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Abstract. The problem of investigation of temporal and/or spatial behavior of highly nonlinear or complex natural systems has long been of fundamental scientific interest. At the same time it is presently well understood that identification of dynamics of processes in complex natural systems, through their qualitative description and quantitative evaluation, is far from a purely academic question and has an essential practical importance. This is quite understandable as systems with complex dynamics abound in nature and examples can be found in very different areas such as medicine and biology (rhythms, physiological cycles, epidemics), atmosphere (climate and weather change), geophysics (tides, earthquakes, volcanoes, magnetic field variations), economy (financial markets behavior, exchange rates), engineering (friction, fracturing), communication (electronic networks, internet packet dynamics) etc. The past two decades of research on qualitative and especially quantitative investigations of dynamics of real processes of different origin brought significant progress in the understanding of behavior of natural processes. At the same time serious drawbacks have also been revealed. This is why exhaustive investigation of

. The standard way to reduce a flow to a time--discrete dynamical system is by the technique of the Poincare surface of section. We discuss the relationship between Poincare maps obtained from di#erent surfaces of section and give some considerations for practical application for this method. PACS numbers: 05.45.Ac, 02.30.Hq Short title: LETTER TO THE EDITOR December 10, 1999 + Present address: Max--Planck--Institut fur Stromungsforschung, Bunsenstrae 10, D--37073 Gottingen, Germany # e--mail: wolfram@chaos.gwdg.de 2 1. Introduction One century ago Poincare introduced the concept of time--discrete dynamical systems in his study of two--dimensional autonomous di#erential equations [1]. Such construction of Poincare maps has become one of the very basic tools in nonlinear dynamics and is nowadays contained in every textbook on nonlinear dynamics (c.f. e.g. [2]). The method is to a good deal at the heart of e.g. data analysis [3], control in nonlinear systems [4], and numerical ...

Transition from chaotic to quasi-periodic phase in modified Lorenz model is analyzed by performing the contact transformation such that the trajectory in R 3 is projected on R 2. The relative torsion number and the characteristics of the template are measured using the eigenvector of the Jacobian instead of vectors on moving frame along the closed trajectory. Application to the circulation of a fluid in a convection loop and oscillation of the electric field in single-mode laser system are performed. The time series of the eigenvalues of the Jacobian and the scatter plot of the trajectory in the transformed coordinate plane X − Z in the former and |X | − |Z | in the latter, allow to visualize characteristic pattern change at the transition from quasi-periodic to chaotic. In the case of single mode laser, we observe the correlation between the critical movement of the eigenvalues of the Jacobian in the complex plane and intermittency.

Abstract. This paper introduces the truncator map as a dynamical system on the space of configurations of an interacting particle system. We represent the symbolic dynamics generated by this system as a non-commutative algebra and classify its periodic orbits using properties of endomorphisms of the resulting algebraic structure. A stochastic model is constructed on these endomorphisms, which leads to the classification of the distribution of periodic orbits for random truncator maps. This framework is applied to investigate the periodic transitions of Bornholdt’s spin market model. 1. Model Description Let Ω = [−1, 1] N for some positive dimension N and consider a set {Sj} M j=1 of mutually exclusive and exhaustive subsets of Ω. A typical example will be the generalized quadrants, i.e. Sj = x ∈ Ω