Meso-scale obstructions to stability of 1D center manifolds for networks of coupled differential equations with symmetric Jacobian

J. Epperlein; A. L. Do; T. Gross; S. Siegmund

doi:10.1016/j.physd.2013.05.010

Meso-scale obstructions to stability of 1D center manifolds for networks of coupled differential equations with symmetric Jacobian

J. Epperlein, A. L. Do, T. Gross, S. Siegmund

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

A linear system ẋ = Ax, A ∈ ℝ^n×n, x ∈ ℝⁿ, with rkA = n - 1, has a one-dimensional center manifold E^c = {v ∈ ℝⁿ : Av = 0}. If a differential equation ẋ = f (x) has a one-dimensional center manifold W^c at an equilibrium x^* then E^c is tangential to W ^c with A = Df (x^*) and for stability of W ^c it is necessary that A has no spectrum in C⁺, i.e. if A is symmetric, it has to be negative semi-definite. We establish a graph theoretical approach to characterize semi-definiteness. Using spanning trees for the graph corresponding to A, we formulate meso-scale conditions with certain principal minors of A which are necessary for semi-definiteness. We illustrate these results by the example of the Kuramoto model of coupled oscillators.

Original language	English
Pages (from-to)	1-7
Number of pages	7
Journal	Physica D: Nonlinear Phenomena
Volume	261
DOIs	https://doi.org/10.1016/j.physd.2013.05.010
State	Published - 1 Jan 2013
Externally published	Yes

Keywords

Definiteness
Minors
Positive spanning tree
Stability

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics
Condensed Matter Physics
Applied Mathematics

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10.1016/j.physd.2013.05.010

Cite this

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title = "Meso-scale obstructions to stability of 1D center manifolds for networks of coupled differential equations with symmetric Jacobian",

abstract = "A linear system ẋ = Ax, A ∈ ℝn×n, x ∈ ℝn, with rkA = n - 1, has a one-dimensional center manifold Ec = {v ∈ ℝn : Av = 0}. If a differential equation ẋ = f (x) has a one-dimensional center manifold Wc at an equilibrium x* then Ec is tangential to W c with A = Df (x*) and for stability of W c it is necessary that A has no spectrum in C+, i.e. if A is symmetric, it has to be negative semi-definite. We establish a graph theoretical approach to characterize semi-definiteness. Using spanning trees for the graph corresponding to A, we formulate meso-scale conditions with certain principal minors of A which are necessary for semi-definiteness. We illustrate these results by the example of the Kuramoto model of coupled oscillators.",

keywords = "Definiteness, Minors, Positive spanning tree, Stability",

author = "J. Epperlein and Do, {A. L.} and T. Gross and S. Siegmund",

note = "Funding Information: All authors are members of the Center for Dynamics, Dresden, Germany. This work is partly supported by the German Research Foundation (DFG) through the Cluster of Excellence {\textquoteleft}Center for Advancing Electronics Dresden{\textquoteright} (cfaed).",

year = "2013",

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doi = "10.1016/j.physd.2013.05.010",

language = "English",

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journal = "Physica D: Nonlinear Phenomena",

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T1 - Meso-scale obstructions to stability of 1D center manifolds for networks of coupled differential equations with symmetric Jacobian

AU - Epperlein, J.

AU - Do, A. L.

AU - Gross, T.

AU - Siegmund, S.

N1 - Funding Information: All authors are members of the Center for Dynamics, Dresden, Germany. This work is partly supported by the German Research Foundation (DFG) through the Cluster of Excellence ‘Center for Advancing Electronics Dresden’ (cfaed).

PY - 2013/1/1

Y1 - 2013/1/1

N2 - A linear system ẋ = Ax, A ∈ ℝn×n, x ∈ ℝn, with rkA = n - 1, has a one-dimensional center manifold Ec = {v ∈ ℝn : Av = 0}. If a differential equation ẋ = f (x) has a one-dimensional center manifold Wc at an equilibrium x* then Ec is tangential to W c with A = Df (x*) and for stability of W c it is necessary that A has no spectrum in C+, i.e. if A is symmetric, it has to be negative semi-definite. We establish a graph theoretical approach to characterize semi-definiteness. Using spanning trees for the graph corresponding to A, we formulate meso-scale conditions with certain principal minors of A which are necessary for semi-definiteness. We illustrate these results by the example of the Kuramoto model of coupled oscillators.

AB - A linear system ẋ = Ax, A ∈ ℝn×n, x ∈ ℝn, with rkA = n - 1, has a one-dimensional center manifold Ec = {v ∈ ℝn : Av = 0}. If a differential equation ẋ = f (x) has a one-dimensional center manifold Wc at an equilibrium x* then Ec is tangential to W c with A = Df (x*) and for stability of W c it is necessary that A has no spectrum in C+, i.e. if A is symmetric, it has to be negative semi-definite. We establish a graph theoretical approach to characterize semi-definiteness. Using spanning trees for the graph corresponding to A, we formulate meso-scale conditions with certain principal minors of A which are necessary for semi-definiteness. We illustrate these results by the example of the Kuramoto model of coupled oscillators.

KW - Definiteness

KW - Minors

KW - Positive spanning tree

KW - Stability

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SN - 0167-2789

VL - 261

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EP - 7

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

ER -

Meso-scale obstructions to stability of 1D center manifolds for networks of coupled differential equations with symmetric Jacobian

Abstract

Keywords

ASJC Scopus subject areas

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