Detecting phase transitions in collective behavior using manifold's curvature

Kelum Gajamannage, Erik M. Bollt
2016 Mathematical Biosciences and Engineering  
If a given behavior of a multi-agent system restricts the phase variable to a invariant manifold, then we define a phase transition as change of physical characteristics such as speed, coordination, and structure. We define such a phase transition as splitting an underlying manifold into two sub-manifolds with distinct dimensionalities around the singularity where the phase transition physically exists. Here, we propose a method of detecting phase transitions and splitting the manifold into
more » ... e transitions free sub-manifolds. Therein, we utilize a relationship between curvature and singular value ratio of points sampled in a curve, and then extend the assertion into higher-dimensions using the shape operator. Then we attest that the same phase transition can also be approximated by singular value ratios computed locally over the data in a neighborhood on the manifold. We validate the phase transitions detection method using one particle simulation and three real world examples.
doi:10.3934/mbe.2017027 pmid:27879108 fatcat:53lu67pixrazbe7zcbrnrxwmuy