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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 10: Singular Perturbations and Averaging 1 This lecture presents results which describe local behavior of parameterdependent ODE models in cases when dependence on a parameter is not continuous in the usual sense. 10.1 Singularly perturbed ODE In this section we consider parameterdependent systems of equations x ˙( t ) = f ( x ( t ) , y ( t ) , t ) , (10.1) y ˙ = g ( x ( t ) , y ( t ) , t ) , where → [0 , ] is a small positive parameter. When > 0, (10.1) is an ODE model. For = 0, (10.1) is a combination of algebraic and differential equations. Models such as (10.1), where y represents a set of less relevant, fast changing parameters, are fre quently studied in physics and mechanics. One can say that singular perturbations is the “classical” approach to dealing with uncertainty, complexity, and nonlinearity. 10.1.1 The Tikhonov’s Theorem A typical question asked about the singularly perturbed system (10.1) is whether its solutions with > converge to the solutions of (10.1) with = as 0. A suﬃ cient condition for such convergence is that the Jacobian of g with respect to its second argument should be a Hurwitz matrix in the region of interest. Theorem 10.1 Let x : [ t , t 1 ] ∞ R n , y : [ t , t 1 ] ∞ R m be continuous functions satisfying equations x ˙ ( t ) = f ( x ( t ) , y ( t ) , t ) , = g ( x ( t ) , y ( t ) , t ) , 1 Version of October 15, 2003 2 where f : R n × R m × R ∞ R n and g : R n × R m × R ∞...
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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
 AlexandreMegretski
 Dynamics

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