Survival and weak chaos

Abstract

Survival analysis in biology and reliability theory in engineering concern the dynamical functioning of bio/electro/mechanical units. Here we incorporate effects of chaotic dynamics into the classical theory. Dynamical systems theory now distinguishes strong and weak chaos. Strong chaos generates Type II survivorship curves entirely as a result of the internal operation of the system, without any age-independent, external, random forces of mortality. Weak chaos exhibits (a) intermittency and (b) Type III survivorship, defined as a decreasing per capita mortality rate: engineering explicitly defines this pattern of decreasing hazard as 'infant mortality'. Weak chaos generates two phenomena from the normal functioning of the same system. First, infant mortality-sensu engineering-without any external explanatory factors, such as manufacturing defects, which is followed by increased average longevity of survivors. Second, sudden failure of units during their normal period of operation, before the onset of age-dependent mortality arising from senescence. The relevance of these phenomena encompasses, for example: no-fault-found failure of electronic devices; high rates of human early spontaneous miscarriage/abortion; runaway pacemakers; sudden cardiac death in young adults; bipolar disorder; and epilepsy.

Keywords: Pomeau–Manneville map; chaos; infant mortality; life-history theory; reliability theory; survival analysis.

Figure 1.

Survivorship curves I, II and III plotted using the Weibull distribution, k = 3, 1, 0.3, visually descending. As in ecological practice, S(t) is on a log scale, allowing the per capita survivorship rates of those still alive, and how this may be changing over time, to be visually inferred from the slope. The Pareto distribution—see text—is plotted for t_min = 0.02, k = 0.3.

Figure 2.

The map equation (3.1), with r = 4. This map was originally suggested by John von Neumann as a random number generator for use in the first computer programs [36].

Figure 3.

The logistic map, r = 4, is the parabola. The tent map, solid black, appears as a straightened version of the logistic, and the Bernoulli map as a broken tent map, in dashed red. We retain the graph wall to create an image of a box for a particle to bounce around in, according to the map rules.

Figure 4.

More maps in a box: Bernoulli map (red) and the Pomeau–Manneville map (blue).

Figure 5.

Three typical time trajectories of the Pomeau–Manneville map with initial conditions close to zero and parameters as described in the text. Repeated simulation suggests that no run ‘takes off’ before about 40 iterations.

Figure 6.

Mean remaining lifetime function of a typical run of the logistic map as previously described. Here, and throughout, the points are plotted at each failure event, as this information may also be informative to the reader.

Figure 7.

The MRL of a typical simulation of the Pomeau–Manneville map, with parameters as described in the text, grows rapidly.