861     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2009.01.025     Document Type: Article

References (52)

1. Akin E. The General Topology of Dynamical Systems (1993), Amer. Math. Soc., Providence, RI
2. Alon U. An Introduction to Systems Biology: Design Principles of Biological Circuits (2006), Chapman & Hall
3. Auslander J., Bhatia N.P., and Seibert P. Attractors in dynamical systems. Bol. Soc. Mat. Mexicana (2) 9 (1964) 55-66
4. Bhatia N. On asymptotic stability in dynamical systems. Math. Syst. Theory 1 (1967) 113-128
5. H. Bothe, Review of Milnor [27], Math. Reviews
6. n. Math. Syst. Theory 1 (1967) 129-133
7. Coddington E.A., and Levinson N. Theory of Ordinary Differential Equations (1955), McGraw-Hill, New York
8. Conley C.C. Isolated Invariant Sets and the Morse Index. Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society. CBMS Reg. Conf. Ser. Math. vol. 38 (1978), Amer. Math. Soc., Providence, RI
9. Constantin P., Foias C., Nicolaenko B., and Temam R. Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations. Appl. Math. Sci. vol. 70 (1989), Springer-Verlag, New York
10. DasGupta B., Enciso G.A., Sontag E.D., and Zhang Y. Algorithmic and complexity aspects of decompositions of biological networks into monotone subsystems. Biosystems 90 (2007) 161-178
11. Feinberg M. Chemical reaction network structure and the stability of complex isothermal reactors, I. The deficiency zero and deficiency one theorems. Chem. Eng. Sci. 2 (1987) 2229-2268
12. Gouze J.L. Positive and negative circuits in dynamical systems. J. Biol. Syst. 6 (1998) 11-15
13. Hadeler K.P., and Glas D. Quasimonotone systems and convergence to equilibrium in a population genetic model. J. Math. Anal. Appl. 95 (1983) 297-303
14. Hale J.K. Asymptotic Behavior of Dissipative Systems. Math. Surveys Monogr. vol. 25 (1988), Amer. Math. Soc., Providence, RI
15. R. Hamming, Review of Stein and Ulam [43], Math. Reviews #6666 29 (1965) 1248
16. Hirsch M.W. Systems of differential equations which are competitive or cooperative. I: Limit sets. SIAM J. Math. Anal. 13 (1982) 167-179
17. Hirsch M.W. The dynamical systems approach to differential equations. Bull. Amer. Math. Soc. 11 (1984) 1-64
18. Hirsch M.W., and Smith H.L. Monotone dynamical systems. In: Cañada A., Drábek P., and Fonda A. (Eds). Handbook of Differential Equations: Ordinary Differential Equations, vol. 2 (2005), Elsevier North-Holland, Boston 239-258
19. Hofbauer J., and So J. Multiple limit cycles for three dimensional competitive Lotka-Volterra equations. Appl. Math. Lett. 7 (1994) 65-70
20. Hüffner F., Betzler N., and Niedermeier R. Optimal edge deletions for signed graph balancing. Proceedings of the 6th Workshop on Experimental Algorithms (WEA07). June 6-8, 2007, Rome (2007), Springer-Verlag, Heidelberg
21. Kamke E. Zur Theorie der Systeme gewöhnlicher Differentialgleichungen II. Acta Math. 58 (1932) 57-85
22. Ladyzhenskaya O. Attractors for Semigroups and Evolution Equations (1991), Cambridge Univ. Press, Cambridge, UK
23. Levinson N., and Smith O. A general equation for relaxation oscillations. Duke Math. J. 9 (1942) 382-403
24. Lorenz E. Deterministic nonperiodic flows. J. Atmospheric Sci. 20 (1963) 130-141
25. Lu Z., and Luo Y. Three limit cycles for a three-dimensional Lotka-Volterra competitive system with a heteroclinic cycle. Comput. Math. Appl. 46 (2003) 231-238
