Stability aspects of the Hayes delay differential equation with scalable hysteresis

Hysteresis models are strongly nonlinear, with slope discontinuities at every load reversal. Stability analysis of hysteretically damped systems is therefore challenging. Recently, a scalable hysteresis model has been reported, motivated by materials with distributed microscopic frictional cracks. In a scalable system, if x(t) is a solution, then so is αx(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha x(t)$$\end{document} for any α>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha > 0$$\end{document}. Scalability in the hysteretic damping allows us to study interesting stability aspects of otherwise linear dynamic systems. In this paper, we study the first-order Hayes delay differential equation with scalable hysteresis. Stability of this system can be examined on a two-dimensional parameter plane. We use Galerkin projection to convert the Hayes delay differential equation with hysteresis to a system of ODEs. We then use numerically obtained Lyapunov-like exponents for stability analysis. Some stability boundaries in the parameter plane contain periodic solutions, which we compute numerically by continuation and also approximately using harmonic balance and related approximations. There is an extended region in the parameter plane for which nonzero equilibria exist (like sticking solutions in scalar dry friction), and infinitesimal stability analysis thereof leads to a pseudolinear delay differential equation. On the stability boundary of the pseudolinear equation, there are solutions with linear drift in one state and periodicity in all other states. Stability regions of the original and the pseudolinearized equation overlap, but are not identical. The reason is explained in terms of differences in sets of initial conditions used for computing Lyapunov-like exponents.