Convergence analysis of Davidchack and Laiʹs algorithm for finding periodic orbits

We rigorously study a recent algorithm due to Davidchack and Lai (DL) [Davidchack RL, Lai Y-C. Phys Rev E 1999;60(5):6172–5] for efficiently locating complete sets of hyperbolic periodic orbits for chaotic maps. We give theorems concerning sufficient conditions on convergence and also describing variable sized basins of attraction of initial seeds, thus pointing out a particularly attractive feature of the DL-algorithm. We also point out the true role of involutary matrices which is different from that implied by Schmelcher and Diakonos [Schmelcher P, Diakonos FK. Phys Rev E 1998;57(3):2739–46] and propagated by Davidchack and Lai.