In this work, we investigate the transition from regular dynamics to chaotic behavior in a one-dimensional quartic anharmonic classical oscillator driven by a time-dependent external square-wave force. Owing to energy conservation, the motion of the undriven quartic anharmonic oscillator is regular, periodic, and stable. For the driven quartic anharmonic oscillator, the equations of motion cannot be solved analytically due to the presence of an anharmonic term in the potential energy function. Using the fourth-order Runge-Kutta method to numerically solve the equations of motion for the driven quartic anharmonic oscillator, we find that the oscillator motion under the influence of a sufficiently small driving force remains regular, while by gradually increasing the driving force, a series of nonlinear resonances can occur, grow, overlap, and ultimately disappear due to the emergence of chaos.