Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 861(ii) For any first–order dynamical equation, there exists a (time–dependent) Riemannian metric such that every solution of this equationis Lyapunov stable.(iii) Moreover, the Lyapunov exponent of any solution of a first–orderdynamical equation can be made equal to any real number with respectto the appropriate (time–dependent) Riemannian metric. It follows thatchaos in dynamical systems described by smooth (C ∞ ) first–order dynamicalequations can be characterized in full by time–dependent Riemannianmetrics.5.6.15 First–Order Dynamical EquationsLet R be the time axis provided with the Cartesian coordinate t andtransition functions t ′ = t+const. In geometrical terms [Mangiarotti andSardanashvily (1998)], a (smooth) first–order dynamical equation in non–autonomous mechanics is defined as a vector–field γ on a smooth fibrebundlewhich obeys the condition γ⌋dt = 1, i.e.,π : Y −→ R (5.174)γ = ∂ t + γ k ∂ k . (5.175)The associated first–order dynamical equation takes the formṫ = 1, ẏ k = γ k (t, y j )∂ k , (5.176)where (t, y k , ṫ, ẏ k ) are holonomic coordinates on T Y . Its solutions are trajectoriesof the vector–field γ (5.175). They assemble into a (regular) foliationF of Y . Equivalently, γ (5.175) is defined as a connection on the fibrebundle (5.174).A fibre bundle Y (5.174) is trivial, but it admits different trivializationsY ∼ = R × M, (5.177)distinguished by fibrations Y −→ M. For example, if there is a trivialization(5.177) such that, with respect to the associated coordinates, the componentsγ k of the connection γ (5.175) are independent of t, one says that γis a conservative first–order dynamical equation on M.Hereafter, the vector–field γ (5.175) is assumed to be complete, i.e.,there is a unique global solution of the dynamical equation γ through each