26. Mendelson P. On unstable attractors. Bol. Soc. Mat. Mexicana (2) 5 (1960) 270-276
27. Milnor J. On the concept of attractor. Comm. Math. Phys. 99 (1985) 177-195
28. Milo R., Shen-Orr S., Itzkovitz S., Kashtan N., Chklovskii D., and Alon U. Network motifs: Simple building blocks of complex networks. Science 298 (2002) 824-827
29. Müller M. Über das Fundamenthaltheorem in der Theorie der gewöhnlichen Differentialgleichungen. Math. Z. 26 (1926) 619-645
30. m, m ≥ 3. Comm. Math. Phys. 64 (1978-1979) 35-40
31. Pigolotti S., Krishna S., and Jensen M.H. Oscillation patterns in negative feedback loops. Proc. Natl. Acad. Sci. USA 104 (2007) 6533-6537
32. Plykin R.V., and Zhirov A. Some problems of attractors of dynamical systems. Topology Appl. 54 (1993) 19-46
33. Ruelle D. Strange attractors as a mathematical explanation of turbulence. Statistical Models and Turbulence (Proc. Sympos., Univ. California). La Jolla, CA, 1971. Lecture Notes in Phys. vol. 12 (1972), Springer-Verlag, Berlin
34. Ruelle D. Turbulence, Strange Attractors, and Chaos. World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises vol. 16 (1995), World Scientific, River Edge, NJ
35. Ruelle D., and Takens F. On the nature of turbulence. Comm. Math. Phys. 20 (1971) 167-192
36. Ruelle D., and Takens F. Note concerning our paper: "On the nature of turbulence". Comm. Math. Phys. 23 (1971) 343-344
37. Sell G. Periodic solutions and asymptotic stability. J. Differential Equations 2 (1966) 143-157
38. Sell G., and You Y. Dynamics of Evolutionary Equations. Appl. Math. Sci. vol. 143 (2002), Springer-Verlag, New York
39. Smale S. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967) 747-817
40. Snoussi E.H. Necessary conditions for multistationarity and stable periodicity. J. Biol. Syst. 6 (1998) 3-9
41. Sontag E.D. Monotone and near-monotone biochemical networks. Systems and Synthetic Biology 1 (2007) 59-87
42. P.R. Stein, S.M. Ulam, Quadratic transformations, part I, Los Alamos Report LA2305, Washington, Office of Technical Services, US Department of Commerce, 1959
43. Stein P.R., and Ulam S.M. Non-Linear Transformation Studies on Electronic Computers. Dissertationes Math. (Rozprawy Mat.) vol. xxxix (1964), Państwowe Wydawnictwo Naukowe, Warsaw
44. Thom R. Stabilité structurelle et morphogénèse; essai d'une théorie generale des modéles (1972), W.A. Benjamin, Reading, MA
45. Thom R. Structural Stability and Morphogenesis: An Outline of a General Theory of Models (1975), W.A. Benjamin, Reading, MA
46. Thomas R. On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillations. Springer Ser. Synergetics vol. 9 (1981) 180-193
47. Turing A.M. The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. Ser. B 237 (1952) 37-72
48. Ura T. Sur les courbes définies par les equations differentielles dans l'éspace à m dimensions. Ann. Sci. Ecole Norm. Sup. (3) 70 (1953) 287-360
49. Xiao D., and Li W. Limit cycles for the competitive three dimensional Lotka-Volterra system. J. Differential Equations 164 (2000) 1-15
50. Ward L. Partially ordered topological spaces. Proc. Amer. Math. Soc. 5 (1954) 144-161
51. R.F. Williams, Expanding attractors, in: Colloque de Topologie Différentielle, Mont-Aigoual, 1969, Université de Montpellier, Montpellier, pp. 79-89
52. http://www.weizmann.ac.il/mcb/UriAlon/Papers/networkMotifs/yeastData.mat