Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j(Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 3: Continuous Dependence On Parameters Arguments based on continuity of functions are common in dynamical system analy They rarely apply to quantitative statements, instead being used mostly for proofs of existence of certain objects(equilibria, open or closed invariant set, etc. ) Alternatively, continuity arguments can be used to show that certain qualitative conditions cannot be satisfied for a class of systems 3.1 Uniqueness Of Solutions In this section our main objective is to establish sufficient conditions under which solutions of ode with given initial conditions are unique. 3.1.1 A counterexample Continuity of the function a: R"HR on the right side of ODE c(t)=a(r(t)), a(to)=io does not guarantee uniqueness of solutions Example 3.1 The ODE i(t)=31x(t)2/3,x(0)=0 has solutions a(t)=0 and x(t)=t(actually, there are infinitely many solutions in this I Version of September 12, 2003Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 3: Continuous Dependence On Parameters1 Arguments based on continuity of functions are common in dynamical system analysis. They rarely apply to quantitative statements, instead being used mostly for proofs of existence of certain objects (equilibria, open or closed invariant set, etc.) Alternatively, continuity arguments can be used to show that certain qualitative conditions cannot be satisfied for a class of systems. 3.1 Uniqueness Of Solutions In this section our main objective is to establish sufficient conditions under which solutions of ODE with given initial conditions are unique. 3.1.1 A counterexample Continuity of the function a : Rn ∈� Rn on the right side of ODE x˙ (t) = a(x(t)), x(t0) = ¯x0 (3.1) does not guarantee uniqueness of solutions. Example 3.1 The ODE x˙ (t) = 3|x(t)| 2/3 , x(0) = 0 has solutions x(t) ≥ 0 and x(t) ≥ t3 (actually, there are infinitely many solutions in this case). 1Version of September 12, 2003