set5_slides_fall09

set5_slides_fall09 - Hartman-Grobman Theorem in n...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Hartman-Grobman Theorem in n Dimensions Definition: An equilibrium point of is said to be hyperbolic if all eigenvalues of the Jacobian matrix have non-zero 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 > & H-1 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: Bendixson-Dulac 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...
View Full Document

This note was uploaded on 12/28/2011 for the course ME 580 taught by Professor Na during the Fall '10 term at Purdue.

Page1 / 46

set5_slides_fall09 - Hartman-Grobman Theorem in n...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online