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Unformatted text preview: HartmanGrobman Theorem in n Dimensions Definition: An equilibrium point of is said to be hyperbolic if all eigenvalues of the Jacobian matrix have nonzero real parts . Theorem If is a hyperbolic equilibrium of = & ¡ ¡ x f(x) ¡ * Df(x ) * x * x = & ¡ ¡ x f(x), & then in a neighborhood of , is topologically equivalent to the linearized vector field (this implies linear stability ⇔ nonlinear stability) = & ¡ ¡ ¡ * x D f(x ) x = & ¡ ¡ x f(x) * x We now want to talk about ways to detect limit cycles & Simulation of van der Pol oscillator & Bendixson criterion & Poincare Bendixson theorem Recall the van der Pol oscillator > = =  μ & & 2 The equation of motion is : x y, y x y(x 1) & for μ > < & H > & H1 1 stable limit cycle > x 1 Bendixson’s (Negative ) Criterion The objective is to prove conditions under which no limit cycles can exist. Theorem : If then there are no closed phase paths in a simply connected (no holes!) region of phase space in which the divergence of the vector field ( ) is of one sign . = = & & x f(x,y), y g(x,y) is the system, + ,x ,y f g & & The proof is by contradiction !! Proof : Let ℑ be such a closed path. Inside the region, the divergence is of one sign x y ℑ n t 2 2 ( , ) / ( , ) Here f x y t f g g x y & ¡ = + ¢ £ ¤ ¥ By Divergence Theorem However, the area integral on left cannot vanish & we have a contradiction & the enclosing curve ℑ is not a closed path. ℑ + = • = ℑ ¡¡ ¡ & & ,x ,y interior of (f g )dx dy t n ds ¡ & Ex: Consider the system We want to prove that no limit cycles are possible any where in phase space. Soln : The system in vector form is = = =  = ¡ ¡ 3 x y f(x,y), y y x g(x,y) ¢ + = = 2 2 ,x ,y Then, f g 3y 3y non positive + + = ¡¡ ¡ 3 x (x) x never changes sign in the entire phase space! i.e., no limit cycles can exist! (Bendixson’s criterion) There are generalizations: BendixsonDulac criterion Poincare Bendixson Theorem (only valid in 2D) Let ℜ be a closed bounded region in the phase plane; Let be continuously differentiable; et ot contain any fixed points; = & ¡ ¡ x f(x) + ,x ,y f g & Let ℜ not contain any fixed points; Suppose that there is a trajectory ℑ which is confined to ℜ (starts and stays always in ℜ ); Then either ℑ is a closed orbit (limit cycle) or it spirals towards it as → ∞ t . Notes: conditions (1) – (3) are easily satisfied;  for condition (4) Trapping region or positively invariant region & no equilibrium points in this region Ex 1: When (a) Show that a closed orbit (limit cycle) exists at r = 1. (b) Show that the limit cycle continues to exist for & > 0 ( & is small) (a) = + μ θ & 2 consider the system r r(1 r ) r cos θ = & 1 μ = 0 : = = θ = & & 2 or 0, r r(1 r ), 1 & Since r and θ equations are decoupled, it is enough to consider the r equation: the velocity function is μ = = θ = For 0, r r(1...
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This note was uploaded on 12/28/2011 for the course ME 580 taught by Professor Na during the Fall '10 term at Purdue.
 Fall '10
 NA

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