Stable oscillations in mathematical models of biological control systems

  • Leon Glass
  • Joel S. Pasternack


Oscillations in a class of piecewise linear (PL) equations which have been proposed to model biological control systems are considered. The flows in phase space determined by the PL equations can be classified by a directed graph, called a state transition diagram, on anN-cube. Each vertex of theN-cube corresponds to an orthant in phase space and each edge corresponds to an open boundary between neighboring orthants. If the state transition diagram contains a certain configuration called a cyclic attractor, then we prove that for the associated PL equation, all trajectories in the regions of phase space corresponding to the cyclic attractor either (i) approach a unique stable limit cycle attractor, or (ii) approach the origin, in the limitt→∞. An algebraic criterion is given to distinguish the two cases. Equations which can be used to model feedback inhibition are introduced to illustrate the techniques.

Key words

Biological control systems Limit cycles Feedback inhibition 


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Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Leon Glass
    • 1
  • Joel S. Pasternack
    • 2
  1. 1.Department of PhysiologyMcGill UniversityMontrealCanada
  2. 2.Department of Mathematics and The School of MedicineThe University of RochesterRochesterUSA

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