Bifurcations of the Hénon map with additive bounded noise

TL;DR

The paper addresses bifurcations of minimal attractors for the discrete-time Hénon map with additive bounded noise, modeled by $z_{i+1}=f(z_i)+\xi_i$ with $\xi_i\in\overline{B_{\varepsilon}(0)}$ and $f(x,y)=(1- a x^2+y, b x)$. It adopts a finite-dimensional boundary map $\beta$ to link boundary dynamics with attractor bifurcations, establishing correspondences between topological bifurcations and boundary-map bifurcations. Key findings include (i) wedge singularities arising when boundary-map eigenvalues transition from real to complex, (ii) fold and heteroclinic bifurcations of the boundary map corresponding to topological changes of attractors, and (iii) cascades of wedges and shallow singularities from non-transversal intersections with quantified scaling and accumulation points; these phenomena are demonstrated across parameter regimes and illustrated via boundary-map projections. The boundary-map approach offers computational advantages over brute-force set-valued methods and provides a pathway toward a general bifurcation theory for attractors in bounded-noise random dynamical systems, with potential relevance to control, uncertainty quantification, and front propagation.

Abstract

We numerically study bifurcations of attractors of the Hénon map with additive bounded noise with spherical reach. The bifurcations are analysed using a finite-dimensional boundary map. We distinguish between two types of bifurcations: topological bifurcations and boundary bifurcations. Topological bifurcations describe discontinuous changes of attractors and boundary bifurcations occur when singularities of an attractor's boundary are created or destroyed. We identify correspondences between topological and boundary bifurcations of attractors and local and global bifurcations of the boundary map.

Figures (16)

• Figure 1: Numerical approximation of minimal attractors of the random Hénon map \ref{['eq:randomdiff']} with $b = 0.3$ and $\varepsilon = 0.0625$ and varying $a$ between $0.59$ and $0.6$, one observes a topological bifurcation from (a) one connected minimal attractor to (b) a minimal attractor that consists of two disjoint parts between which orbits alternate.
• Figure 2: Numerical approximation of the minimal attractor of the random Hénon map \ref{['eq:randomdiff']} with $b = 0.06$ and $\varepsilon = 0.6$. A boundary bifurcation is observed between $a = 0.06$ and $a = 0.18$, where (a) the boundary of the minimal attractor is smooth at $a=0.06$, and (b) singularities have appeared at $a=0.18$, see also the magnification in (c).
• Figure 3: Schematic illustration of iterations of the set-valued map $F(x) = \overline{B_{\varepsilon}(f(x))}$.
• Figure 4: The boundary map sends a boundary point $m_1 \in \partial M_1$ with unit normal vector $n$ to a boundary point $m_2 \in \partial M_2$ with unit normal vector $n_1 = \frac{(f'(m_1)^T)^{-1}n}{\|(f'(m_1)^T)^{-1}n\|}$.
• Figure 5: Illustration of contributors to boundary points using the sets $M_1, M_2$ and $M_3$ in Figure \ref{['fig:set-valued_illustrate']}. Each point on the smooth boundary $\partial M_2$ has a unique contributor on $\partial f(M_1)$. Conversely, the boundary point $x_1 \in \partial M_3$ has two distinct contributors $y_1,y_2 \in \partial f(M_2)$, resulting in a singular (non-differentiable) boundary point of wedge type. There are also points on $\partial f(M_2)$ which do not contribute to any boundary points on $\partial M_3$, for example, the point $y_3 \in \partial f(M_2)$.

Theorems & Definitions (7)

• Definition 3.1: Boundary Map kourliouros2023persistence
• Proposition 3.2
• Proposition 3.3
• Lemma A.1
• proof
• Proposition B.1
• proof