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Set propagation in dynamical systems with generalised polynomial algebra and its computational complexity

Vasile, Massimiliano and Ortega Absil, Carlos and Riccardi, Annalisa (2019) Set propagation in dynamical systems with generalised polynomial algebra and its computational complexity. Communications in Nonlinear Science and Numerical Simulation, 75. pp. 22-49. ISSN 1007-5704

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Abstract

This paper presents an approach to propagate sets of initial conditions and model parameters through dynamical systems. It is assumed that the dynamics is dependent on a number of model parameters and that the state of the system evolves from some initial conditions. Both model parameters and initial conditions vary within a set Ω. The paper presents an approach to approximate the set Ω with a polynomial expansion and to propagate, under some regularity assumptions, the polynomial representation through the dynamical system. The approach is based on a generalised polynomial algebra that replaces algebraic operators between real numbers with operators between polynomials. The paper first introduces the concept of generalised polynomial algebra and its use to propagate sets through dynamical systems. Then it analyses, both theoretically and experimentally, its time complexity and compares it against the time complexity of a non-intrusive counterpart. Finally, the paper provides an empirical convergence analysis on two illustrative examples of linear and non-linear dynamical systems.