Stability: The Critical Case
In the previous chapter two procedures to check the stability of the origin of an autonomous vector differential equation were described. In the first approach, the Liapunov direct method, the stability of the origin is inferred by a function V(x) with suitable properties. In the second approach, the stability of the origin is determined by the linearized version of (5.1). This method is very simple because it requires knowledge of the eigenvalues of the Jacobian matrix A of the right-hand side of (5.1), evaluated at the origin. However, this method is applicable when the origin is asymptotically stable or unstable for the linearized equation; that is, if A has no eigenvalue whose real part vanishes. When the real part of at least one eigenvalue vanishes, we are faced with a critical case (see, e.g., [111,)
KeywordsFunction Versus Jacobian Matrix Canonical Form Phase Portrait Parametric Equation
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