It has been quite common in the analysis of single- or multi- fractal signals originating from complex nonlinear systems to make a time-delayed construction of the state space attractor in which the dynamics can be qualitatively viewed. This involves the calculations of the embedding dimension and an appropriate time delay based on the signal nonlinear correlation behavioral pattern. This is usually followed by a sub-optimal short-term linear prediction in the signal time/subspace domain instead of the optimal nonlinear prediction in the time/frequency domain. In this paper, we propose to alleviate the sub-optimality problem and exploit the nonlinear signal dynamics embedded in the attractor and integrate them in the design of a new family of temporal multiple- step Volterra predictors. Essentially, this is done by including relevant past information preserved in the signal up to a time td equals the embedded dimension X embedded time delay, and sampled at instances coincident with the embedded time delays, to predict one-step ahead, adaptively, using the LMS criteria. The results obtained using several nonlinear (chaotic or non-chaotic) synthetic and measured biomedical signals and performing the novel quadratic- and cubic-Volterra predictions show superior MSE performance, of as much as 30 dB, over those obtained using an optimized conventional one-step Volterra predictor of the same order, particularly in the case of electrocardiogram signals. This is not achieved at the expense of any increase in the CPU time as the algorithm is designed to cater for new parallel Volterra structures, with progressive delayed inputs.