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A Generalized 2D-Dynamical Mean-Field Ising Model with a Rich Set of Bifurcations (Inspired and Applied to Financial Crises)

12 Pages Posted: 16 Nov 2017  

Damian Smug

University of Exeter

Didier Sornette

ETH Zürich - Department of Management, Technology, and Economics (D-MTEC); Swiss Finance Institute

Peter Ashwin

University of Exeter

Date Written: October 31, 2017

Abstract

We analyse an extended version of the dynamical mean-field Ising model. Instead of classical physical representation of spins and external magnetic field, the model describes traders’ opinion dynamics. The external field is endogenised to represent a smoothed moving average of the past state variable. This model captures in a simple set-up the interplay between instantaneous social imitation and past trends in social coordinations. We show the existence of a rich set of bifurcations as a function of the two parameters quantifying the relative importance of instantaneous versus past social opinions on the formation of the next value of the state variable. Moreover, we present thorough analysis of chaotic behaviour, which is exhibited in certain parameter regimes. Finally, we examine several transitions through bifurcation curves and study how they could be understood as specific market scenarios. We find that the amplitude of the corrections needed to recover from a crisis and to push the system back to “normal” is often significantly larger than the strength of the causes that led to the crisis itself.

Keywords: Ising model, dynamic map, social opinion dynamics, bifurcation diagram, chaos, regime shifts, bifurcation delay

JEL Classification: C32, G01, C73

Suggested Citation

Smug, Damian and Sornette, Didier and Ashwin, Peter, A Generalized 2D-Dynamical Mean-Field Ising Model with a Rich Set of Bifurcations (Inspired and Applied to Financial Crises) (October 31, 2017). Swiss Finance Institute Research Paper No. 17-34. Available at SSRN: https://ssrn.com/abstract=3064673

Damian Smug (Contact Author)

University of Exeter ( email )

Northcote House
The Queen's Drive
Exeter, Devon EX4 4QJ
United Kingdom

Didier Sornette

ETH Zürich - Department of Management, Technology, and Economics (D-MTEC) ( email )

Scheuchzerstrasse 7
Zurich, ZURICH CH-8092
Switzerland
41446328917 (Phone)
41446321914 (Fax)

HOME PAGE: http://www.er.ethz.ch/

Swiss Finance Institute ( email )

c/o University of Geneve
40, Bd du Pont-d'Arve
1211 Geneva, CH-6900
Switzerland

Peter Ashwin

University of Exeter ( email )

Northcote House
The Queen's Drive
Exeter, Devon EX4 4QJ
United Kingdom

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