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Stability of dynamical systems in the plane. (English) Zbl 0724.34060
The authors investigate the stability and the asymptotic stability of an ordinary differential equation in the plane, \(\dot x(t)=A(t)x(t)+h(t,x(t)),\) where \(A(t)=(a_{ij}(t))\), \(i,j=1,2\), is a square matrix and \(h(t,x)=(h_ 1(t,x),h_ 2(t,x))\) is a vector. The above system is first transformed into an equation with complex conjugate coordinates, \(\dot z=az+b\bar z.\) This transformed system is then studied using a Lyapunov-like function. The stability results obtained are tested on basic mathematical models which represent damped oscillatory phenomena. In this case the result is more general than the results of Z. Artstein and E. F. Infante [Quart. Appl. Math. 34, 195-199 (1976; Zbl 0336.34048)] and T. Tatarkiewicz [Ann. Univ. Mariae Curie-Skłodowska, Sect. A 7, 19-81 (1954; Zbl 0058.071)].
Reviewer: E.Chukwu (Raleigh)

34D20 Stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations