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Stability and asymptotic properties of dynamical systems in the plane. (English) Zbl 0728.34052
Differential equations and their applications, Proc. 7th Conf., Equadiff 7, Prague/Czech. 1989, Teubner-Texte Math. 118, 103-105 (1990).
[For the entire collection see Zbl 0704.00019.]
By means of the transformation \(y=Bx\) with \(y=(z,\bar z)\), \(x=(x_ 1,x_ 2)\), and \(B=\left[\begin{matrix} 1&i \\ 1&-i \end{matrix}\right]\), the plane dynamical system (1) \(x'=A(t)x+h(t,x)\), with \(A(t)=(a_{ij}(t))\), \(t\in [t_ 0,\infty)\), \(h(t,x)=(h_ 1(t,x),h_ 2(t,x))\), and \(x=(x_ 1,x_ 2)\), is converted to the equation (2) \(y'=BA(t)B^{- 1}y+Bh(t,B^{-1}y)\) where \[ BAB^{-1}= \left[\begin{matrix} a&b \\ \bar b&\bar a \end{matrix}\right], \] a\(=1/2(a_{11}+a_{22})+i/2(a_{21}- a_{12})\), \(b=1/2(a_{11}-a_{22})+i/2(a_{21}+a_{12})\). The first equation of (2) is given by (3) \(z'=a(t)z+b(t)\bar z+g(t,z,\bar z)\) with \(g(t,z,\bar z)=h_ 1(t,1/2(z+\bar z),1/2i(z-\bar z))+ih_ 2(t,1/2(z+\bar z),1/2i(z-\bar z))\). The asymptotic properties of (3) are related to those of (1). In the paper, sufficient conditions are given for the trivial solution of (3) to be stable or asymptotically stable.

34D20 Stability of solutions to ordinary differential equations
34M99 Ordinary differential equations in the complex domain
Zbl 0704.00019