Relating high dimensional stochastic complex systems to low-dimensional intermittency

Abstract

We evaluate the implication and outlook of an unanticipated simplification in the macroscopic behavior of two high-dimensional sto-chastic models: the Replicator Model with Mutations and the Tangled Nature Model (TaNa) of evolutionary ecology. This simplification consists of the apparent display of low-dimensional dynamics in the nonstationary intermittent time evolution of the model on a coarse-grained scale. Evolution on this time scale spans generations of individuals, rather than single reproduction, death or mutation events. While a local one-dimensional map close to a tangent bifurcation can be derived from a mean-field version of the TaNa model, a nonlinear dynamical model consisting of successive tangent bifurcations generates time evolution patterns resembling those of the full TaNa model. To advance the interpretation of this finding, here we consider parallel results on a game-theoretic version of the TaNa model that in discrete time yields a coupled map lattice. This in turn is represented, a la Langevin, by a onedimensional nonlinear map. Among various kinds of behaviours we obtain intermittent evolution associated with tangent bifurcations. We discuss our results.

A zodiac of studies on complex systems

“…Surprisingly for us, the same replicator map in Eq. ( 29) was obtained after a major reduction [46] is made on a gametheoretic adaptation [85] of the Tangled Nature (TaNa) model [86] for evolutionary dynamics of ecological systems. The game-theoretic version of the TaNa model consists of a large payoff matrix so that the replicator equation involves a large frequency vector [85].…”

“…This model exhibits macroscopic nonstationary intermittent evolution similar to that in the TaNa model, and, interestingly, its discrete-time version that operates in the limit of many strategies constitutes a coupled map lattice (CML). Subsequently, a drastic simplification of these CML replicator-mutation equations was considered to reduce the model all the way down to a one-dimensional nonlinear map [46]. With this approximation, the possible connection was examined between the macroscopic intermittent behaviors of the above-mentioned high-dimensional models with the known low-dimensional sources of intermittency, such as the tangent bifurcation.…”

“…Games. The initial consideration of the discretetime version of the replicator equation of conventional games leads straightway to novel bifurcation diagrams and valuable coupled-map models [44][45][46]. Partitions.…”

Manifestations of the onset of chaos in condensed matter and complex systems

“…This model exhibits macroscopic non-stationary intermittent evolution similar to that in the TaNa model, and, interestingly, its discrete time version that operates in the limit of many strategies constitutes a coupled map lattice (CML). Subsequently, a major simplification of these CML replicatormutation equations was considered to reduce the model into a one-dimensional nonlinear map [48]. With this approximation the possible connection was examined between the macroscopic intermittent behaviors of the above-mentioned high-dimensional models with the low dimensional established sources of intermittency, such as the tangent bifurcation [1] with known 1/f noise spectra [31].…”

“…(b) Bifurcation diagram and Lyapunov exponent along the parameter line A ≡ S = T [47]. (c) Five-times-composed replicator map at the tangent bifurcation of its period five window with a close view inset[48]. (d) Trajectory for the conditions in (c) that imitates the behavioral patterns of a high-dimensional model for ecological evolution[48].…”