Non-determinism in the limit of nonsmooth dynamics

TL;DR

This paper investigates how discontinuous derivatives in dynamical systems can lead to non-deterministic outcomes, especially near grazing points, and explores their implications in physical systems like superconducting resonators.

Contribution

It demonstrates the existence of well-defined solution sets for multi-valued flows caused by grazing discontinuities and analyzes the loss of determinism in such systems.

Findings

Multi-valued solution sets exist near grazing points.

Loss of determinism quantifies infinite sensitivity to initial conditions.

Applications to superconducting resonators and oscillators show practical relevance.

Abstract

Discontinuous time derivatives are used to model threshold-dependent switching in such diverse applications as dry friction, electronic control, and biological growth. In a continuous flow, a discon- tinuous derivative can generate multiple outcomes from a single initial state. Here we show that well defined solution sets exist for flows that become multi-valued due to grazing a discontinuity. Loss of determinism is used to quantify dynamics in the limit of infinite sensitivity to initial conditions, then applied to the dynamics of a superconducting resonator and a negatively damped oscillator.