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131F08.classnotes4

131F08.classnotes4 - ∈ G Note that the last condition is...

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Class notes #4 for MTH G131: Fall 2008 We consider the two-dimensional autonomous system ˙ x = f ( x,y ) , ˙ y = g ( x,y ) Lyapunov function Suppose the origin (0 , 0) is a stationary point for the system. Let G be an open neighborhood of the origin, and let V ( x,y ) be a continuously differentiable function on G . Then V ( x,y ) is a Lyapunov function for the system if (i) V (0 , 0) = 0, (ii) V ( x,y ) > 0 for all ( x,y ) G \ { (0 , 0) } , and (iii) ˙ V 0 for all ( x,y
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Unformatted text preview: ) ∈ G . Note that the last condition is equivalent to (iv): f ( x,y ) ∂V ∂x + g ( x,y ) ∂V ∂y ≤ Poincare-Bendixson Theorem Suppose that for some initial condition the solution ( x ( t ) ,y ( t )) enters and does not leave some closed, bounded domain D , and that there are no stationary points in D . Then there is at least one periodic orbit in D ....
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