Fabio Tramontana

BORDER-COLLISION BIFURCATIONS IN 1D PIECEWISE-LINEAR MAPS AND LEONOV'S APPROACH

50 years ago (1959) in a series of publications by Leonov, a detailed analytical study of the nested period adding bifurcation structure occurring in piecewise-linear discontinuous 1D maps was presented. The results obtained by Leonov are barely known, although they allow the analytical calculation of border-collision bifurcation subspaces in an elegant and much more efficient way than it is usually done. In this work we recall Leonovs approach and explain why it works. Furthermore, we slightly improve the approach by avoiding an unnecessary coordinate transformation, and also demonstrate that the approach can be used not only for the calculation of border-collision bifurcation curves.

PERIOD ADDING IN PIECEWISE LINEAR MAPS WITH TWO DISCONTINUITIES

In this work we consider the border collision bifurcations occurring in a one-dimensional piecewise linear map with two discontinuity points. The map, motivated by an economic application, is written in a generic form and considered in the stable regime, with all slopes between zero and one. We prove that the period adding structures occur in maps with more than one discontinuity points and that the Leonovs method to calculate the bifurcation curves forming these structures is applicable also in this case. We demonstrate the existence of particular codimension-2 bifurcation (big-bang bifurcation) points in the parameter space, from which infinitely many bifurcation curves are issuing associated with cycles involving several partitions. We describe how the bifurcation structure of a map with one discontinuity is modified by the introduction of a second discontinuity point, which causes orbits to appear located on three partitions and organized again in a period-adding structure. We also describe particular codimension-2 bifurcation points which represent limit sets of doubly infinite sequences of bifurcation curves and appear due to the existence of two discontinuities.

One-Dimensional Discontinuous Piecewise-Linear Maps and the Dynamics of Financial Markets

We present a simple one-dimensional discontinuous piecewise-linear agent-based financial market model in which prices evolve with respect to the trading activity of heterogeneous speculators. In line with empirical evidence, speculators rely on technical or fundamental trading rules to determine their orders. The general setup that comes out of our assumptions can be subdivided into various sub-models. We survey some analytical results obtained for these sub-models and illustrate how their deterministic skeletons are able to produce some important stylized facts of financial markets, including bubbles, crashes and excess volatility. We also develop and calibrate a stochastic version of the model that matches the dynamics of actual financial markets quite well. In fact, simulated returns are virtually unpredictable and display features like volatility clustering and long memory effects.

Original article: Border collision bifurcations in one-dimensional linear-hyperbolic maps

Abstract: In this paper we consider a continuous one-dimensional map, which is linear on one side of a generic kink point and hyperbolic on the other side. This kind of map is widely used in the applied context. Due to the simple expression of the two functions involved, in particular cases it is possible to determine analytically the border collision bifurcation curves that characterize the dynamic behaviors of the model. In the more general model we show that the steps to be performed are the same, although the analytical expressions are not given in explicit form.

A simple financial market model with chartists and fundamentalists

We present a simple financial market model with interacting chartists and fundamentalists. Since some speculators only become active when a certain misalignment level has been crossed, the model dynamics is driven by a discontinuous piecewise linear map. Recent mathematical techniques allow a comprehensive study of the models dynamical system. One of its surprising features is that model simulations may appear to be chaotic, although only regular dynamics can emerge. While our deterministic model is able to produce stylized bubbles and crashes we also show that a stochastic version of our model is able to match the finer details of financial market dynamics.

Two different routes to complex dynamics in an heterogeneous triopoly game

We study a triopoly game with heterogeneous players. The market is characterized by a nonlinear (isoelastic) demand function and three competitors. The main novelty is the double route to complex dynamics that we find and is quite rare in heterogeneous triopoly models. We show that the two routes have important implications for the economic interpretation of the dynamics emerging when the Cournot–Nash equilibrium becomes locally unstable. Moreover the model displays multistability of different attractors, requiring a global analysis of the dynamical system.

