Ivancevic_Applied-Diff-Geom

506 Applied Differential Geometry: A Modern IntroductionUsingL = 1 [(ẋ 1 ) 2 + (ẋ 2 ) 2] + x 2 ẋ 1 − x 1 ẋ 2 + f,2H = 1 [(ẋ 1 ) 2 + (ẋ 2 ) 2] − f, g ij = (H + f)δ ij ,2N j i = −F j i = −δ ih F jh ,F ij = ∂X j∂x i− ∂X i∂x j, (i, j, h = 1, 2),the solutions of the differential system (4.26) are horizontal pregeodesics ofthe Riemann–Jacobi–Lagrangian manifold (R 2 \ {0}, g ij , N j i ).Geometry of the Lorenz FlowWe use the Riemannian manifold (R 3 , δ ij ). The Lorenz flow is a first dissipativemodel with chaotic behavior discovered in numerical experiment.Its state equations are (see [Lorenz (1963); Sparrow (1982)])ẋ 1 = −σx 1 + σx 2 , ẋ 2 = −x 1 x 3 + rx 1 − x 2 , ẋ 3 = x 1 x 2 − bx 3 ,where σ, r, b are real parameters. Usually σ, b are kept fixed whereas r isvaried. Atr > r 0 =σ(σ + b + 3)σ − b − 1chaotic behavior is observed [Udriste (2000)].Let X = (X 1 , X 2 , X 3 ), X 1 (x 1 , x 2 , x 3 ) = −σx 1 + σx 2 ,X 2 (x 1 , x 2 , x 3 ) = −x 1 x 3 + rx 1 − x 2 , X 3 (x 1 , x 2 , x 3 ) = x 1 x 2 − bx 3 ,f = 1 2 [(−σx 1 + σx 2 ) 2 + (−x 1 x 3 + rx 1 − x 2 ) 2 + (x 1 x 2 − bx 3 ) 2 ],curl X = (2x 1 , −x 2 , r − x 3 − σ).The Lorenz dynamics is described byẍ i = ∂f ( ∂Xi+ − ∂X )jẋ j , (i, j = 1, 2, 3), or∂x i ∂x j ∂x iẍ 1 = ∂f∂x 1+ (σ + x 3 − r)ẋ 2 − x 2 ẋ 3 ,ẍ 2 = ∂f∂x 2+ (r − x 3 − σ)ẋ 1 − 2x 1 ẋ 3 ,ẍ 3 = ∂f∂x 3+ x 2 ẋ 1 + 2x 1 ẋ 2 . (4.27)