2022/09/12 by Tristan Gowdridge, Nikolaos Dervilis, Gowdridge, Tristan +3
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Force Microscopy Techniques and Applications #Fractional Differential Equations Solutions #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2209.05169
openalex publication_date 2022/09/12 · openalex created_date 2022/09/14 · openalex updated_date 2026/07/28
Analytical solutions to nonlinear differential equations -- where they exist at all -- can often be very difficult to find. For example, Duffing's equation for a system with cubic stiffness requires the use of elliptic functions in the exact solution. A system with general polynomial stiffness would be even more difficult to solve analytically, if such a solution was even to exist. Perturbation and series solutions are possible, but become increasingly demanding as the order of solution increases. This paper aims to revisit, present and discuss a geometric/algebraic method of determining system response which lends itself to automation. The method, originally due to Fliess and co-workers, makes use of the generating series and shuffle product, mathematical ideas founded in differential geometry and abstract algebra. A family of nonlinear differential equations with polynomial stiffness is considered; the process of manipulating a series expansion into the generating series follows and is shown to provide a recursive schematic, which is amenable to computer algebra. The inverse Laplace-Borel transform is then applied to derive a time-domain response. New solutions are presented for systems with general polynomial stiffness, both for deterministic and Gaussian white-noise excitation