Generating chaos via .x vertical bar x vertical bar

This paper investigates the role of the function x \x \ as a chaos generato r in nonautonomous systems. A Duffing-like nonautonomous oscillator is used for illustration. It is rigorously proven via the Melnikov function method that this particular quadratic function induces Smale horseshoes to the Du ffing-Like system. Moreover, its physical meaning as an energy function is demonstrated, which provides a critical value for the emerge of chaos. Simu lations with bifurcation analysis are given for better understanding of the underlying dynamics.