EE222 - Spring 16 - Lecture 2 Notes 1 Murat Arcak January 21 2016 1 Licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Essentially Nonlinear Phenomena Continued 1. Finite escape time 2. Multiple isolated equilibria 3. Limit cycles: Linear oscillators exhibit a continuum of periodic orbits; e.g., every circle is a periodic orbit for ẋ = Ax where 0 β A = (λ 1,2 = jβ). β 0 In contrast, a limit cycle is an isolated periodic orbit and can occur only in nonlinear systems. harmonic oscillator limit cycle van der Pol oscillator i L i R C = i L + v 3 C L i L = C + L i R = + v 3 C "negative resistance" i L
ee222 - spring 16 - lecture 2 notes 2 4. Chaos: Irregular oscillations, never exactly repeating. Lorenz system (derived by Ed Lorenz in 1963 as a simplified model of convection rolls in the atmosphere): ẋ = σ(y x) ẏ = rx y xz ż = xy bz. Chaotic behavior with σ = 10, b = 8/3, r = 28: 30 50 45 20 40 10 0-10 y(t) z 35 30 25 20 15-20 10 5-30 0 10 20 30 40 50 60 t 0-20 -15-10 -5 0 5 10 15 20 x For continuous-time, time-invariant systems, n 3 state variables required for chaos. n = 1: x(t) monotone in t, no oscillations: f (x) x n = 2: Poincaré-Bendixson Theorem (to be studied in Lecture 3) guarantees regular behavior. Poincaré-Bendixson does not apply to time-varying systems and n 2 is enough for chaos (Homework problem). For discrete-time systems, n = 1 is enough (we will see an example in Lecture 5). Planar (Second Order) Dynamical Systems Phase Portraits of Linear Systems: ẋ = Ax Chapter 2 in both Sastry and Khalil Distinct real eigenvalues T 1 AT = λ 1 0 0 λ 2
ee222 - spring 16 - lecture 2 notes 3 In z = T 1 x coordinates: ż 1 = λ 1, ż 2 = λ 2. The equilibrium is called a node when λ 1 and λ 2 have the same sign (stable node when negative and unstable when positive). It is called a saddle point when λ 1 and λ 2 have opposite signs. stable node unstable node saddle λ 1 < λ 2 < 0 λ 1 > λ 2 > 0 λ 2 < 0 < λ 1 Complex eigenvalues: λ 1,2 = α jβ T 1 α AT = β β α ż 1 = α β polar coordinates ṙ = αr ż 2 = α + β θ = β stable focus unstable focus α < 0 α > 0 α = 0 center Phase Portraits of Nonlinear Systems Near Hyperbolic Equilibria hyperbolic equilibrium: linearization has no eigenvalues on the imaginary axis Phase portraits of nonlinear systems near hyperbolic equilibria are qualitatively similar to the phase portraits of their linearization. According to the Hartman-Grobman Theorem (below) a continuous deformation maps one phase portrait to the other.
ee222 - spring 16 - lecture 2 notes 4 x h Hartman-Grobman Theorem: If x is a hyperbolic equilibrium of ẋ = f (x), x R n, then there exists a homeomorphism 2 z = h(x) defined in a neighborhood of x that maps trajectories of ẋ = f (x) to those of ż = Az where A f. x=x x 2 a continuous map with a continuous inverse The hyperbolicity condition can t be removed: ẋ 1 = x 2 + ax 1 (x1 2 + x2 2 ) ẋ 2 = x 1 + ax 2 (x1 1 + x2 2 ) = ṙ = ar θ = 1 x = (0, 0) A = f 0 1 x = x=x 1 0 There is no continuous deformation that maps the phase portrait of the linearization to that of the original nonlinear model: 3 ẋ = Ax ẋ = f (x) (a > 0) Periodic Orbits in the Plane Bendixson s Theorem: For a time-invariant planar system ẋ 1 = f 1 (x 1, x 2 ) ẋ 2 = f 2 (x 1, x 2 ), if f (x) = f 1 x 1 + f 2 x is not identically zero and does not change 2 sign in a simply connected region D, then there are no periodic orbits lying entirely in D.
ee222 - spring 16 - lecture 2 notes 5 Proof: By contradiction. Suppose a periodic orbit J lies in D. Let S denote the region enclosed by J and n(x) the normal vector to J at x. Then f (x) n(x) = 0 for all x J. By the Divergence Theorem: f (x) n(x) x Trace(A) = 0, f (x) n(x)dl = f (x)dx. J } {{ } } S {{ } = 0 = 0 ẋ = Ax, x R 2 can have periodic orbits only if e.g., A = 0 β β 0. J S ẋ 1 = x 2 ẋ 2 = δx 2 + x 1 x 3 1 + x2 1 x 2 δ > 0 f (x) = f 1 x 1 + f 2 x 2 = x 2 1 δ Therefore, no periodic orbit can lie entirely in the region x 1 δ where f (x) 0, or δ x 1 δ where f (x) 0, or x 1 δ where f (x) 0. not possible: x 2 x 1 x 1 = δ x 1 = δ possible: x 2 x 1 x 1 = δ x 1 = δ