Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j(Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 4: Analysis Based On Continuity I This lecture presents several techniques of qualitative systems analysis based on what is frequently called topological arguments, i.e. on the arguments relying on continuity of functions involved 4.1 Analysis using general topology arguments This section covers results which do not rely specifically on the shape of the state space and thus remain valid for very general classes of systems. We will start by proving gener alizations of theorems from the previous lecture to the case of discrete-time autonomous 4.1.1 Attractor of an asymptotically stable equilibrium Consider an autonomous time invariant discrete time system governed by equation r(t+1)=f(x(t),r(t)∈X,t=0.,1,2, (4.1) where X is a given subset of R",f: X H X is a given function. Remember that f is called continuous if f(xk)→f(x)ask→ o whenever k,x∈ X are such that k-Doo as k-oo). In particular, this means that every function defined on a finite set X is continuous One important source of discrete time models is discretization of differential equations Assume that function a: RhR is such that solutions of the ODe I Version of September 17, 2003Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 4: Analysis Based On Continuity 1 This lecture presents several techniques of qualitative systems analysis based on what is frequently called topological arguments, i.e. on the arguments relying on continuity of functions involved. 4.1 Analysis using general topology arguments This section covers results which do not rely specifically on the shape of the state space, and thus remain valid for very general classes of systems. We will start by proving gener￾alizations of theorems from the previous lecture to the case of discrete-time autonomous systems. 4.1.1 Attractor of an asymptotically stable equilibrium Consider an autonomous time invariant discrete time system governed by equation x(t + 1) = f(x(t)), x(t) ⊂ X, t = 0, 1, 2, . . . , (4.1) where X is a given subset of Rn, f : X ∞� X is a given function. Remember that f is called continuous if f(xk) � f(x�) as k � → whenever xk, x� ⊂ X are such that xk � x� as k � →). In particular, this means that every function defined on a finite set X is continuous. One important source of discrete time models is discretization of differential equations. Rn Assume that function a : ∞� Rn is such that solutions of the ODE x˙ (t) = a(x(t)), (4.2) 1Version of September 17, 2003