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From two-dimensional continuous maps to one-dimensional discontinuous maps: a novel reduction explaining complex bifurcation structures in piecewise-linear families of maps

D. J. W. Simpson, V. Avrutin

TL;DR

This work develops a rigorous reduction of the two-dimensional border-collision normal form to a three-parameter one-dimensional discontinuous map near subsumed homoclinic bifurcations. It proves (Theorem th:main) that the first-return dynamics of the 2D map are well approximated by the 1D family $h(z;\eta,\nu,\sigma)$, and it systematically analyzes four canonical bifurcation structures—period-incrementing, period-adding, bandcount-adding, and bandcount-incrementing—in this reduced setting. Through explicit examples and numerical verification, the paper shows that $h$ reproduces the qualitative and, in many cases, quantitative features of the 2D dynamics, including the organization of attractors and the emergence of chaotic regimes. The findings offer a simple, illuminating lens on complex bifurcation patterns in piecewise-linear maps and suggest analogous reductions for higher-dimensional piecewise-linear systems near similar codimension-two phenomena.

Abstract

Piecewise-linear maps describe dynamical phenomena that switch between distinct states and readily generate complex bifurcation structures due to their strong nonlinearity. We show that two-dimensional continuous piecewise-linear maps near certain codimension-two homoclinic bifurcations are well approximated by a three-parameter family of one-dimensional maps. Each member of the one-dimensional family is discontinuous, because the family is constructed from the first return of iterates to a subset of phase space, and comprised of infinitely many linear pieces, where each piece corresponds to a fixed number of iterations near the saddle associated with the homoclinic bifurcation. The one-dimensional family exhibits period-incrementing, period-adding, bandcount-incrementing, and bandcount-adding structures (all typical for two-piece maps), as well as unique features caused by orbits repeatedly visiting more than two pieces of the map. These structures carry through to the two-dimensional maps with only minor differences in the arrangement of the bifurcations developing with the distance from the codimension-two bifurcations. This leads to a novel and vivid elucidation of the dynamics of the two-dimensional border-collision normal form.

From two-dimensional continuous maps to one-dimensional discontinuous maps: a novel reduction explaining complex bifurcation structures in piecewise-linear families of maps

TL;DR

This work develops a rigorous reduction of the two-dimensional border-collision normal form to a three-parameter one-dimensional discontinuous map near subsumed homoclinic bifurcations. It proves (Theorem th:main) that the first-return dynamics of the 2D map are well approximated by the 1D family , and it systematically analyzes four canonical bifurcation structures—period-incrementing, period-adding, bandcount-adding, and bandcount-incrementing—in this reduced setting. Through explicit examples and numerical verification, the paper shows that reproduces the qualitative and, in many cases, quantitative features of the 2D dynamics, including the organization of attractors and the emergence of chaotic regimes. The findings offer a simple, illuminating lens on complex bifurcation patterns in piecewise-linear maps and suggest analogous reductions for higher-dimensional piecewise-linear systems near similar codimension-two phenomena.

Abstract

Piecewise-linear maps describe dynamical phenomena that switch between distinct states and readily generate complex bifurcation structures due to their strong nonlinearity. We show that two-dimensional continuous piecewise-linear maps near certain codimension-two homoclinic bifurcations are well approximated by a three-parameter family of one-dimensional maps. Each member of the one-dimensional family is discontinuous, because the family is constructed from the first return of iterates to a subset of phase space, and comprised of infinitely many linear pieces, where each piece corresponds to a fixed number of iterations near the saddle associated with the homoclinic bifurcation. The one-dimensional family exhibits period-incrementing, period-adding, bandcount-incrementing, and bandcount-adding structures (all typical for two-piece maps), as well as unique features caused by orbits repeatedly visiting more than two pieces of the map. These structures carry through to the two-dimensional maps with only minor differences in the arrangement of the bifurcations developing with the distance from the codimension-two bifurcations. This leads to a novel and vivid elucidation of the dynamics of the two-dimensional border-collision normal form.

Paper Structure

This paper contains 22 sections, 6 theorems, 57 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Numerical methods
  3. Construction of the one-dimensional approximation
  4. First return to the third quadrant
  5. A saddle fixed point
  6. Coordinates relative to the eigendirections
  7. Main result
  8. Verification of the one-dimensional approximation
  9. Examples
  10. A minimal example
  11. An example with $\delta_L = 0$
  12. An example for a saddle periodic solution
  13. Basic properties of the one-dimensional family
  14. Scale invariance
  15. Number of branches over the absorbing interval
  16. ...and 7 more sections

