Matrix logistic map: fractal spectral distributions and transfer of chaos

• URL: http://arxiv.org/abs/2303.06176v1
• Date: Fri, 10 Mar 2023 19:19:56 GMT
• Title: Matrix logistic map: fractal spectral distributions and transfer of chaos
• Authors: {\L}ukasz Pawela and Karol \.Zyczkowski
• Abstract summary: We show that for an arbitrary initial ensemble of hermitian random matrices with a continuous level density supported on the interval $[0,1]$, the level density converges to the invariant measure of the logistic map. This approach generalizes the known model of coupled logistic maps, and allows us to study the transition to chaos in complex networks and multidimensional systems.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The standard logistic map, $x'=ax(1-x)$, serves as a paradigmatic model to demonstrate how apparently simple non-linear equations lead to complex and chaotic dynamics. In this work we introduce and investigate its matrix analogue defined for an arbitrary matrix $X$ of a given order $N$. We show that for an arbitrary initial ensemble of hermitian random matrices with a continuous level density supported on the interval $[0,1]$, the asymptotic level density converges to the invariant measure of the logistic map. Depending on the parameter $a$ the constructed measure may be either singular, fractal or described by a continuous density. In a broader class of the map multiplication by a scalar logistic parameter $a$ is replaced by transforming $aX(\mathbb{I}-X)$ into $BX(\mathbb{I}-X)B^{\dagger}$, where $A=BB^{\dagger}$ is a fixed positive matrix of order $N$. This approach generalizes the known model of coupled logistic maps, and allows us to study the transition to chaos in complex networks and multidimensional systems. In particular, associating the matrix $B$ with a given graph we demonstrate the gradual transfer of chaos between subsystems corresponding to vertices of a graph and coupled according to its edges.

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