О КАЛИБРОВКЕ АВТОНОМНОЙ МОДЕЛИ ТУНДРОВОЙ БИОЛОГИЧЕСКОЙ ПОПУЛЯЦИИ ЛЕММИНГОВ

The paper presents an autonomous model of the biological population of lemmings of a phenomenological type, designed in complex studies of tundra communities. In the model, the dynamics of a population is described through a difference equation relating the population in two neighboring years that depends on three parameters of the biological and ecological genesis. The combination of parameter values included in the equation under consideration determines a class of one-dimensional unimodal mappings of a dynamical system in which the bifurcation properties, asymptotics, and stability of trajectories were analytically and numerically studied. In this paper the main focus is made on model identification criterion. The method of identification sets is proposed to be used for calibration of the model. The identification sets method is based on the approximation and visualization of small-dimensional projections of a multidimensional graph of the error function given in the space of three environmental and two population parameters. This paper describes a case study of model identification using data on the tundra lemming population on the Taimyr Peninsula. It is shown that in this case two biological and ecological parameters allow for stable location distribution.

THE BUNDLE PLOT: EVOLUTION OF SYMBOLIC SPACE UNDER THE SYSTEM PARAMETER CHANGES

This paper provides a topological dynamics perspective on the full bifurcation unfolding in unimodal mappings. We present a bundle structure, visualized as a bundle plot, to show the evolution of symbolic space as we vary a system parameter. The bundle plot can be viewed as a limit process of an assignment plot, which are line assignments between points from two dynamical systems. Such line assignments are determined by a commuter, which is a coordinates transformation function that satisfies a commuting relationship but not necessarily a homeomorphism. The bundle structure is studied by understanding the implication of the system's qualitative changes. In addition, the case of the bundle plot with higher dimensional parameter variation is also considered. A main concern in the bundle plot is a special structure, called "joint", which determines a critical value of the parameter where the kneading sequence becomes periodic.

BRANCHING PROCESSES AND COMPUTATIONAL COLLAPSE OF DISCRETIZED UNIMODAL MAPPINGS

In computer simulations of smooth dynamical systems, the original phase space is replaced by machine arithmetic, which is a finite set. The resulting spatially discretized dynamical systems do not inherit all functional properties of the original systems, such as surjectivity and existence of absolutely continuous invariant measures. This can lead to computational collapse to fixed points or short cycles. The paper studies loss of such properties in spatial discretizations of dynamical systems induced by unimodal mappings of the unit interval. The problem reduces to studying set-valued negative semitrajectories of the discretized system. As the grid is refined, the asymptotic behavior of the cardinality structure of the semitrajectories follows probabilistic laws corresponding to a branching process. The transition probabilities of this process are explicitly calculated. These results are illustrated by the example of the discretized logistic mapping.

Asymptotic rigidity of scaling ratios for critical circle mappings

Let $f$ be a smooth homeomorphism of the circle having one cubic-exponent critical point and irrational rotation number of bounded combinatorial type. Using certain pull-back and quasiconformal surgical techniques, we prove that the scaling ratios of $f$ about the critical point are asymptotically independent of $f$. This settles in particular the golden mean universality conjecture. We introduce the notion of a holomorphic commuting pair, a complex dynamical system that, in the analytic case, represents an extension of $f$ to the complex plane and behaves somewhat as a quadratic-like mapping. We define a suitable renormalization operator that acts on such objects. Through careful analysis of the family of entire mappings given by $z\mapsto z+\theta -(1/2\pi)\sin{2\pi z}$, $\theta$ real, we construct examples of holomorphic commuting pairs, from which certain necessary limit set pre-rigidity results are extracted. The rigidity problem for $f$ is thereby reduced to one of renormalization convergence. We handle this last problem by means of Teichmüller extremal methods made available through the recent work of Sullivan on Riemann surface laminations and renormalization of unimodal mappings.