Areas of attraction for nonautonomous differential equations on finite time intervals (Q408247)

In this interesting paper, the authors establish new concepts for (exponential) attractivity in nonautonomous ODEs NEWLINE\[NEWLINE\dot x=f(t,x) \tag{\(*\)} NEWLINE\]NEWLINE on finite time-intervals \([0,T]\). Due to the general nonuniqueness of finite-time objects, whole areas of attraction rather than individual solutions are studied. NEWLINENEWLINENEWLINE NEWLINEMore precisely, (exponential) attractivity of solutions is characterized in terms of the time-\(T\)-map \(x(T,0,\cdot):{\mathbb R}^n\to{\mathbb R}^n\), where \(x\) is the general solution to \((\ast)\), as well as by means of an appropriate finite-time Lyapunov-function. Furthermore, using a time-varying Riemannian metric, necessary and sufficient conditions for the existence of an area of exponential attraction are deduced.NEWLINENEWLINEVarious examples carefully illustrate and motivate the introduced concepts and results.