Temporal Order pp 197-200

# 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

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