Deterministic chaos within the transfer space - An unstable fixed point as a narrow ford to complexity through chaos

The complete reinvestment of the net profit of a previous production cycle as substrate of the next cycle of a single party may result in deterministic chaos. The dynamics of such a feedback loop is controlled by the size relation of a benefit factor (serves also as complexity factor) and a cost factor. An increasing benefit factor or decreasing cost factor trigger bifurcation and deterministic chaos at certain size relations. In deterministic chaos the size of the net profit of the reinvestment is no longer reliable. Thus, a limit to the evolution of complexity via an increasing benefit factor and complete reinvestment would be expected. Chaos already starts when benefit exceeds cost; a sink. In a source cost exceeds benefit. Both conditions met, source and sink form an ensemble, peacefully transfer substrate when in contact, and produce superadditivity. At low substrate concentrations a sink has to pass through the region of chaos to become a source. To suppress chaotic behaviour, an ensemble could become active when on both sides benefit still exceeds cost; two sinks. The emerging superadditivity supports such a behaviour. In addition, the mathematical analysis of my model identifies a unique substrate concentration leading to an unstable fixed point. Notably, this concentration is independent of an increasing benefit factor and thus does not collide with evolution towards complexity. Moreover, this concentration is a turning point as the result of a further complete reinvestment no longer grows. This limit guides the ensemble through chaos towards complexity and division of labour by a sink and a source.