The asymptotically stable equilibrium point. The stable equilibrium (Xeq) is illustrated by trajectories and phase orbits in the 18D space with 10% perturbations. The red line: the original curve with respect to the experimental initial value ; the yellow/blue lines: the curves whose initial value has a -10%/+10% perturbation from the experimental one; the triangular spots: projections of Xeq on the corresponding dimensions. (A) Asymptotical stability shown by trajectories. Each subplot represents a dimension in the 18D space, i.e. the kinetics of a metabolite. All trajectories eventually and consistently converge to the Xeq (projection on the corresponding dimension) although a 10% perturbation is in the initial value. The x-axis: time (s); y-axis: concentration (mM). (B) Asymptotical stability shown by phase orbits. Stability is more clearly illustrated in such presentation. We randomly chose 2 state variables (metabolites #7 and #8 in the plot) to form the phase orbit in the 2D subspace. Arrows denote the directions of orbits and they eventually and consistently converge to Xeq (projection on the 2D subspace, marked by the triangular spot). For other 2D subspaces, the orbit profiles are the same.
Li and Liu BMC Systems Biology 2012 6(Suppl 1):S11 doi:10.1186/1752-0509-6-S1-S11