Journal of Mathematical Biology

, Volume 63, Issue 3, pp 593–600 | Cite as

Stable periodicity and negative circuits in differential systems

  • Adrien Richard
  • Jean-Paul Comet


We provide a counter-example to a conjecture of René Thomas on the relationship between negative feedback circuits and stable periodicity in ordinary differential equation systems (Kaufman et al. in J Theor Biol 248:675–685, 2007). We also prove a weak version of this conjecture by using a theorem of Snoussi.


Feedback circuit Regulatory network Oscillation Jacobian matrix Interaction graph 

Mathematics Subject Classification (2000)

34C10 92C42 


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  1. Braun M (1993) Differential equations and their applications. Springer, BerlinzbMATHGoogle Scholar
  2. Cinquin P, Demongeot J (2002) Positive and negative feedback: striking a balance between necessary antagonists. J Theor Biol 216: 229–241MathSciNetCrossRefGoogle Scholar
  3. de Jong H (2002) Modeling and simulation of genetic regulatory systems: a literature review. J Comp Biol 9: 67–103CrossRefGoogle Scholar
  4. Gouzé JL (1998) Positive and negative circuits in dynamical systems. J Biol Syst 6: 11–15zbMATHCrossRefGoogle Scholar
  5. Kaufman M, Soulé C, Thomas R (2007) A new necessary condition on interaction graphs for multistationarity. J Theor Biol 248: 675–685CrossRefGoogle Scholar
  6. Perko L (2002) Differential equations and dynamical systems. Springer, BerlinGoogle Scholar
  7. Plahte E, Mestl T, Omholt WS (1995) Feedback circuits, stability and multistationarity in dynamical systems. J Biol Syst 3: 409–413CrossRefGoogle Scholar
  8. Remy E, Ruet P, Thieffry D (2008) Graphics requirement for multistability and attractive cycles in a boolean dynamical framework. Adv Appl Math 41: 335–350MathSciNetzbMATHCrossRefGoogle Scholar
  9. Richard A (2010) Negative circuits and sustained oscillations in asynchronous automata networks. Adv Appl Math 44: 378–392MathSciNetzbMATHCrossRefGoogle Scholar
  10. Richard A, Comet JP (2007) Necessary conditions for multistationarity in discrete dynamical systems. Discrete Appl Math 155: 2403–2413MathSciNetzbMATHCrossRefGoogle Scholar
  11. Snoussi EH (1998) Necessary conditions for multistationarity and stable periodicity. J Biol Syst 6: 3–9zbMATHCrossRefGoogle Scholar
  12. Soulé C (2003) Graphic requirements for multistationarity. ComPlexUs 1: 123–133CrossRefGoogle Scholar
  13. Thomas R (1981) On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillations. Synergetics, vol 9, pp 180–193. Springer, BerlinGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.I3S, UMR 6070 CNRS, Université de Nice-Sophia AntipolisSophia AntipolisFrance

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