  There are presented examples of the rather sudden and violent explosion of
the strange attractor of a one-dimensional driven damped anharmonic oscillator
induced by a relatively small change of the amplitude of the strongly
nonperturbative periodic driving force. A phenomenologic characterization of
the explosion of the strange attractor has been given in terms of the behavior
of the average maximal Lyapunov exponent ${\bar \lambda}$ and that of the
fractal dimension $D_{q}$ for $q=-4$. It is shown that the building up of the
exploding strange attractor is accompanied by a nearly linear increase of the
maximal average Lyapunov exponent ${\bar \lambda}$. A sudden jump of the
fractal dimension $D_{-4}$ is detected when the explosion starts off from an
attractor consisting of disjoint bunches separated by an empty phase-space
region.
