Temporal Order pp 197-200 | Cite as

Computation of Bifurcation Diagrams for Selkov’s Model of Glycolytic Oscillations

  • Bruno Eckhardt
  • Peter Richter
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 29)

Abstract

Consider a system of N first order differential equations
$$ \frac{d}{{dt}}{x_i} = {f_i}\left( {{x_1}, \ldots, {x_N},\lambda, t} \right) $$
where fi(t+τ) = fi(t) and λ is an additional parameter. For certain ranges of λ and τ the long-time behaviour of the solution may be stationary or periodic, for others it may be quasi-periodic or chaotic. We study bifurcations of periodic orbits by using the following mapping (stroboscopic map; Poincaré section): Let xi(x(o), λ, t) be the solution to initial condition x(o). Then
$$ {T_{\lambda }}:{x^{{(0)}}} \to x\left( {{x^{{(0)}}},\lambda, \tau } \right) $$
The Jacobian DTλ = ∂xi/∂xj (o) gives information on the stability of the orbit:

all eigenvalues|<1:stable

|at least one eigenvalue|>1: unstable

Different bifurcations arise from different ways in which eigenvalues of DTλ cross the unit circle:
  1. Type D

    one eigenvalue = +1; saddle-node bifurcation: a stable and an unstable orbit collide and annihilate each other; intermittency (Pomeau-Manneville)

     
  2. Type I

    one eigenvalue = -1; period doubling: an orbit of period n becomes unstable while a stable orbit of period 2n appears (Feigenbaum)

     
  3. Type H

    two complex conjugate eigenvalues e±iφ, φ 0,π; a new frequency shows up in the motion (Hopf)

     

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References

  1. [1]
    H. Kawakami, IEEE Circuits and Systems, 31, 248–260 (1984)CrossRefMATHADSMathSciNetGoogle Scholar
  2. [2]
    P.H. Richter, P. Rehmus, J. Ross, Prog. Theor. Phys. 66, 285 (1981)CrossRefGoogle Scholar
  3. [3]
    P.H. Richter, Physica 10D, 353 (1984)ADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Bruno Eckhardt
    • 1
  • Peter Richter
    • 1
  1. 1.Forschungsschwerpunkt “Dynamische Systeme” Fachbereich PhysikUniversität BremenBremen 33Fed. Rep. of Germany

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