Cutting states associated with the orthogonal cutting of stiff cylinders are identified through an analysis of the singular values of a Toeplitz matrix of third order cumulants of acceleration measurements. The ratio of the two pairs of largest singular values is shown to differentiate between light cutting, medium cutting, pre-chatter and chatter states. Sequences of cutting experiments were performed in which either depth of cut or turning frequency was varied. Two sequences of experiments with variable turning frequency and five with variable depth of cut, 42 cutting experiments in all, provided a database for the calculation of third order cumulants. Ratios of singular values of cumulant matrices find application in the analysis and control of orthogonal cutting.

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Abstract. Differential equations are derived which describe the evolution of area tensors and normals associated with the subspaces of an ^-dimensional Euclidean phase space, En. These provide computational methods for determining the Lyapunov exponents of continuous dynamical systems.1. Introduction. In [7] and [14] a method is given for the computation of the Lyapunov exponents. A subspace, Em, of ^-dimensional Euclidean phase space, En, is defined by a set of m orthogonal vectors which are evolved. These are repeatedly orthonormalized to prevent their collapse into a set of lower dimension.In the following, subspaces are defined by the area tensor and its dual, the normal tensor. Sets of o.d.e. are derived which describe the evolution of these quantities, some of which have three-dimensional counterparts in the theory of continuum kinematics, [4], [5], and [9], The Lyapunov exponents may be expressed as limits of functions of m-dimensional area elements. [7,8], Evaluation of the Lyapunov exponents was effected through the integration of the derived o.d.e. for area elements of dimension r, 1 < r < n, along a trajectory. A numerical study indicates the utility of the derived algorithms.