A new class of Hamiltonian flows with random-walk behavior originating from zero-sum games and Fictitious Play

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Title: A new class of Hamiltonian flows with random-walk behavior originating from zero-sum games and Fictitious Play
Authors: Strien, SV
Item Type: Journal Article
Abstract: In this paper we introduce Hamiltonian dynamics, inspired by zero-sum games (best response and fictitious play dynamics). The Hamiltonian functions we consider are continuous and piecewise affine (and of a very simple form). It follows that the corresponding Hamiltonian vector fields are discontinuous and multi-valued. Differential equations with discontinuities along a hyperplane are often called 'Filippov systems', and there is a large literature on such systems, see for example (di Bernardo et al 2008 Theory and applications Piecewise-Smooth Dynamical Systems (Applied Mathematical Sciences vol 163) (London: Springer); Kunze 2000 Non-Smooth Dynamical Systems (Lecture Notes in Mathematics vol 1744) (Berlin: Springer); Leine and Nijmeijer 2004 Dynamics and Bifurcations of Non-smooth Mechanical Systems (Lecture Notes in Applied and Computational Mechanics vol 18) (Berlin: Springer)). The special feature of the systems we consider here is that they have discontinuities along a large number of intersecting hyperplanes. Nevertheless, somewhat surprisingly, the flow corresponding to such a vector field exists, is unique and continuous. We believe that these vector fields deserve attention, because it turns out that the resulting dynamics are rather different from those found in more classically defined Hamiltonian dynamics. The vector field is extremely simple: outside codimension-one hyperplanes it is piecewise constant and so the flow phgrt piecewise a translation (without stationary points). Even so, the dynamics can be rather rich and complicated as a detailed study of specific examples show (see for example theorems 7.1 and 7.2 and also (Ostrovski and van Strien 2011 Regular Chaotic Dynf. 16 129–54)). In the last two sections of the paper we give some applications to game theory, and finish with posing a version of the Palis conjecture in the context of the class of non-smooth systems studied in this paper.
Issue Date: 30-Jun-2011
Date of Acceptance: 25-Mar-2011
URI: http://hdl.handle.net/10044/1/64002
DOI: https://dx.doi.org/10.1088/0951-7715/24/6/002
ISSN: 0951-7715
Publisher: IOP Publishing
Start Page: 1715
End Page: 1742
Journal / Book Title: Nonlinearity
Volume: 24
Issue: 6
Copyright Statement: © 2011 IOP Publishing Ltd. This is an author-created, un-copyedited version of an article accepted for publication in Nonlinearity. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher authenticated version is available online at https://dx.doi.org/10.1088/0951-7715/24/6/002.
Keywords: math.DS
37Jxx, 37Dxx, 91Axx
0102 Applied Mathematics
General Mathematics
Publication Status: Published
Online Publication Date: 2011-04-21
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences

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