Handbook of operation and flight instructions for the model by United States. Army Air Forces. Materiel Division. Field

By United States. Army Air Forces. Materiel Division. Field Service Section.; Glenn L. Martin Company

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Additional info for Handbook of operation and flight instructions for the model B-10B bombardment airplane, manufactured by The Glenn L. Martin Co

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A) Solutions track the stable branch x− (y) at a distance growing up to order ε1/3 . They jump after a delay of order ε2/3 . (b) For multidimensional slow variables y, a saddle–node bifurcation corresponds to a fold in the slow manifold. 15) there is an unstable slow manifold M+ = {(x, y) ∈ N : x = x+ (y), y < 0} , √ where x+ (y) = − −y [1 + Oy (1)]; • there are no other slow manifolds in N ; • g(x, y) > 0 in N . 16) M− being uniformly asymptotically stable for negative y, bounded away from 0, there is an adiabatic manifold associated with it which, in this (1 + 1)– dimensional setting, is simply a particular solution of the system.

27) 26 2 Deterministic Slow–Fast Systems where the last line follows from the implicit-function theorem, applied to the equation f (x (y), y) = 0. 9 (Overdamped motion of a particle in a potential – continued). 3. 29) ¯(y, ε) where ∂yy U (y) denotes the Hessian matrix of U . The dynamics of x = x on the adiabatic manifold is thus governed by the equation y−x ε ¯(y, ε)∇U (x) = −∂y x x˙ = = − 1l + ε∂xx U (x) + O(ε2 ) ∇U (x) . 30) This is indeed a small correction to the limiting equation x˙ = −∇U (x).

27) for t τ and yt < 0. We remark in passing that the corresponding argument for the lower bound in particular shows that zt > 0. Now choosing c1 = (3KM/c− )2/3 , we find that for all t τ satisfying yt √ −c1 ε2/3 , we also have zt c− −yt /3M , so that actually t < τ whenever yt −c1 ε2/3 . 19). The lower bound is obtained in a similar way. 28) holds. 29) which is a small perturbation of a solvable Riccati equation. 29) without the error term yields the linear second order equation ϕ (s) = −sϕ(s), whose solution can be expressed in terms of Airy functions.

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