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Matrix stability and bifurcation analysis by a network-based approach

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Abstract

In this paper, we develop a network-based methodology to investigate the problems related to matrix stability and bifurcations in nonlinear dynamical systems. By matching a matrix with a network, i.e., interaction graph, we propose a new network-based matrix analysis method by proving a theorem about matrix determinant under which matrix stability can be considered in terms of feedback loops. Especially, the approach can tell us how a node, a path, or a feedback loop in the interaction graph affects matrix stability. In addition, the roles played by a node, a path, or a feedback loop in determining bifurcations in nonlinear dynamical systems can also be revealed. Therefore, the approach can help us to screen optimal node or node combinations. By perturbing them, unstable matrices can be stabilized more efficiently or bifurcations can be induced more easily to realize desired state transitions. To illustrate feasibility and efficiency of the approach, some simple matrices are used to show how single or combinatorial perturbations affect matrix stability and induce bifurcations. In addition, the main idea is also illustrated through a biological problem related to T cell development with three nodes: TCF-1, GATA3, and PU.1, which can be considered to be a three-variable nonlinear dynamical system. The approach is especially helpful in understanding crucial roles of single or molecule combinations in biomolecular networks. The approach presented here can be expected to analyze other biological networks related to cell fate transitions and systematic perturbation strategy selection.

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Funding

This research is supported by the National Natural Science Foundation of China (Grants No. 11971297 and No. 12371497).

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Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Zhenzhen Zhao, Ruoyu Tang & Ruiqi Wang

  2. Newtouch Center for Mathematics of Shanghai University, Shanghai, 200444, China

    Ruiqi Wang

Authors
  1. Zhenzhen Zhao
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  2. Ruoyu Tang
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  3. Ruiqi Wang
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Contributions

ZZ: matrix perturbation analysis and theorem proof; write and edit. RT: perturbation analysis of the nonlinear dynamical system. RW: provide ideas; formulate overall research goals and aims; supervise and revise this paper.

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Correspondence to Ruiqi Wang.

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Zhao, Z., Tang, R. & Wang, R. Matrix stability and bifurcation analysis by a network-based approach. Theory Biosci. 142, 401–410 (2023). https://doi.org/10.1007/s12064-023-00405-0

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