Higher-order discontinuity mappings in piecewise-smooth dynamical systems

Rohit Chawla, Aasifa Rounak and Vikram Pakrashi

Many physical systems in engineering applications, like gear assemblies, impact print hammers, and mechanical and electrical devices, are subjected to instantaneous changes during their operation. These rapid non-linear changes can also be found in several other diverse fields like biology, finance, ecology, epidemiology, etc. Such events can cause unwanted and chaotic vibrations, potentially leading to increased wear and tear. Analysis of these highly non-linear phenomena is carried out using the concepts of piecewise-smooth (PWS) dynamical systems – such non-linear models allow discontinuous/rapid changes to occur within the system by constructing “discontinuity mappings”. However, the existing methods to construct discontinuity mappings rely on first-order local linearization techniques, which may not correctly capture the true behaviour of physical systems during the occurrences of such rapid events. This work presents a generalized closed-form derivation of a higher-order discontinuity mapping, which highlights and resolves the drawbacks of using the widely accepted first-order approaches. The higher-order discontinuity mapping has been derived for two types of PWS systems – hybrid impacting and Filippov-type systems which can cater to or model a larger class of physical systems where such extreme events can occur. The higher-order correction terms provide a better estimation of how PWS systems should behave during the occurrences of these non-linear and non-smooth phenomena. The methods proposed here can effectively capture and predict when such chaotic and unstable vibrations occur so as to avoid any undesirable motion.