247 research outputs found
Smoothing tautologies, hidden dynamics, and sigmoid asymptotics for piecewise smooth systems
Switches in real systems take many forms, such as impacts, electronic relays,
mitosis, and the implementation of decisions or control strategies. To
understand what is lost, and what can be retained, when we model a switch as an
instantaneous event, requires a consideration of so-called hidden terms. These
are asymptotically vanishing outside the switch, but can be encoded in the form
of nonlinear switching terms. A general expression for the switch can be
developed in the form of a series of sigmoid functions. We review the key steps
in extending the Filippov's method of sliding modes to such systems. We show
how even slight nonlinear effects can hugely alter the behaviour of an
electronic control circuit, and lead to `hidden' attractors inside the
switching surface.Comment: 12 page
Non-Filippov dynamics arising from the smoothing of nonsmooth systems, and its robustness to noise
Switch-like behaviour in dynamical systems may be modelled by highly
nonlinear functions, such as Hill functions or sigmoid functions, or
alternatively by piecewise-smooth functions, such as step functions. Consistent
modelling requires that piecewise-smooth and smooth dynamical systems have
similar dynamics, but the conditions for such similarity are not well
understood. Here we show that by smoothing out a piecewise-smooth system one
may obtain dynamics that is inconsistent with the accepted wisdom --- so-called
Filippov dynamics --- at a discontinuity, even in the piecewise-smooth limit.
By subjecting the system to white noise, we show that these discrepancies can
be understood in terms of potential wells that allow solutions to dwell at the
discontinuity for long times. Moreover we show that spurious dynamics will
revert to Filippov dynamics, with a small degree of stochasticity, when the
noise magnitude is sufficiently large compared to the order of smoothing. We
apply the results to a model of a dry-friction oscillator, where spurious
dynamics (inconsistent with Filippov's convention or with Coulomb's model of
friction) can account for different coefficients of static and kinetic
friction, but under sufficient noise the system reverts to dynamics consistent
with Filippov's convention (and with Coulomb-like friction).Comment: submitted to: Nonlinear Dynamic
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