## Abstract

A linear system ẋ = Ax, A ∈ ℝ^{n×n}, x ∈ ℝ^{n}, with rkA = n - 1, has a one-dimensional center manifold E^{c} = {v ∈ ℝ^{n} : 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 |
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Pages (from-to) | 1-7 |

Number of pages | 7 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 261 |

DOIs | |

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