Perturbation and determinacy of nonsmooth systems

Take a system where several variables $x_i$ (for $i=1,2,\dots$) cause decision states $h_i$ to be set independently to values $A_i$, at any instant, and the outcome then affects how each $x_i$ evolves according to a differential equation. We show here that the probability that the system lies in a given decision state at any instant cannot be determined solely from these differential equations, but is determined by the emergence of a dynamical attractor. Moreover this attractor is sensitive to small perturbations in how the decisions are enacted, and even how the system’s evolution is calculated. If the probability that $x_1$ decides ‘$A_1$’ is $P\left(A_1\right)$ and $x_2$ decides ‘$A_2$’ is $P\left(A_2\right)$, for instance, the probability that $x_1$ decides ‘$A_1$’ and $x_2$ decides ‘$A_2$’ at any moment is not generally $P\left(A_1\right)P\left(A_2\right)$, despite the independence of their decisions (nor is it any other determinable quantity such as $P\left(A_1\right)P\left(A_2\right|A_1)$). Only certain weighted sums of probabilities of being in different decision states are determined by the logic of the system.

This result comes from formulating this simple decision-making scenario as a dynamical system with discontinuities (or piecewise-smooth or nonsmooth system), and exposes a need to better understand the indeterminacy of discontinuous models, and how they behave under perturbation. The perturbations of interest might represent physical properties neglected in an idealised model with discontinuities, or imperfections introduced in simulations, perhaps by discretising the system, by smoothing out a discontinuity, or delaying a discontinuity’s effect on the system.

We define concepts here that permit us to characterise the determinacy of discontinuous systems and compare them under such perturbations. We find that although the overall dynamics of a system is indeterminable at a discontinuity, certain measures of occupancy either side of a discontinuity are determinable. These give a refined insight into Filippov’s differential inclusions, and give more precision to Utkin’s notion of equivalent dynamics, interestingly allowing us to treat discontinuities in dynamical systems similarly to Markov processes.