Tangent map intermittency as an approximate analysis of intermittency in a high dimensional fully stochastic dynamical system: The Tangled Nature model

Abstract

It is well known that low-dimensional nonlinear deterministic maps close to a tangent bifurcation exhibit intermittency and this circumstance has been exploited, e.g., by Procaccia and Schuster [Phys. Rev. A 28, 1210 (1983)], to develop a general theory of 1/f spectra. This suggests it is interesting to study the extent to which the behavior of a high-dimensional stochastic system can be described by such tangent maps. The Tangled Nature (TaNa) Model of evolutionary ecology is an ideal candidate for such a study, a significant model as it is capable of reproducing a broad range of the phenomenology of macroevolution and ecosystems. The TaNa model exhibits strong intermittency reminiscent of punctuated equilibrium and, like the fossil record of mass extinction, the intermittency in the model is found to be non-stationary, a feature typical of many complex systems. We derive a mean-field version for the evolution of the likelihood function controlling the reproduction of species and find a local map close to tangency. This mean-field map, by our own local approximation, is able to describe qualitatively only one episode of the intermittent dynamics of the full TaNa model. To complement this result, we construct a complete nonlinear dynamical system model consisting of successive tangent bifurcations that generates time evolution patterns resembling those of the full TaNa model in macroscopic scales. The switch from one tangent bifurcation to the next in the sequences produced in this model is stochastic in nature, based on criteria obtained from the local mean-field approximation, and capable of imitating the changing set of types of species and total population in the TaNa model. The model combines full deterministic dynamics with instantaneous parameter random jumps at stochastically drawn times. In spite of the limitations of our approach, which entails a drastic collapse of degrees of freedom, the description of a high-dimensional model system in terms of a low-dimensional one appears to be illuminating.

A zodiac of studies on complex systems

“…In short, the many-strategy gametheoretic problem was reduced to a classic version of two strategies, where one of them represents a selected agent or species and the other groups all the others. One advantage in using this approximation is that a one-dimensional nonlinear dynamical model had been recently constructed [45] such that its time evolution consists of successive tangent bifurcations that generate patterns resembling those of the full TaNa model in macroscopic scales. See 13b) and d).…”

“…We had already made use of the windows in the bifurcation diagram, but only marginally, by using the properties in the chaotic neighborhood of their tangent bifurcation borders, the existence of the infinite set of windows was used to construct a simple model [45,46] to reproduce the punctuated equilibrium [86] observed in the nature of evolutionary ecology, that is captured by Jensen's TaNa model [86]. But other insights are awaiting for the more complete use of the features of the sets of windows, for example, for the construction of simple enough models that would describe complex systems where the most significant property is embedding, nested systems within systems.…”

“…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.…”

Relating high dimensional stochastic complex systems to low-dimensional intermittency

“…and depends on the probability for no mutation P (0) mut , one mutation p (1) mut and the J coupling averaged over the highly occupied types of agentsJ and the average of the the couplings over these types and the set of types they are connected to via single mutations J. The details are in [5].…”

“…Namely, the emergence of low-dimensional behavior in the coarse-grained description of the high-dimensional dynamics. But very recently another example of this occurrence has been exhibited for the high-dimensional TaNa model under mean-field approximations [5], such that effective low-dimensional dynamics is displayed in the macroscopic non-stationary intermittent evolution of the model. Recently, a game-theoretic rendering of the TaNa model described by a set of coupled replicator equations that incorporate stochastic mutations has been derived and studied [6], and found to exhibit macroscopic non-stationary intermittent evolution similar to that in the TaNa model.…”