Almost sure asymptotic stability of an oscillator with delay feedback when excited by finite-state Markov noise Nishanth Lingala a,n , N. Sri Namachchivaya a , Oliver M. O’Reilly b , Volker Wihstutz c a University of Illinois, Urbana-Champaign, USA b University of California at Berkeley, USA c University of North Carolina at Charlotte, USA article info Article history: Received 14 March 2012 Accepted 5 December 2012 Available online 11 January 2013 Keywords: Almost-sure stability Delay differential equation Lyapunov exponent Chatter abstract An oscillator of the form € qðtÞþ2z _ qðtÞþ qðtÞ¼k½qðtÞqðtrÞ is unstable when the strength of the feedback (k) is greater than a critical value (k c ). Oscillations of constant amplitude persist when k ¼ k c . We study the almost-sure asymptotic stability of the oscillator when k ¼ k c and the system is excited by a two-state Markov noise. For small intensity noise, we construct an asymptotic expansion for the maximal Lyapunov exponent. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction Delay differential equations (DDEs), where the time derivative can depend on both past and present values of the variable, arise in a variety of areas such as manufacturing systems, biological systems, and control systems. Oscillators with delay feedback, which are second-order DDE, arise in physical applications such as machining processes [1,2] and study of human balance control [3]. Of particular interest in this paper is the critical behavior of a simple model for regenerative chatter in machining processes. The model of interest is a second-order delay differential equation for the position q of a point on a machine tool which is cutting material from a shaft which is rotating with a time period of revolution r € qðtÞþ 2z _ qðtÞþ qðtÞ¼k½qðtÞqðtrÞ: ð1Þ The term k½qðtÞqðtrÞ in (1) represents the assumption that the force acting on the tool is proportional to the variation in the width of the chip being cut, and the variation is the difference between the present position (q(t)) of the tool and its position one revolution earlier (qðtrÞ). It is known that, for a fixed r, there exists a critical k c such that the amplitude q of the oscillator decreases exponentially if k ok c and increases exponentially if k 4k c . When k ¼ k c oscillations of constant amplitude persist. In machining, this oscillatory behavior is called chatter. Building on the earlier work [4], we study the asymptotic stability of the system (1) at k ¼ k c when it is excited by a two- state Markov noise. Specifically € qðtÞþ 2z _ qðtÞþ qðtÞ¼k½qðtÞqðtrÞþ esðxðtÞÞ½F o 21 qðtÞ þ F o 22 _ qðtÞþ F r 21 qðtrÞþ F r 22 _ qðtrÞ, ð2Þ here s is a mean zero function of the two-state Markov noise xðtÞ, that is, E½sðxÞ ¼ 0, and e is a small parameter signifying that the noise perturbation is weak. The maximal Lyapunov exponent characterizes the asymptotic stability of the system. It is defined as l e : ¼ lim sup t-1 ð1=tÞ log 9qðtÞ9 and signifies the exponential growth rate of the amplitude, i.e., amp[q(t)]  e l e t . Thus, the system would be asymptotically stable under the noise excitation if l e o0 and unstable if l e 40. Our analysis is based on an expansion to third order in e for the maximal Lyapunov exponent: l e ¼ l 0 þ el 1 þ e 2 l 2 þ : ð3Þ For the problem of interest, l 0 ¼ 0, because, in the absence of the noise (i.e., when e ¼ 0), oscillations are of constant amplitude when k ¼ k c . In other words, the corresponding deterministic system has no eigenvalues with positive real parts and a pair of eigenvalues on the imaginary axis. Because l 1 ¼ 0 due to the mean zero nature of the noise, we are forced to examine l 2 and we give an explicit expression for this term. Using the methods in [5], it can be shown that the remaining term in the expansion (3) is bounded and is of order e 3 (see [6]). Hence, for sufficiently weak noise, l 2 characterizes the asymptotic stability. Our primary result is to show that stabilization is possible by the noise in (2), i.e., by suitably selecting the parameters in (2), it Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/probengmech Probabilistic Engineering Mechanics 0266-8920/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.probengmech.2012.12.008 n Corresponding author. Tel.: þ1 217 418 4635. E-mail address: lingala1@illinois.edu (N. Lingala). Probabilistic Engineering Mechanics 32 (2013) 21–30