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Unformatted text preview: Lectures on Periodic Orbits 11 February 2009 Most of the contents of these notes can found in any typical text on dynamical systems, most notably Strogatz [ 1994 ], Perko [ 2001 ] and Verhulst [ 1996 ]. Complete proofs have been omitted and wherever possible, references to the literature have been given instead. As these notes are being written during the term, they will be frequently updated. Weekbyweek changes are detailed on the course web page. 1 Ruling Out Periodic Orbits Gradient Systems. A gradient system is a dynamical system of the form ˙ x =∇ V ( x ) (1.1) for a given function V ( x ) in R n . Theorem 1.1. Gradient systems cannot have periodic orbits. Proof. Suppose to the contrary that γ : t 7→ x ( t ) is a periodic orbit of the gradient system ( 1.1 ) with period T . Then V ( x ( T )) V ( x (0)) = 0, but on the other hand V ( x ( T )) V ( x (0)) = Z T dV dt dt = Z T ∇ V · ˙ xdt = Z T k ˙ x k 2 dt < , a contradiction. Note that this result is valid for gradient systems in arbitrary dimensions, in contrast to the subsequent results, which are specific for the plane. Dulac’s Criterion. Recall that a region R of the plane is said to be simply connected if every closed loop within R can be shrunk to a point without leaving R (intuitively, R contains no holes). Dulac’s criterion gives sufficient conditions for the nonexistence of periodic orbits of dy namical systems in simply connected regions of the plane. The downside of this method is that it depends on the choice of an appropriate multiplier, which might be hard to find. Theorem 1.2. Let R be a simply connected region in R 2 and consider a planar dynamical system in R given by ˙ x = f ( x,y ) and ˙ y = g ( x,y ) , (1.2) 1 2 Index Theory 2 where f,g are C 1 functions in R . Suppose that there exists a C 1 function h ( x,y ) in R so that ∇ · h ( fe x + ge y ) has a definite sign in R . Then the dynamical system ( 1.2 ) cannot have any periodic orbits in R . Proof. Assume that ( 1.2 ) has a periodic orbit γ contained in R and let A be the area enclosed by γ . Since R is simply connected, A lies entirely in R . By Green’s theorem, we have ZZ A ∇ · h ( fe x + ge y ) dxdy = I γ h ( fe x + ge y ) · ndl, where n is the outward normal to γ . Now, note that the lefthand side of this expression is different from zero because of the signdefiniteness. However, the righthand side is zero (since fe x + ge y is tangent to γ ), a contradiction. A special case of this theorem, where the multiplier h = 1, is known as Bendixson’s criterion . When asked to verify whether a vector field can have periodic orbits, the first thing to check is whether Bendixson’s criterion is applicable: checking that the divergence of the vector field is sign definite can be a quick and straightforward way to rule out periodic orbits. When this fails, Dulac’s criterion or other techniques can be used....
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This note was uploaded on 01/04/2012 for the course CDS 140b taught by Professor List during the Fall '10 term at Caltech.
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