Using chaos to shadow the quadratic map for all time

We present a new way of proving that a computer-generated orbit for the cha otic attractor outside the periodic windows of the quadratic map f(a) = ax (1-x) can be shadowed for ail time (i.e., there exist true orbits {x(ij)}(i =0)(kj-1) which slay near a numerical orbit {p(i)}(i=0)(N) for all time). T his is done by computing a numerical orbit for a particular value of a and show that {pi}(i=0)(N)approximate to boolean ORj=1m{x(ij)}(i=1)(kj-1) where Sigma(j=1 )(m) k(j)=N The true orbits are found using slightly different maps f(aj) =a(j)x(1 -x), where max (1 less than or equal to j less than or equal to m) (a(j) - a) < root delta(p). This technique can therefore be applied to other chaotic differential equat ion and discrete systems.