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

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 f_i(t+τ) = f_i(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 x_i(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 DT_λ = ∂x_i/∂x_j ^(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 DT_λ cross the unit circle:

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

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

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