MATH528

MATH528 Lesson11: Qualitative nonlinear ODE system analysis

Algorithm ODE system qualitative analysis

For ODE $𝒚^{'} = 𝒇 (𝒚)$ , find roots $𝒚^{*}$ , $𝒇 (𝒚^{*}) = 𝟎$ .
Taylor series expand $𝒇 (𝒚) = 𝒇 (𝒚^{*}) + 𝑨 (𝒚 - 𝒚^{*}) + 𝒪 ({|| 𝒚 - 𝒚^{*} ||}^{2})$
Determine type of critical point from eigenvalues of the Jacobian
$𝑨 = 𝑱 (𝒇 (𝒚^{*})) = \frac{\partial 𝒇}{\partial 𝒚} (𝒚^{*}) .$

Free undampened pendulum

θ^{''} + k \sin θ = 0, (k = g / l)

𝒚^{'} = (\begin{array}{l} θ^{'} \\ ω^{'} \end{array}) = 𝒇 (𝒚) = (\begin{array}{l} ω \\ - k \sin θ \end{array})

Critical points are $𝒚_{j}^{*} = (\begin{array}{l} j π \\ 0 \end{array})$ . Near a critical point the linearization of the system gives

𝒚^{'} = 𝑨 (𝒚 - 𝒚_{j}^{*}), 𝑨 = \frac{\partial 𝒇}{\partial 𝒚} (𝒚_{j}^{*}) = (\begin{array}{ll} 0 & 1 \\ - k \cos (j π) & 0 \end{array}) = (\begin{array}{ll} 0 & 1 \\ {(- 1)}^{j + 1} k & 0 \end{array})

Eigenvalues of $𝑨$ :