  Parametric modulation in nonlinear dynamical systems can give rise to
attractors on which the dynamics is aperiodic and nonchaotic, namely with
largest Lyapunov exponent being nonpositive. We describe a procedure for
creating such attractors by using random modulation or pseudo-random binary
sequences with arbitrarily long recurrence times. As a consequence the
attractors are geometrically fractal and the motion is aperiodic on
experimentally accessible timescales. A practical realization of such
attractors is demonstrated in an experiment using electronic circuits.
