Abstract
Low pass filters, which are used to remove high frequency noise from time series data, smooth the signals they are applied to. In this paper we examine the action of low pass filters on discontinuous or non-differentiable signals from non-smooth dynamical systems. We show that the application of such a filter is equivalent to a change of variables, which transforms the non-smooth system into a smooth one. We examine this smoothing action on a variety of examples and demonstrate how it is useful in the calculation of a non-smooth system’s Lyapunov spectrum.
Highlights
Non-smooth dynamical systems are used to model mechanical systems with impacts or friction, as well as control systems with switching between distinct modes of operation
We might find the smoothing action useful or interesting in its own right and secondly such systems are already being investigated whenever experimental data is smoothed with a low pass filter
We have shown that low-pass filters can be used to formulate smoothing transformations that map discontinuous or non-differentiable systems to smooth systems
Summary
Non-smooth dynamical systems are used to model mechanical systems with impacts or friction, as well as control systems with switching between distinct modes of operation. In this paper we introduce the notion of smoothing a non-smooth system with a low-pass filter. To apply a smoothing transformation numerically to an orbit we apply the associated low pass filter to the time-series. We might find the smoothing action useful or interesting in its own right (we will show that it is useful for computing Lyapunov exponents) and secondly such systems are already being investigated whenever experimental data is smoothed with a low pass filter.
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