Matrix logistic map: fractal spectral distributions and transfer of chaos
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915455130664960 |
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| author | Pawela, Łukasz Życzkowski, Karol |
| author_facet | Pawela, Łukasz Życzkowski, Karol |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_06176 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Matrix logistic map: fractal spectral distributions and transfer of chaos Pawela, Łukasz Życzkowski, Karol Chaotic Dynamics Quantum Physics 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. |
| title | Matrix logistic map: fractal spectral distributions and transfer of chaos |
| topic | Chaotic Dynamics Quantum Physics |
| url | https://arxiv.org/abs/2303.06176 |