Abstract
A linear system x' = Ax, with A in n×n, x in R, has a one-dimensional center manifold Ec = {v in Rn : Av = 0}. If a differential equation x' = f(x) has a one-dimensional center manifold Wc at an equilibrium x* then Ec is tangential to Wc with A = Df(x?) and for stability of Wc 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 mesoscale 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 |
Early online date | 11 Jun 2013 |
DOIs | |
Publication status | Published - 15 Oct 2013 |
Research Groups and Themes
- Engineering Mathematics Research Group
Keywords
- Definiteness
- Positive spanning tree
- Stability
- Minors