▁general ▁characteristic ▁of ▁cha otic ▁time ▁series . ▁This ▁is ▁a ▁property ▁of ▁the ▁sensitive ▁dependence ▁on ▁initial ▁conditions ▁common ▁to ▁cha otic ▁time ▁series . ▁A ▁small ▁initial ▁error ▁is ▁ampl ified ▁with ▁time . ▁A ▁measure ▁of ▁the ▁diver gence ▁of ▁time ▁series ▁with ▁nearly ▁identical ▁initial ▁conditions ▁is ▁known ▁as ▁the ▁Ly ap un ov ▁exponent . ▁ ▁Control ▁of ▁a ▁cha otic ▁time ▁series ▁ ▁We ▁assume ▁the ▁output ▁of ▁the ▁log istic ▁map ▁can ▁be ▁manip ulated ▁through ▁a ▁control ▁parameter ▁ ▁such ▁that ▁ ▁. ▁ ▁The ▁goal ▁is ▁to ▁choose ▁the ▁control ▁parameter ▁in ▁such ▁a ▁way ▁as ▁to ▁drive ▁the ▁time ▁series ▁to ▁a ▁desired ▁output ▁. ▁This ▁can ▁be ▁done ▁if ▁we ▁choose ▁the ▁control ▁param er ▁to ▁be ▁ ▁where ▁ ▁is ▁an ▁approximation ▁to ▁the ▁underlying ▁natural ▁dynamics ▁of ▁the ▁system . ▁ ▁The ▁learning ▁algorithm ▁is ▁given ▁by ▁ ▁where ▁ ▁. ▁ ▁See ▁also ▁ ▁Rad ial ▁basis ▁function ▁kernel ▁ ▁In ▁Situ ▁Ada pt ive ▁Tab ulation ▁ ▁Pred ict ive ▁analyt ics ▁ ▁Cha os ▁theory ▁ ▁Hier arch ical ▁R BF ▁ ▁C ere bell ar ▁model ▁art ic ulation ▁controller ▁ ▁Inst ant ane ously ▁trained ▁neural ▁networks ▁ ▁References ▁ ▁Further ▁reading ▁ ▁J . ▁Mo ody ▁and ▁C . ▁J . ▁Dark en , ▁" F ast ▁learning ▁in ▁networks ▁of ▁locally ▁tun ed ▁processing ▁units ," ▁Ne ural ▁Comput ation , ▁ 1 , ▁ 2 8 1 - 2 9 4 ▁(