Max Planck Institute for Dynamics and Self-Organization -- Department for Nonlinear Dynamics and Network Dynamics Group
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MPI Kolloquium

Wednesday, 16.01.2013 15 c.t.

Discrete scale invariance and log-periodicity in material rupture, growth processes, turbulence, finance and other systems

by Prof. Dr. Didier Sornette
from ETH Zurich, Department of Management, Technology and Economics (D-MTEC), Zurich Switzerland

Contact person: Jan Nagler

Location

Ludwig Prandtl lecture hall

Abstract

We discuss the concept of discrete scale invariance and how it leads to complex critical exponents (or dimensions), i.e. to the log-periodic corrections to scaling. After their initial suggestion as formal solutions of renormalization group equations in the seventies, complex exponents have been studied in the eighties in relation to various problems of physics embedded in hierarchical systems. Only recently has it been realized that discrete scale invariance and its associated complex exponents may appear 'spontaneously' in euclidean systems, i.e. without the need for a pre-existing hierarchy. Examples are diffusion-limited-aggregation clusters, rupture in heterogeneous systems, earthquakes, animals (a generalization of percolation) among many other systems. We review the known mechanisms for the spontaneous generation of discrete scale invariance and provide an extensive list of situations where complex exponents have been found. This is done in order to provide a basis for a better fundamental understanding of discrete scale invariance. The main motivation to study discrete scale invariance and its signatures is that it provides new insights in the underlying mechanisms of scale invariance. We will describe the operation of the Financial Crisis Observatory, whose goal is to diagnose financial bubbles and crashes in advance, based on a technology derived from discrete scale invariance as fingerprints of critical instabilities.

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