Hysteresis in the trade cycle

Chaos in a seasonally perturbed SIR model: avian influenza in a seabird colony as a paradigm

“…The technique involves the construction of the translation operator, which in this case was the time-one map corresponding to the seasonally perturbed SIR model. The topological hyperbolicity method has been applied to other highly nonlinear systems through Poincaré maps (Pokrovskii et al 2007;Rasskazov 2003) and through mappings with strong nonlinearities (McNamara and Pokrovskii 2006). Therefore, while our study was motivated by a particular case of avian influenza in a seabird colony, we wish to emphasise that the methods employed in this paper are highly flexible and can be readily applied to a broad range of comparable "pathogen in population" deterministic models.…”

“…The topological hyperbolicity method can be applied to prove chaos in the Smale sense, see Guckenheimer and Holmes (1983) or Wiggins (2003) for an introduction to Smale-type chaos. Computer-assisted applications of the method were used to prove the existence of chaotic trajectories for the extended Korteweg-de Vries-Burgers wave equations (Cox et al 2005), for the Kaldor model of the trade cycle with a hysteretic nonlinearity (McNamara and Pokrovskii 2006) and for a piecewise linear oscillator (Pokrovskii et al 2007). A proof of the existence of chaos in a singularly perturbed system using the method, which was not computer-aided, was provided by Pokrovskii and Zhezherun (2008).…”

“…The topological hyperbolicity method can be applied to prove chaos in the Smale sense, i.e., to show the existence of a correspondence between the left shift mapping on the set of symbolic sequences and a restriction of some fixed iterate of the translation operator of the dynamical system under consideration to an invariant set. The method has been applied to prove the existence of chaotic trajectories for a wide variety of strongly nonlinear systems (Cox et al 2005;McNamara and Pokrovskii 2006;Pokrovskii et al 2007;Pokrovskii and Zhezherun 2008). We will illustrate the theory using the seasonally perturbed SIR model of the seabird-avian influenza system.…”

Hysteresis and economics

“…The implications of Preisach-based models for business cycles are considered in [37]. Methods for analyzing how recessions leave curses in their wake in the form of a lower growth-path for outputs, and how booms can leave blessings, in the form of a higher growth-path, are presented in [38] and [39], providing a framework for incorporating Preisach memory- It is unfortunate that the most common usage of the term "hysteresis" in economics differs significantly from that defined in this article or employed in the physical sciences.…”

Alexei Vadimovich Pokrovskii, 1948–2010

“…He aspired to make the theory available to, and useful for, problems in these new areas of application, in the same way, as it has already proved to be successful in more traditional fields such as magnetism, plasticity, material science, mechanical engineering and control design. He had a very productive collaboration focusing on modelling hysteresis in hydrology [43][44][45][46] with J. Philip O'Kane, Professor and Head of Department of Civil & Environmental Engineering in UCC with whom he co-supervised two PhD students; and with Rod Cross, Professor of Economics at University of Strathclyde with whom he developed models of hysteresis in macroeconomics [47][48][49][50]. Their memoirs [2,51] reflecting on Alexei's impact on these subjects are available in open access.…”