This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 timevarying 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 nby k matrixvalued 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 ....
View
Full
Document
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
 AlexandreMegretski
 Dynamics

Click to edit the document details