Diaz-Ruelas, A; Jensen, HJ; Piovani, D; Robledo, A; (2016) Tangent map intermittency as an approximate analysis of intermittency in a high dimensional fully stochastic dynamical system: The Tangled Nature model. Chaos , 26 (12) , Article 123105. 10.1063/1.4968207.

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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. Intermittent dynamics in the form of long periods of little change separated by relatively short time intervals of hectic activity is observed in many complex systems. Examples include snoring, mass extinctions, financial crashes, and brain activity. Such systems contain large numbers of components and very often involve stochastic processes. It was more than 40 years ago suggested by Procaccia and Schuster that important aspects of such intermittent dynamics can be captured by a simple essentially deterministic equation. It is not in general straightforward to connect in detail the complex system to the mathematics considered by Procaccia and Schuster. We consider a multi-component model of evolutionary ecology and derive how the single component equation of the type considered by Procaccia and Schuster is related to the parameters of the stochastic many component dynamics. The single component description enables us to describe aspects of the intermittent extinction dynamics that so far has eluded mathematical analysis. We also demonstrate that the single component mathematics is able to qualitatively mimic the evolution and extinction dynamics of consecutive ecologies generated by the full many component model. We think that our results expands the applicability of the analysis put forward by Procaccia and Schuster and thereby help connect the methodology developed for deterministic and typically low dimensional dynamics to the stochastic dynamics of complex systems.

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