Limit Cycles and the Poincaré-Bendixson Theorem 1. Show that the nonlinear two dimensional systenm z'(t) x(t) + y(t)-エ(t)(z"(t)-r(t)), has the single critical point (0,0). Hint: show that the only solution to is y 0, by first changing to polar coordinates and then proving that-ea+v' = 0 is the only solution, i.e that there are no solutions (r.0) with r>0

Linear Algebra: A Modern Introduction
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Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
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Limit Cycles and the Poincaré-Bendixson Theorem
1. Show that the nonlinear two dimensional systenm
z'(t)
x(t) + y(t)-エ(t)(z"(t)-r(t)),
has the single critical point (0,0). Hint: show that the only solution to
is
y 0, by first changing to polar coordinates
and then proving that-ea+v' = 0 is the only solution, i.e that there are no
solutions (r.0) with r>0
Transcribed Image Text:Limit Cycles and the Poincaré-Bendixson Theorem 1. Show that the nonlinear two dimensional systenm z'(t) x(t) + y(t)-エ(t)(z"(t)-r(t)), has the single critical point (0,0). Hint: show that the only solution to is y 0, by first changing to polar coordinates and then proving that-ea+v' = 0 is the only solution, i.e that there are no solutions (r.0) with r>0
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