Global Bifurcations in a Three-Dimensional Financial Model of Bull and Bear Interactions

In a previous paper Tramontana et al. (2009), we developed a three-dimensional discrete-time dynamic model in which two stock markets of two countries, say H(ome) and A(broad), are linked via and with the foreign exchange market. The latter is modelled in the sense of Day and Huang (1990), i.e. it is characterized by a nonlinear interplay between technical traders (or chartists) and fundamental traders (or fundamentalists). In the absence of connections, the foreign exchange market is driven by the iteration of a one-dimensional cubic map, which has the potential to produce a regime of alternating and unpredictable bubbles and crashes for sufficiently large values of a key parameter, which captures the speculative behavior of chartists. Such a dynamic feature, first observed and explained by Day and Huang (1990) in their stylized model of financial market dynamics, can be understood with the help of bifurcation analysis: an initial situation of bi-stability (two coexisting, attracting non-fundamental steady states around an unstable fundamental equilibrium) evolves into coexistence of cycles or chaotic intervals within two disjoint bull and bear regions, which eventually merge via a homoclinic bifurcation. By introducing connections between markets (i.e. by allowing stock market traders to be active abroad), the endogenous fluctuations originating in one of the markets spread throughout the whole three-dimensional system. It therefore becomes interesting to investigate how the coupling of the markets affects the bull and bear dynamics of the model. With regard to this, in Tramontana et al. (2009) we already performed a thorough analytical and numerical study of two simplified lower-dimensional cases, where connections are either totally absent (each market evolves according to an independent one-dimensional map) or occur in one direction (a two-dimensional system evolves independently of the third dynamic equation). Also a short analysis of the stability of the equilibria of the three-dimensional model was there started, arguing that the global (homoclinic) bifurcations may still be a characteristic of the dynamics. This investigation is precisely the object of the present paper. We shall analyze the dynamic behavior of the complete three-dimensional model, following the approach adopted in Tramontana et al. (2009), based mainly on the numerical and graphical detection of the relevant global bifurcations. Although analytical conditions for such global bifurcation, mainly homoclinic bifurcations, are difficult to be formalized, their existence and occurrence can be numerically detected. As it is standard in the qualitative study of dynamic behaviors, the transverse crossing between stable and unstable sets of unstable cycles, leading to homoclinic trajectories, give numerical tools which may be considered as proofs in a given numerical example.

Original article: Bifurcation analysis of an inductorless chaos generator using 1D piecewise smooth map

In this work we investigate the dynamics of a one-dimensional piecewise smooth map, which represents the model of a chaos generator circuit. In a particular (symmetric) case analytic results can be given showing that the chaotic region is wide and robust. In the general model only the border collision bifurcation can be analytically determined. However, the dynamics behave in a similar way, leading effectively to robust chaos.

Consumo e consumatori di prodotti alimentari nella società postmoderna

The concept of the ‘postmodern consumer’ plays a central role in the debate, started in the early 80s, about economic, social and cultural changes in developed countries in the years following the end of the second world war. These changes were interpreted as a passage from modern to postmodern society. According to this literature, postmodern conditions have had a significant impact on the consumer, especially with regard to his/her psychological characteristics. In this new framework the consumer is viewed as someone more interested in the symbolic or cultural value of products and services than in their functional and utility value. At the same time, he/she is represented as an active player in the market scenario, exercising the freedom to move in search of trademarks, symbols and experiences through which he/she can communicate his/her own identity. The figure of the postmodern consumer is difficult to place in the framework of standard neoclassical theories on consumerism, which highlights the shortcomings of this theoretical approach in studying the behavior of the postmodern consumer. These shortcomings are likely to be more relevant when considering the consumer of food products, given the strong nexus between consumption and the well-being of the consumer and the symbolic and cultural value that food products project. The main goal of the paper is to provide an interdisciplinary overview of the postmodern consumer of food products by means of an analysis of scientific literature, mainly in the areas of behavioral economics, sociology and psychology. Following this, the paper focuses on questions regarding information and the rational behavior of consumers as being the main hypothesis upon which standard neoclassical theories are based, adding to the traditional approach to consumer choice the new insights provided by this different perspective. Finally, the implications of this type of analysis for food safety and quality policies are considered, together with a discussion on further research needed to define more effective policies.

Period adding structure in a 2D discontinuous model of economic growth

We study the dynamics of a growth model formulated in the tradition of Kaldor and Pasinetti where the accumulation of the ratio capital/workers is regulated by a two-dimensional discontinuous map with triangular structure. We determine analytically the border collision bifurcation boundaries of periodicity regions related to attracting cycles, showing that in a two-dimensional parameter plane these regions are organized in the period adding structure. We show that the cascade of flip bifurcations in the base one-dimensional map corresponds for the two-dimensional map to a sequence of pitchfork and flip bifurcations for cycles of even and odd periods, respectively.