Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 8635.6.16 Lyapunov Tensor and Stability5.6.16.1 Lyapunov TensorThe collection of coefficientsl j k = ∂ j γ k (5.180)of the variation equation (5.179) is called the Lyapunov matrix. Clearly, itis not a tensor under bundle coordinate transformations of the fibre bundleY (5.174). Therefore, we introduce a covariant Lyapunov tensor as follows.Let a fibre bundle Y → R be provided with a Riemannian fibre metricg, defined as a section of the symmetrized tensor product∨ 2 V ∗ Y → Y (5.181)of the vertical cotangent bundle V ∗ Y of Y → R. With respect to theholonomic coordinates (t, y k , y k ) on V ∗ Y , it takes the coordinate formg = 1 2 g ij(t, y k )dy i ∨ dy j ,where {dy i } are the holonomic fibre bases for V ∗ Y .Given a first–order dynamical equation γ, letV ∗ γ = γ − ∂ j γ k y k ∂ j , where ∂ j = ∂∂y j. (5.182)be the canonical lift of the vector–field γ (5.175) onto V ∗ Y . It is a connectionon V ∗ Y −→ R. Let us consider the Lie derivative L γ g of the Riemannianfibre metric g along the vector–field V ∗ γ (5.182). It readsL ij = (L γ g) ij = ∂ t g ij + γ k ∂ k g ij + ∂ i γ k g kj + ∂ j γ k g ik . (5.183)This is a section of the fibre bundle (5.181) and, consequently, a tensor withrespect to any bundle coordinate transformation of the fibre bundle (5.174).We agree to call it the covariant Lyapunov tensor. If g is an Euclideanmetric, it becomes the following symmetrization of the Lyapunov matrix(5.180),L ij = ∂ i γ j + ∂ j γ i = l i j + l j i .Let us point the following two properties of the covariant Lyapunovtensor.