The steady states of coupled dynamical systems compose according to matrix arithmetic

Abstract

Open dynamical systems are mathematical models of machines that take input, change their internal state, and produce output. For example, one may model anything from neurons to robots in this way. Several open dynamical systems can be arranged in series, in parallel, and with feedback to form a new dynamical system---this is called compositionality---and the process can be repeated in a fractal-like manner to form more complex systems of systems. One issue is that as larger systems are created, their state space grows exponentially. In this paper a technique for calculating the steady states of an interconnected system of systems, in terms of the steady states of its component dynamical systems, is provided. These are organized into "steady state matrices" which generalize bifurcation diagrams. It is shown that the compositionality structure of dynamical systems fits with the familiar monoidal structure for the steady state matrices, where serial, parallel, and feedback composition of matrices correspond to multiplication, Kronecker product, and partial trace operations. The steady state matrices of dynamical systems respect this compositionality structure, exponentially reducing the complexity involved in studying the steady states of composite dynamical systems.

48 pages

References in corpus (2)

• Algebras of Open Dynamical Systems on the Operad of Wiring Diagrams
• The operad of temporal wiring diagrams: formalizing a graphical language for discrete-time processes

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• Compositionality of the Runge-Kutta Method
• Compositional Cyber-Physical Systems Theory
• Radically Compositional Cognitive Concepts