This Research Topic is for invited contributions from Frontiers Applied Mathematics and Statistics' speciality section Dynamical Systems.

University of Urbino Carlo Bo

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Frontiers in Applied Mathematics and Statistics

Dynamical Systems

Frontiers in Dynamical Systems: Invited Contributions

In this paper we present an overview of the results concerning dynamics of a piecewise linear bimodal map. The organizing principles of the bifurcation structures in both regular and chaotic domains of the parameter space of the map are discussed. In addition to the previously reported structures, a family of regions closely related to the so-called U-sequence is described. The boundaries of distinct regions belonging to these structures are obtained analytically using the skew tent map and the map replacement technique.

In this paper we present an overview of the results concerning dynamics of a piecewise linear bimodal map. The organizing principles of the bifurcation structures in both regular and chaotic domains of the parameter space of the map are discussed. In addition to the previously reported structures, a family of regions closely related to the so-called U-sequence is described. The boundaries of distinct regions belonging to these structures are obtained analytically using the skew tent map and the map replacement technique.

Bifurcation Structures in a Bimodal Piecewise Linear Map

Single-product oligopolies without product differentiation are examined with linear production, production adjustment, flexible workforce and investment costs. The price function is assumed to be hyperbolic which makes the non-linearity of the model much stronger than in the case of linear price function examined earlier in the literature. The best responses of the firms are determined which are not monotonic in contrast to the linear case. The set of all steady states is then characterized and in the case of a duopoly it is illustrated. The asymptotical behavior of the steady states is examined by using simulation. We analyze the effects of the different types of costs on the industry dynamics and compare them to the prediction by the well known model with hyperbolic price function and no product adjustment and investments costs.

Single-product oligopolies without product differentiation are examined with linear production, production adjustment, flexible workforce and investment costs. The price function is assumed to be hyperbolic which makes the non-linearity of the model much stronger than in the case of linear price function examined earlier in the literature. The best responses of the firms are determined which are not monotonic in contrast to the linear case. The set of all steady states is then characterized and in the case of a duopoly it is illustrated. The asymptotical behavior of the steady states is examined by using simulation. We analyze the effects of the different types of costs on the industry dynamics and compare them to the prediction by the well known model with hyperbolic price function and no product adjustment and investments costs.

Extended Dynamic Oligopolies with Flexible Workforce and Isoelastic Price Function

This paper considers a continuous-time heterogeneous agent model of a financial market with one risky asset, two types of agents (i.e., the fundamentalists and the chartists), and three time delays. The chartist's demand is determined through a nonlinear function of the difference between the current price and a weighted moving average of the delayed prices whereas the fundamentalist's demand is governed by the difference between the current price and the fundamental value. The asset price dynamics is described by a nonlinear delay differential equation. Two main results are analytically and numerically shown: (i) a single delay destabilizes the market price and generates cyclic oscillations around the equilibrium;</list-item>(i) under multiple delays, stability loss and gain repeatedly occur as the length of the delay increases.</list-item></list>

This paper considers a continuous-time heterogeneous agent model of a financial market with one risky asset, two types of agents (i.e., the fundamentalists and the chartists), and three time delays. The chartist&apos;s demand is determined through a nonlinear function of the difference between the current price and a weighted moving average of the delayed prices whereas the fundamentalist&apos;s demand is governed by the difference between the current price and the fundamental value. The asset price dynamics is described by a nonlinear delay differential equation. Two main results are analytically and numerically shown:(i) a single delay destabilizes the market price and generates cyclic oscillations around the equilibrium;(i) under multiple delays, stability loss and gain repeatedly occur as the length of the delay increases.

A Heterogeneous Agent Model of Asset Price with Three Time Delays

Let I = (a, b) and L be a nowhere dense perfect set containing the ends of the interval I and let φ : I → ℝ be a non-increasing continuous surjection constant on the components of I \ L and the closures of these components be the maximal intervals of constancy of φ. The family {Ft, t ∈ ℝ} of the interval-valued functions Ft(x): = φ−1[t + φ(x)], x ∈ I forms a set-valued iteration group. We determine a maximal dense subgroup T ⊊ ℝ such that the set-valued subgroup {Ft, t ∈ T} has some regular properties. In particular, the mappings T ∍ t → Ft(x) for t ∈ T possess selections ft(x) ∈ Ft(x), which are disjoint group of continuous functions.

Let I = (a, b) and L be a nowhere dense perfect set containing the ends of the interval I and let φ : I → ℝ be a non-increasing continuous surjection constant on the components of I \ L and the closures of these components be the maximal intervals of constancy of φ. The family {Ft, t ∈ ℝ} of the interval-valued functions Ft(x): = φ−1[t + φ(x)], x ∈ I forms a set-valued iteration group. We determine a maximal dense subgroup T ⊊ ℝ such that the set-valued subgroup {Ft, t ∈ T} has some regular properties. In particular, the mappings T ∍ t → Ft(x) for t ∈ T possess selections ft(x) ∈ Ft(x), which are disjoint group of continuous functions.

On Singular Interval-Valued Iteration Groups

The goal of this paper is to review some work on agent-based financial market models in which the dynamics is driven by piecewise-linear maps. As we will see, such models allow deep analytical insights into the functioning of financial markets, may give rise to unexpected dynamics effects, allow explaining a number of important stylized facts of financial markets, and offer novel policy recommendations. However, much remains to be done in this rather new research field. We hope that our paper attracts more scientists to this area.

The goal of this paper is to review some work on agent-based financial market models in which the dynamics is driven by piecewise-linear maps. As we will see, such models allow deep analytical insights into the functioning of financial markets, may give rise to unexpected dynamics effects, allow explaining a number of important stylized facts of financial markets, and offer novel policy recommendations. However, much remains to be done in this rather new research field. We hope that our paper attracts more scientists to this area.