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Arnoldi JF, Haegeman B. Unifying dynamical and structural stability of equilibria. Proc Math Phys Eng Sci. 2016;472(2193):20150874doi: 10.1098/rspa.2015.0874.
Arnoldi, J. F., & Haegeman, B. (2016). Unifying dynamical and structural stability of equilibria. Proceedings. Mathematical, physical, and engineering sciences, 472(2193), 20150874. https://doi.org/10.1098/rspa.2015.0874
Arnoldi, Jean-François, and Haegeman, Bart. "Unifying dynamical and structural stability of equilibria." Proceedings. Mathematical, physical, and engineering sciences vol. 472,2193 (2016): 20150874. doi: https://doi.org/10.1098/rspa.2015.0874
Arnoldi JF, Haegeman B. Unifying dynamical and structural stability of equilibria. Proc Math Phys Eng Sci. 2016 Sep;472(2193):20150874. doi: 10.1098/rspa.2015.0874. PMID: 27713656; PMCID: PMC5046980.
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Proc Math Phys Eng Sci. 2016 Sep;472(2193):20150874. doi: 10.1098/rspa.2015.0874.

Unifying dynamical and structural stability of equilibria.

Proceedings. Mathematical, physical, and engineering sciences

Jean-François Arnoldi, Bart Haegeman

Affiliations

  1. Center for Biodiversity Theory and Modelling , Station d'Écologie Théorique et Expérimentale , 2 route du CNRS, 09200 Moulis, France.

PMID: 27713656 PMCID: PMC5046980 DOI: 10.1098/rspa.2015.0874

Abstract

We exhibit a fundamental relationship between measures of dynamical and structural stability of linear dynamical systems-e.g. linearized models in the vicinity of equilibria. We show that dynamical stability, quantified via the response to external perturbations (i.e. perturbation of dynamical variables), coincides with the minimal internal perturbation (i.e. perturbations of interactions between variables) able to render the system unstable. First, by reformulating a result of control theory, we explain that harmonic external perturbations reflect the spectral sensitivity of the Jacobian matrix at the equilibrium, with respect to constant changes of its coefficients. However, for this equivalence to hold, imaginary changes of the Jacobian's coefficients have to be allowed. The connection with dynamical stability is thus lost for real dynamical systems. We show that this issue can be avoided, thus recovering the fundamental link between dynamical and structural stability, by considering stochastic noise as external and internal perturbations. More precisely, we demonstrate that a linear system's response to white-noise perturbations directly reflects the intensity of internal white-noise disturbance that it can accommodate before becoming stochastically unstable.

Keywords: external perturbations; internal perturbations; linear systems; non-normal matrices; stability radius; white-noise perturbations

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