Dynamical Inference of Simple Heteroclinic Networks - Voit, Maximilian; Meyer-Ortmanns, Hildegard (Frontiers in Applied Mathematics and Statistics, 2019)
Heteroclinic networks are structures in phase space that consist of multiple saddle fixed points as nodes, connected by heteroclinic orbits as edges. They provide a promising candidate attractor to generate reproducible sequential series of metastable states. While from an engineering point of view it is known how to construct heteroclinic networks to achieve certain dynamics, a data based approach for the inference of heteroclinic dynamics is still missing. Here, we present a method by which a template system dynamically learns to mimic an input sequence of metastable states. To this end, the template is unidirectionally, linearly coupled to the input in a master-slave fashion, so that it is forced to follow the same sequence. Simultaneously, its eigenvalues are adapted to minimize the difference of template dynamics and input sequence. Hence, after the learning procedure, the trained template constitutes a model with dynamics that are most similar to the training data. We demonstrate the performance of this method at various examples, including dynamics that differ from the template, as well as a regular and a random heteroclinic network. In all cases the topology of the heteroclinic network is recovered precisely, as are most eigenvalues. Our approach may thus be applied to infer the topology and the connection strength of a heteroclinic network from data in a dynamical fashion. Moreover, it may serve as a model for learning in systems of winnerless competition.