4. Mathematically, this corresponds to nonlinear dynamics in which equilibrium solutions change stability as parameters cross critical values.
5. Red Queen cycles as oscillatory attractors.
In some regimes, the system does not converge to a fixed point but exhibits limit cycles, where RNA and protein continually adapt in response to each other. These cycles reflect perpetual coevolutionary motion---molecular Red Queen dynamics.
6. Evolutionary implications.
Attractors explain why RNA--protein partnerships, once stabilized, are highly resilient to perturbations. Bifurcations explain why evolutionary innovation often appears abrupt, as systems transition between attractor basins rather than through gradual, linear improvement. Through this lens, the RNA--protein system is best understood as a nonlinear dynamical landscape with multiple attractors, oscillatory trajectories, and bifurcations. These emergent properties resolve the paradox of synchronized codes: coadaptation is not improbable, but a natural outcome of complex feedback dynamics.
V. ResultsÂ
A. Analytical results: stability and bifurcations
1. Reduced dynamical model (minimal CAS core)
To make analytic progress we study a low-dimensional reduction that retains the essential co-dependence between RNA and protein populations. Let R(t)R(t)R(t) denote the effective abundance (or normalized mean frequency) of RNA genotypes that contribute to coding capacity, and let P(t)P(t)P(t) denote the effective abundance of protein genotypes that provide structural/catalytic support. A compact phenomenological CAS model combining replicator--type growth, mutational loss, and mutual coupling is
R=R(sRFR(R,P)R),P=P(sPFP(R,P)P),(1)\begin{aligned} \dot R &= R\Big( s_R\,F_R(R,P) - \mu_R \Big),\\[4pt] \dot P &= P\Big( s_P\,F_P(R,P) - \mu_P \Big), \end{aligned} \tag{1}RP=R(sRFR(R,P)R),=P(sPFP(R,P)P),(1)
where
