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15 February 2025
 
  » arxiv » 1508.0237

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Gradient and passive circuit structure in a class of non-linear dynamics on a graph
Herbert Mangesius ; Jean-Charles Delvenne ; Sanjoy K. Mitter ;
Date 2 Aug 2015
AbstractWe consider a class of non-linear dynamics on a graph with non-symmetric weighting and provide a systematic method to formulate the governing ODE system in gradient descent form of sum-separable strictly convex energy functions. By that we establish a family of sum-separable Lyapunov functions for this class, it coincides with Csisz’ar’s information divergences. Our approach bases on structure results from passive network synthesis and on a transformation of the original problem to a mass-preserving transport problem. We solve a conjecture of P. Moylan related to the existence of a function inverse in the context of realizing a non-linear ODE system as non-linear circuit. This work brings together in novel ways Markov chain theory, circuit theory, and network systems, in the framework of gradient flows on graphs. We discuss possible applications in the context of coupled oscillator and electric power grid studies and propose a discrete version of De Bruijn’s identity relating entropy production to Fisher information.
Source arXiv, 1508.0237
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