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Dynamics of 1D discontinuous maps with multiple partitions and linear functions having the same fixed point. An application to financial market modeling

Laura Gardini, Davide Radi, Noemi Schmitt, Iryna Sushko, Frank Westerhoff

TL;DR

This paper analyzes a broad class of one-dimensional piecewise-linear discontinuous maps with multiple partitions that share a fixed point, showing that their bounded non-fixed-point dynamics reduce to those of a PWL circle map. Consequently, these dynamics comprise either nonhyperbolic cycles or quasiperiodic orbits densely filling intervals, with possible coexistence, and no true chaos (zero Lyapunov exponent). The authors apply the framework to a financial-market model, demonstrating endogenous price oscillations that appear chaotic but are fundamentally quasiperiodic. They also explore how perturbations can create true chaotic regimes, highlighting the distinction between apparent complexity and underlying regularity, with implications for modeling and policy in economics and beyond.

Abstract

Piecewise smooth systems are intensively studied today in many application areas, such as economics, finance, engineering, biology, and ecology. In this work, we consider a class of one-dimensional piecewise linear discontinuous maps with a finite number of partitions and functions sharing the same real fixed point. We show that the dynamics of this class of maps can be analyzed using the well-known piecewise linear circle map and prove that their bounded behavior, unrelated to the fixed point, may consist of either nonhyperbolic cycles or quasiperiodic orbits densely filling certain segments, with possible coexistence. A corresponding model describing the price dynamics of a financial market serves as an illustrative example. While simulated model dynamics may be mistaken for chaotic behavior, our results demonstrate that they are quasiperiodic.

Dynamics of 1D discontinuous maps with multiple partitions and linear functions having the same fixed point. An application to financial market modeling

TL;DR

This paper analyzes a broad class of one-dimensional piecewise-linear discontinuous maps with multiple partitions that share a fixed point, showing that their bounded non-fixed-point dynamics reduce to those of a PWL circle map. Consequently, these dynamics comprise either nonhyperbolic cycles or quasiperiodic orbits densely filling intervals, with possible coexistence, and no true chaos (zero Lyapunov exponent). The authors apply the framework to a financial-market model, demonstrating endogenous price oscillations that appear chaotic but are fundamentally quasiperiodic. They also explore how perturbations can create true chaotic regimes, highlighting the distinction between apparent complexity and underlying regularity, with implications for modeling and policy in economics and beyond.

Abstract

Piecewise smooth systems are intensively studied today in many application areas, such as economics, finance, engineering, biology, and ecology. In this work, we consider a class of one-dimensional piecewise linear discontinuous maps with a finite number of partitions and functions sharing the same real fixed point. We show that the dynamics of this class of maps can be analyzed using the well-known piecewise linear circle map and prove that their bounded behavior, unrelated to the fixed point, may consist of either nonhyperbolic cycles or quasiperiodic orbits densely filling certain segments, with possible coexistence. A corresponding model describing the price dynamics of a financial market serves as an illustrative example. While simulated model dynamics may be mistaken for chaotic behavior, our results demonstrate that they are quasiperiodic.

Paper Structure

This paper contains 9 sections, 2 theorems, 13 equations, 13 figures.

Table of Contents

  1. Introduction
  2. A simple nonlinear financial market model
  3. Preliminary properties of a PWL Lorenz map
  4. 1D discontinuous PWL homogeneous maps
  5. One discontinuity point
  6. Two discontinuity points
  7. Proof of Theorem 1
  8. Class of maps separating regular/chaotic behaviors
  9. Conclusions and future works

Key Result

Theorem 1

Let $x^{\prime}=G(x)$ be a 1D discontinuous PWL homogeneous map as in Definition 1. Then the following statements hold:

Figures (13)

  • Figure 1: One-dimensional bifurcation diagram of the map in (\ref{['new']}) as a function of slope $s_{R}$. In (a), $Z_{L}=0.6,$$Z_{R}=1,$$s_{L}=-0.9,$$s_{M}=-1.3.$ In (b), $Z_{L}=1,$$Z_{R}=0.6,$$s_{L}=-0.5,$$s_{M}=-1.1.$
  • Figure 2: PWL Lorenz map. In (a), a gap map; in (b), a circle map; in (c), an overlapping map.
  • Figure 3: Map $F$ in (\ref{['MapF']}) with $h=1$. Function $F$ is shown in black and its second iterate in azure. In (a), case (i1); in (b), case (i2); in (c), case (i3).
  • Figure 4: Map $G_{3}$ in (\ref{['new']}) with $Z_{L}=1,$$Z_{R}=0.5$, $s_{L}=-1.2,$$s_{M}=1.2,$$s_{R}=-1.1.$ In (b), first return map in a neighborhood of $Z_{L}=1$.
  • Figure 5: Map $G_{3}$ in (\ref{['new']}) with $Z_{L}=1,$$Z_{R}=0.7,$$s_{L}=0.9,$$s_{M}=-1.1,$$s_{R}=-0.5.$ In (b), first return map in a neighborhood of $Z_{R}=0.7$.
  • ...and 8 more figures

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Lemma 1