Matrix logistic map: fractal spectral distributions and transfer of chaos

Technical Summary: Matrix Logistic Map: Fractal Spectral Distributions and Transfer of Chaos

Problem Statement
The standard logistic map, $x_{n+1} = ax_n(1-x_n)$, serves as a fundamental paradigm for understanding the transition from regular to chaotic dynamics in one-dimensional systems. While the behavior of this scalar map is well-characterized—including period doubling, the Feigenbaum universality, and the emergence of fractal invariant measures—its extension to higher-dimensional spaces remains an open area of investigation. Previous attempts to generalize the logistic map to matrices have focused on non-Hermitian cases leading to explosive divergence or specific two-dimensional cases. This work addresses the need to define and analyze a robust matrix analogue of the logistic map acting on arbitrary $N \times N$ matrices, specifically investigating the asymptotic spectral distributions of these ensembles and the mechanisms of chaos transfer in coupled systems defined by graph topologies.

Methodology
The authors propose two primary generalizations of the scalar logistic map:

1. Matrix Logistic Map on Hermitian Matrices:
   The map is defined on the set of Hermitian positive matrices $H$ with bounded operator norm ($0 \le H \le I$). The evolution is given by:
   $$H_{t+1} = aH_t(I - H_t)$$
   By diagonalizing $H_t = U D_t U^\dagger$, the authors demonstrate that the unitary matrix $U$ remains invariant, and the dynamics reduce to the parallel iteration of the scalar logistic map on the eigenvalues (the diagonal elements of $D_t$).

2. Matrix Logistic Map on Non-Hermitian Matrices:
   For an arbitrary matrix $X$, the authors utilize the singular value decomposition (SVD), $X = U E V^\dagger$. The map is constructed to act on the singular values while preserving the unitary bases:
   $$X_{t+1} = a U [E_t(I - E_t)] V^\dagger$$
   This formulation ensures that the dynamics of the singular values follow the scalar logistic map, while the singular vectors remain fixed.

3. Graph-Dependent Coupled Maps:
   The scalar parameter $a$ is generalized to a fixed positive matrix $A = BB^\dagger$, where $B$ is associated with the adjacency matrix of a directed graph. The evolution equation becomes:
   $$H_{t+1} = B H_t (I - H_t) B^\dagger$$
   This setup models coupled logistic maps where the interaction topology is determined by the graph structure. The authors analyze linear and star graphs, varying the coupling strength $\epsilon$ to observe the propagation of chaos from a chaotic node to regular nodes.

Key Contributions and Results

• Convergence to Invariant Measures:
  The paper establishes that for an initial ensemble of random Hermitian matrices with a continuous spectral density supported on $[0,1]$, the asymptotic spectral density converges to the invariant measure $\mu^*_a$ of the scalar logistic map as $t \to \infty$. Depending on the parameter $a$, this limiting measure can be singular, fractal, or continuous (e.g., the arcsine law for $a=4$). A similar convergence is proven for the singular value density of non-Hermitian random matrices.

• Fractal Spectral Distributions:
  By selecting parameters $a$ where the scalar map exhibits fractal invariant measures, the authors construct novel ensembles of random matrices with fractal spectral distributions. Numerical simulations (using $N=1000$) demonstrate the rapid convergence of initial distributions (such as the shifted Wigner semicircle for GUE or the quarter-circle law for Ginibre matrices) to these fractal measures.

• Spectral Geometry of Non-Hermitian Ensembles:
  For non-Hermitian matrices, the authors find that while the singular value distribution becomes fractal, the eigenvalues in the complex plane form a single ring. This holds for cases of period-two dynamics and fully chaotic dynamics ($a=4$), consistent with the "single ring theorem." Analytic expressions for these spectral densities are derived using the theory of free random matrices and the Haagerup-Larsen theorem.

• Transfer of Chaos in Networks:
  In the coupled model defined by graph $B$, the authors demonstrate the gradual transfer of chaos. Starting with a system where one vertex exhibits chaotic dynamics ($a_1 \approx 3.91$) and others are regular ($a_i = 2.5$), increasing the coupling strength $\epsilon$ induces chaos in neighboring vertices. The results show that the transition to chaos propagates outward from the source, with vertices further from the chaotic node requiring significantly higher coupling strengths to become chaotic. This is verified using Lyapunov characteristic exponents and the $0-1$ test for chaos.

Significance and Claims
The authors position this work as an initiation of further studies on the dynamics of matrix logistic maps. The primary significance lies in:

1. Bridging Dynamical Systems and Random Matrix Theory: The work establishes a direct link between the logistic map and the theory of random matrices, generating new ensembles with fractal spectral measures.
2. Modeling Complex Networks: The generalized matrix map with $A=BB^\dagger$ provides a versatile tool for analyzing the transition to chaos in complex networks and multidimensional systems, specifically demonstrating how chaos transfers between subsystems based on graph topology.
3. Quantum Applications: The authors note that the model is relevant to nonlinear quantum evolution of density matrices, where the quadratic term corresponds to quantum measurements on multiple copies of a state. The iterated positive matrix can be interpreted as a quantum state undergoing nonlinear evolution.

The paper concludes that these matrix generalizations offer a powerful framework for studying nonlinear dynamics in spaces of matrices, offering insights into both classical coupled systems and potential nonlinear quantum processes.