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Abstract: We state precisely and demonstrate two conjectures of R. Thomas following which a) the existence of a positive circuit in the oriented interaction graph of a dioeerential system is a necessary condition for the existence of several steady states, and b) the existence of a negative non-oriented circuit of length at least two is a necessary condition for the existence of a stable periodic orbit. (Update)
Context of citations to this paper: More
.... strictly greater than 1) in the generation of oscillations resulting in a stable limit cycle has in fact already been shown in [7] and [8] (boolean systems whose feedback circuits are all negative have no steady states, but stable cyclic trajectories [9] 3.2 Positive...
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BibTeX entry: (Update)
Gouze J.L., Positive and negative circuits in dynamical systems, J. Biol. Sys. 6 (1998), 11-15 21 http://citeseer.ist.psu.edu/gouze98positive.html More
@misc{ gouze98positive,
author = "J. Gouze",
title = "Positive and negative circuits in dynamical systems",
text = "Gouze J.L., Positive and negative circuits in dynamical systems, J. Biol.
Sys. 6 (1998), 11-15 21",
year = "1998",
url = "citeseer.ist.psu.edu/gouze98positive.html" }
Citations (may not include all citations):238 Nonnegative matrices in the mathematical sciences (context) - Berman, Plemmons - 1979
22 Structural Models : an introduction to the theory of directe.. (context) - Harary, Norman et al. - 1965
16 Monotone dynamical systems: an introduction to the theory of.. (context) - Smith - 1995
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3 volume 29 of Lecture Notes in Biomathematics (context) - Thomas, Logic - 1979
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2 stability and multistationarity in dynamical systems (context) - Plahte, Mestl et al. - 1995
2 Systems of ordinary dioeerential equations which generates a.. (context) - Smith - 1988
2 Necessary conditions for multistationarity and stable period.. (context) - Snoussi - 1997
2 the relation between the logical structure of systems and th.. (context) - Thomas - 1981
1 Dioeerential equations and dynamical systems (context) - Perko - 1991
1 Systems of dioeerential equations that are competitive or co.. (context) - Hirsch - 1985
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