Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 3.14

Nonautonomous Linear Systems

Mathematics — Calculus · stability-2x2-linear-systems

Problem

Define f:R2→R2f : \mathbb{R}^2 \to \mathbb{R}^2 by f(x)=Axf(x) = Ax, where A=[a11a12a21a22]A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} is a nonzero matrix with real entries. Let tr A=a11+a22anddet⁡A=a11a22−a12a21.\mathrm{tr}\, A = a_{11} + a_{22} \quad \text{and} \quad \det A = a_{11}a_{22} - a_{12}a_{21}. Prove the following statements. (a) If tr A<0\mathrm{tr}\, A < 0 and det⁡A>0\det A > 0, then the origin is asymptotically stable for ff. (b) If tr A<0\mathrm{tr}\, A < 0 and det⁡A=0\det A = 0, then the origin is stable for ff. (c) If tr A=0\mathrm{tr}\, A = 0 and det⁡A>0\det A > 0, then the origin is stable for ff. (d) In all other cases, the origin is unstable for ff.

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