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 This lecture presents results which describe local behavior of parameter-dependent 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 parameter-dependent systems of equations i(t)=f(ar(t),y(t),t), g(x(t),y(t),t), 10.1 where e E[0, Eo] is a small positive parameter. When e>0,(10.1)is an ODE model For e=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- "classicall"approach to dealing with uncertainty, complexity, and nonlineari ons is the quently studied in physics and mechanics. One can say that singular perturbatie 10.1.1 The Tikhonov's heorem A typical question asked about the singularly perturbed system(10. 1)is whether its solutions with e >0 converge to the solutions of(10.1)with e =0 as E-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 o: to, t1]H+ R", yo: [to, til brm be continuous functions tisfying equations io(t)=f(aro(t), yo(t), t),0=g(ao(t), yo(t), t) Version of October 15. 2003Massachusetts 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 Averaging1 This lecture presents results which describe local behavior of parameter-dependent 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 parameter-dependent systems of equations x˙ (t) = f(x(t), y(t), t), (10.1) �y˙ = g(x(t), y(t), t), where � → [0, �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 � > 0 converge to the solutions of (10.1) with � = 0 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 x0 : [t0, t1] ∞� Rn, y0 : [t0, t1] ∞� Rm be continuous functions satisfying equations x˙ 0(t) = f(x0(t), y0(t), t), 0 = g(x0(t), y0(t), t), 1Version of October 15, 2003