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M334Lec40

# M334Lec40 - Math 334 Lecture#40 9.6 Liapunovs Second Method...

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Math 334 Lecture #40 § 9.6: Liapunov’s Second Method When the linearization matrix at an isolated critical point has purely imaginary eigen- values, the Hartman-Grobman Theorem does not determine the type nor the stability of the equilibrium. Liapunov’s Second, or direct, Method provides a tool to determine the stability of such an equilibrium. We will describe this method under the assumption that the critical point is (or has been translated) to the origin. This method uses an auxiliary function V ( x, y ) defined on some open domain D contain- ing the origin, with V (0 , 0) = 0. We define four different types of V : 1. positive definite on D if V ( x, y ) > 0 for all nonzero ( x, y ) in D , 2. positive semidefinite on D if V ( x, y ) 0 for all nonzero ( x, y ) in D , 3. negative definite on D if V ( x, y ) < 0 for all nonzero ( x, y ) in D , and 4. negative semidefinite on D if V ( x, y ) 0 for all nonzero ( x, y ) in D . Lemma. On any domain D containing (0 , 0), the function V ( x, y ) = ax 2 + bxy + cy 2 is 1. positive definite on D if and only if a > 0 and 4 ac - b 2 > 0, and 2. negative definite on D if and only if a < 0 and 4 ac - b 2 > 0. For a solution x = φ ( t ) and y = ψ ( t ) of the system dx/dt = F ( x, y ) , dy/dy = G ( x, y ) near the isolated critical point (0 , 0), and for a function V defined on an open domain D containing (0 , 0) with V (0 , 0) = 0, we compute (by the chain rule), d dt V ( φ ( t ) , ψ ( t )) = V x ( φ ( t ) , ψ ( t )) φ ( t ) + V y ( φ ( t ) , ψ ( t )) ψ ( t ) = V x ( φ ( t ) , ψ ( t )) F ( φ ( t ) , ψ ( t )) + V y

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