Ivancevic_Applied-Diff-Geom

864 Applied Differential Geometry: A Modern Introduction(i) Written with respect to the initial–date coordinates, the covariantLyapunov tensor is given byL ab = ∂ t g ab .(ii) Given a solution s of the dynamical equation γ and a solution s ofthe variation equation (5.179), we haveL ij (t, s k (t))s i s j = ∂ t (g ij (t, s k (t))s i s j ).The definition of the covariant Lyapunov tensor (5.183) depends on thechoice of a Riemannian fibre metric on the fibre bundle Y .If the vector–field γ is complete, there is a Riemannian fibre metric onY such that the covariant Lyapunov tensor vanishes everywhere. Let uschoose the atlas of the initial–date coordinates. Using the fibration ζ : Y−→ Y t=0 , one can give Y with a time–independent Riemannian fibre metricg ab (t, y c ) = h(t)g ab (0, y c ) (5.184)where g ab (0, y c ) is a Riemannian metric on the fibre Y t=0 and h(t) is apositive smooth function on R. The covariant Lyapunov tensor with respectto the metric (5.184) is given byPutting h(t) = 1, we get L = 0.L ab = ∂ t hg ab .5.6.16.2 Lyapunov StabilityWith the covariant Lyapunov tensor (5.183), we get the following criterionof the stability condition of Lyapunov.Recall that, given a Riemannian fibre metric g on a fibre bundle Y−→ R, the instant–wise distance ρ t (s, s ′ ) between two solutions s and s ′ ofa dynamical equation γ on Y at an instant t is the distance between thepoints s(t) and s ′ (t) in the Riemannian space (Y t , g(t)).Let s be a solution of a first–order dynamical equation γ. If there existsan open tubular neighborhood U of the trajectory s where the covariantLyapunov tensor (5.183) is negative-definite at all instants t ≥ t 0 , thenthere exists an open tubular neighborhood U ′ of s such thatlim [ρt ′ t ′(s, →∞ s′ ) − ρ t (s, s ′ )] < 0