Asymptotic stability of gradient homogeneous systems

The homogeneous eigenvalue is a useful tool for analyzing homogeneous systems. Although we obtained necessary conditions and sufficient conditions for asymptotic stability in our previous paper, there remains the natural question of whether the condition “all homogeneous eigenvalues must be negative” becomes a necessary and sufficient condition. In this paper, we define “gradient homogeneous systems.” In addition, we analyze the structure of gradient homogeneous systems and clarify the gradient homogeneous condition is very severe. Moreover, we describe the main theorem of this paper: that gradient homogeneous systems are asymptotically stable if and only if all their homogenous eigenvalues are negative. Finally, we demonstrate the effectiveness of the proposed theorem via an example.