Characteristics of the Invariant Measure of the Strange Attractor of the Bacteria Mathematical Model

TL;DR

The paper treats bacterial metabolism as an open dissipative system and analyzes its chaotic dynamics by constructing a 10-variable nonlinear ODE model. It defines and computes an invariant measure for the system's strange attractor, examining convergence, robustness to sampling, and the density of trajectory intersections with the maximal-measure region. The key finding is that the invariant measure and its convergence reflect adaptive transitions in metabolism, maintaining metabolite levels near their average despite environmental changes. This provides a quantitative framework for characterizing transitional metabolic adaptation in bacteria through phase-space measures of strange attractors.

Abstract

The bacteria metabolic process of open nonlinear dissipative system far from equilibrium point is modeled using classical methods of synergetics. The invariant measure and its convergence in the phase space of the system was obtained in strange attractor mode. The distribution of point density of trajectory intersection of phase space cells with maximum invariant measure and convergence in time of its average value was obtained. The result concluded is that the value of an invariant measure can be a characteristic of the transitional process of adaptation of cell metabolic process to change outside environment.

Figures (4)

• Figure 1: The main scheme of a cell metabolic process
• Figure 2: Graph of convergence of the invariant measure of the strange attractor for the system $13 \times 2^x$ ($\alpha=0.03217)$, where: a -- a projection of the phase portrait of the attractor in 3d phase space $E_1, G, B$; b -- a histogram of the projection of the invariant measure of the strange attractor onto the plane $G, E_1$
• Figure 3: The evolution for density points distribution of intersection strange attractor trajectory $13 \times 2^x$ cells of phase space with maximum invariant measure for $N=200^{10}$ cells: $a$ ($\sum n =465$, $t=4\times 10^6$), $b$ ($\sum n =2298$, $t=2\times 10^7$), $c$ ($\sum n =4592$, $t=4\times 10^7$); for $N=1000^{10}$ cells: $d$ ($\sum n =598$, $t=4\times 10^7$), $e$ ($\sum n =3103$, $t=2\times 10^8$), $f$ ($\sum n =6110$, $t=4\times 10^8$)
• Figure 4: The graph of convergence of the mean over time for strange attractor of the system $13 \times 2^x$ in the cell with maximum invariant measure ($N=1000^{10}$, $\sum n =3103$, $t=2\times 10^8$)