Using chaos to shadow the quadratic map for all time

We present a new way of proving that a computer-generated orbit for the chaotic attractor outside the periodic windows of the quadratic map f_a = ax (1 - x) can be shadowed for all time (i-e., there exist true orbits {x_y}^kj-1_i=0 which stay near a numerical orbit {p_i}^N_i=0 for all time). This is done by computing a numerical orbit for a particular value of a and show that {p_i}^N_i=0 ≈ ∪^m_j=1 {x_ij}^kj-1_i=0 where Σ^m_j=1 k_j = N. The true orbits are found using slightly different maps f_a, = a_jx(1 - x), where max_1≤j≤m (a_j - a) < √δ_p. This technique can therefore be applied to other chaotic differential equation and discrete systems.