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 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 i(t=a(a(t), t, u),. (to)=To(a), (9.1) here u is a parameter. When a and To are differentiable with respect to u, the solution c(t)=a(t, u)is differentiable with respect to u as well. Moreover, the derivative of r(t, u) with respect to u can be found by solving linear ODE with time-varying coefficients Theorem9.1Leta:Rn×R×R→ r be a continuous function,p∈R.Let o: to, ti b+ r be a solution of(9.1)with u= Ho. Assume that a is continuously differentiable with respect to its first and third arguments on an open set X such that (o(t), t, Ho)E X for allt e [ to, t1. Then for all u in a neigborhood of po the ODE in(9.1) has a unique solution a(t)=r(t, u. This solution is a continuously differentiable function of u, and its derivative with respect to u at u= Ho equals A(t), where 4: to, t HR is the n-by-k matric-valued solution of the ODE △(t)=A(t)△(t)+B(t),△(to)=△o, (9 where A(t) is the derivative of the mapiHa(i, t, uo) with respect to i atI=ro(t), B(t) is the derivative of the map u Ha(ao(t),t, u)at A=Ho, and Ao is the derivative of the mp口o(1)atu=p Version of october 10. 2003Massachusetts 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 Trajectories1 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(t0) = ¯x0(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 : Rn × R × Rk ∞� Rn be a continuous function, µ0 ≤ Rk. Let x0 : [t0, t1] ∞� Rn be a solution of (9.1) with µ = µ0. Assume that a is continuously differentiable with respect to its first and third arguments on an open set X such that (x0(t), t, µ0) ≤ X for all t ≤ [t0, t1]. Then for all µ in a neigborhood of µ0 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 = µ0 equals �(t), where � : [t0, t1] ∞� Rn,k is the n-by-k matrix-valued solution of the ODE �(˙ t) = A(t)�(t) + B(t), �(t0) = �0, (9.2) where A(t) is the derivative of the map x¯ ∞� a(¯x, t, µ0) with respect to x¯ at x¯ = x0(t), B(t) is the derivative of the map µ ∞� a(x0(t), t, µ) at µ = µ0, and �0 is the derivative of the map µ ∞� x¯0(µ) at µ = µ0. 1Version of October 10, 2003