Key Result

Theorem 1

Suppose for $\xi_0 \in \Xi$ there exists $m \ge 2$ such that $f_R^m(U) = S$ and $f_R^k(U) \in \Omega_R$ for all $k = 1,2,\ldots,m-1$. Then there exists a neighbourhood $\mathcal{N} \subset \mathbb{R}^4$ of $\xi_0$ and constants $C_1, C_2 > 0$ such that for all $\xi \in \mathcal{N} \cap \Xi$ we have where $z = b(P)$, $z' = b(F(P))$, and $c \in \mathbb{R} \cup \{ \infty \}$ is such that $\lambda \s

Figures (24)

  • Figure 1: A large-scale bifurcation set of the two-dimensional border-collision normal form $f$, given by \ref{['eq:f']}, with $\tau_L = 2$ and $\delta_L = 0.75$. In yellow regions $f$ has a periodic attractor, in blue regions $f$ has a chaotic attractor, and in white regions $f$ has no attractor. The orange curves are boundaries of regions where $f$ has a periodic attractor of fixed period. The red X's indicate parameter points of subsumed homoclinic connections. Refer to §\ref{['sec:numericalMethods']} for a description of the numerical methods used to generate this figure.
  • Figure 2: Subsumed homoclinic connections of the two-dimensional border-collision normal form \ref{['eq:f']}. Panel (a) uses $(\tau_L,\delta_L,\tau_R,\delta_R) = (2,0.75,-0.5,1.5)$ corresponding to the right-most X in Fig. \ref{['fig:bifSetLargeScale']}. The black square $Y$ is a saddle fixed point and its stable and unstable sets are coloured blue and red respectively. The vertical line is the switching line $\Sigma$; the horizontal line is its image $f(\Sigma)$; the dashed curves indicate the action of the map. Panel (b) uses $\mathopen{}\mathclose{\left( \tau_L, \delta_L, \tau_R, \delta_R \right) = \mathopen{}}\mathclose{\left( -\frac{23}{33}, \frac{13}{66}, -\frac{5}{2}, 2 \right)$, taken from Si20. The three black squares form a saddle period-three solution ($RLR$-cycle) and its stable and unstable sets are coloured blue and red respectively. There is also a stable period-three solution ($RLL$-cycle) indicated with blue squares.
  • Figure 3: Panel (a) is a magnification of Fig. \ref{['fig:bifSetLargeScale']}; panel (b) is a bifurcation set of the corresponding one-dimensional family $h$, given by \ref{['eq:h']}. Specifically, panel (b) uses $\sigma = 1.5$ and $\eta$ and $\nu$ given by \ref{['eq:eta2']} and \ref{['eq:nu2']}, with also $\lambda = 0.5$. For panel (a) [panel (b)], in yellow regions $f$ [$h$] has a periodic attractor, in blue regions $f$ [$h$] has a chaotic attractor, and in white regions $f$ [$h$] has no attractor. The orange curves are boundaries of regions where $f$ [$h$] has a periodic attractor of fixed period; the blue curves are boundaries of regions where $f$ [$h$] has a chaotic attractor with a fixed number of connected components. In (a) the roughly triangular regions $\mathcal{P}_k'$ are where $f$ has a stable periodic solution of period $p = k+2$; in (b) the triangular regions $\mathcal{P}_k$ are where $h$ has a stable fixed point $z_k^* \in I_k$. Both panels show the homoclinic corner curves $\eta = 0$ and $\nu = 0$. These curves intersect at the right-most X in Fig. \ref{['fig:bifSetLargeScale']}. In (a) the parameter points (i), (ii), (iii), and (iv) are examined in Figs. \ref{['fig:basins']}a, \ref{['fig:basins']}b, \ref{['fig:bandexA']}, and \ref{['fig:bandexB']} respectively.
  • Figure 4: Two instances of the one-dimensional map $h$, given by \ref{['eq:h']}. If $\eta > \nu$, as in (a), then every branch of $h$ is increasing. If $\eta < \nu$, as in (b), then every branch of $h$ is decreasing.
  • Figure 5: A sketch of the phase space of the two-dimensional border-collision normal form $f$ with $(\tau_L,\delta_L,\tau_R,\delta_R) = (2,0.75,-0.35,1.05)$. The $x \le 0$ parts of the stable and unstable subspaces $E^s(Y)$ and $E^u(Y)$ are coloured blue and red respectively. We also show $\Gamma$ (the line segment from $U$ to $f(U)$) and some of its images under $f$ (these form part of the unstable set $W^u(Y)$). The third quadrant $\mathcal{Q}_3$ is shaded, and we illustrate the forward orbit of a typical point $P \in \mathcal{Q}_3$ located just above $E^s(Y)$.
  • ...and 19 more figures

Theorems & Definitions (16)

  • Theorem 1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • proof : Proof of Theorem \ref{['th:main']}
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • ...and 6 more