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Unformatted text preview: 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 ) = ¯ x ( µ ) , (9.1) where µ is a parameter. When a and ¯ x 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 ≤ R k . Let x : [ t , t 1 ] ∞ R n be a solution of (9.1) with µ = µ . Assume that a is continuously differentiable with respect to its first and third arguments on an open set X such that ( x ( t ) , t, µ ) ≤ X for all t ≤ [ t , t 1 ] . Then for all µ in a neigborhood of µ 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 equals ( t ) , where : [ t , t 1 ] ∞ R n,k is the n-by- k matrix-valued solution of the ODE ˙ ( t ) = A ( t ) ( t ) + B ( t ) , ( t ) = , (9.2) x ∞ a (¯ x at ¯ where A ( t ) is the derivative of the map ¯ x, t, µ ) with respect to ¯ x = x ( t ) , B ( t ) is the derivative of the map µ ∞ a ( x ( t ) , t, µ ) at µ = µ , and is the derivative of the map µ ∞ ¯ x ( µ ) at µ = µ ....
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This note was uploaded on 11/07/2011 for the course AERO 16.36 taught by Professor Alexandremegretski during the Spring '09 term at MIT.
- Spring '09