Locating the Sets of Exceptional Points in Dissipative Systems and the Self-Stability of Bicycles

Kirillov, Oleg N.

doi:10.3390/e20070502

Cited by 6 publications

(4 citation statements)

References 39 publications

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“…At a 1 = 4 there is a quadruple real eigenvalue −1. So, we refer to the set (55) as a set of exceptional points [65,67,68] (abbreviated as the EP-set).…”

Section: Abscissa Minimization and Multiple Rootsmentioning

confidence: 99%

From rotating fluid masses and Ziegler’s paradox to Pontryagin- and Krein spaces and bifurcation theory

Kirillov

Verhulst

2022

Mathematics in Industry

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Four classical systems, the Kelvin gyrostat, the Maclaurin spheroids, the Brouwer rotating saddle, and the Ziegler pendulum have directly inspired development of the theory of Pontryagin and Krein spaces with indefinite metric and singularity theory as independent mathematical topics, not to mention stability theory and nonlinear dynamics. From industrial applications in shipbuilding, turbomachinery, and artillery to fundamental problems of astrophysics, such as asteroseismology and gravitational radiation -that is the range of phenomena involving the Krein collision of eigenvalues, dissipation-induced instabilities, and spectral and geometric singularities on the neutral stability surfaces, such as the famous Whitney's umbrella.

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“…At a 1 = 4 there is a quadruple real eigenvalue −1. So, we refer to the set (55) as a set of exceptional points [65,67,68] (abbreviated as the EP-set).…”

Section: Abscissa Minimization and Multiple Rootsmentioning

confidence: 99%

From rotating fluid masses and Ziegler’s paradox to Pontryagin- and Krein spaces and bifurcation theory

Kirillov

Verhulst

2022

Mathematics in Industry

View full text Add to dashboard Cite

show abstract

“…At a 1 = 4 there is a quadruple real eigenvalue −1. So, we refer to the set (55) as a set of exceptional points [64,66,67] (abbreviated as the EP-set).…”

Section: Abscissa Minimization and Multiple Rootsmentioning

confidence: 99%

From rotating fluid masses and Ziegler's paradox to Pontryagin- and Krein spaces and bifurcation theory

Kirillov,

Verhulst

2019

Preprint

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Three classical systems, the Kelvin gyrostat, the Maclaurin spheroids, and the Ziegler pendulum have directly inspired development of the theory of Pontryagin and Krein spaces with indefinite metric and singularity theory as independent mathematical topics, not to mention stability theory and nonlinear dynamics. From industrial applications in shipbuilding, turbomachinery, and artillery to fundamental problems of astrophysics, such as asteroseismology and gravitational radiation -that is the range of phenomena involving the Krein collision of eigenvalues, dissipation-induced instabilities, and spectral and geometric singularities on the neutral stability surfaces, such as the famous Whitney's umbrella.

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“…Substantial effort has been invested into developing computational methods for finding the EPs explicitly for a given nonhermitian Hamiltonian [57][58][59][60]. In spite of great ingenuity and insightfulness of such algorithms, their application to concrete systems is not always straightforward.…”

Section: Introductionmentioning

confidence: 99%

Equations of motion governing the dynamics of the exceptional points of parameterically dependent nonhermitian Hamiltonians

Šindelka

Stránský

Cejnar

2023

J. Phys. A: Math. Theor.

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We study exceptional points (EPs) of a nonhermitian Hamiltonian $\hat{H}\n(\lambda,\delta)$ whose parameters $\lambda \in {\mathbb C}$and $\delta \in {\mathbb R}$. As the real control parameter $\delta$ is varied, the $k$-th EP (or $k$-th cluster of simultaneously existing EPs)of $\hat{H}\n(\lambda,\delta)$ moves in the complex plane of $\lambda$ along a continuous trajectory, $\lambda_k(\delta)$.Using an appropriate non-hermitian formalism (based upon the $c$-product and not upon the conventional Dirac product), wederive a self contained set of equations of motion (EOM) for the trajectory $\lambda_k(\delta)$, while interpreting $\delta$ as the propagationtime. Such EOM become of interest whenever one wishes to study the response of EPs to external perturbations or continuous parametric changes ofthe pertinent Hamiltonian. This is e.g.~the case of EPs emanating from hermitian curve crossings/degeneracies (which turn into avoidedcrossings/near-degeneracies when the Hamiltonian parameters are continuously varied). The presented EOM for EPs have not only their theoreticalmerits, they possess also a substantial practical relevance. Namely, the just presented approach can be regarded even as an efficient numericalmethod, useful for generating EPs for a broad class of complex quantum systems encountered in atomic, nuclear and condensed matter physics. Performance of such a method is tested here numerically on a simple yet nontrivial toy model.

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Locating the Sets of Exceptional Points in Dissipative Systems and the Self-Stability of Bicycles

Cited by 6 publications

References 39 publications

From rotating fluid masses and Ziegler’s paradox to Pontryagin- and Krein spaces and bifurcation theory

From rotating fluid masses and Ziegler’s paradox to Pontryagin- and Krein spaces and bifurcation theory

From rotating fluid masses and Ziegler's paradox to Pontryagin- and Krein spaces and bifurcation theory

Equations of motion governing the dynamics of the exceptional points of parameterically dependent nonhermitian Hamiltonians

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