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lec9_6243_2003 - Massachusetts Institute of Technology...

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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 9: Local Behavior Near Trajectories 1 This lecture presents results which describe local behavior of ODE models in a neigbor- hood of a given trajectory, with main attention paid to local stability of periodic solutions. 9.1 Smooth Dependence on Parameters In this section we consider an ODE model x ˙( t ) = a ( x ( t ) , t, µ ) , x ( t 0 ) = ¯ x 0 ( µ ) , (9.1) where µ is a parameter. When a and ¯ x 0 are differentiable with respect to µ , the solution x ( t ) = x ( t, µ ) is differentiable with respect to µ as well. Moreover, the derivative of x ( t, µ ) with respect to µ can be found by solving linear ODE with time-varying coefficients. Theorem 9.1 Let a : R n × R × R k ∞� R n be a continuous function, µ 0 R k . Let x 0 : [ t 0 , t 1 ] ∞� R n be a solution of (9.1) with µ = µ 0 . Assume that a is continuously differentiable with respect to its first and third arguments on an open set X such that ( x 0 ( t ) , t, µ 0 ) X for all t [ t 0 , t 1 ] . Then for all µ in a neigborhood of µ 0 the ODE in (9.1) has a unique solution x ( t ) = x ( t, µ ) . This solution is a continuously differentiable function of µ , and its derivative with respect to µ at µ = µ 0 equals �( t ) , where : [ t 0 , t 1 ] ∞� R n,k is the n -by- k matrix-valued solution of the ODE ˙ �( t ) = A ( t )�( t ) + B ( t ) , �( t 0 ) = 0 , (9.2) x ∞� a x at ¯ where A ( t ) is the derivative of the map ¯ x, t, µ 0 ) with respect to ¯ x = x 0 ( t ) , B ( t ) is the derivative of the map µ ∞� a ( x 0 ( t ) , t, µ ) at µ = µ 0 , and 0 is the derivative of the map µ ∞� ¯ x 0 ( µ ) at µ = µ 0 . 1 Version of October 10, 2003
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2 Proof Existence and uniqueness of x ( t, µ ) and �( t ) follow from Theorem 3.1. Hence, in order to prove differentiability and the formula for the derivative, it is sufficient to show that there exist a function C : R + ∞� R + such that C ( r ) /r 0 as r 0 and δ > 0 such that | x ( t, µ ) �( t )( µ µ 0 ) x 0 ( t ) | ≈ C ( | µ µ 0 | ) whenever | µ µ 0 | ≈ δ .
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