Damage spreading for one-dimensional, nonequilibrium models with parity conserving phase transitions

The damage spreading (DS) transitions of two one-dimensional stochastic cellular automata suggested by Grassberger ($A$ and $B$) and the nonequilibrium kinetic Ising model of Menyhárd (NEKIM) have been investigated. These nonequilibrium models exhibit nondirected percolation universality class continuous phase transitions to absorbing states, exhibit parity conservation (PC) law of kinks, and have chaotic to nonchaotic DS phase transitions, too. The relation between the critical point and the damage spreading point has been explored with numerical simulations. For model $B$ the two transition points are well separated and directed percolation universality was found both for spin damage and kink damages in spite of the conservation of damage variables modulo 2 in the latter case. For model $A$ and NEKIM the two transition points coincide with drastic effects on the damage of spin and kink variables showing different time dependent behaviors. While the kink DS transition of these two models shows regular PC class universality, their spin damage exhibits a discontinuous phase transition with compact clusters and PC-like spreading exponents. In the latter case the static exponents determined by finite size scaling are consistent with that of the spins of the NEKIM model at the PC transition point. The generalized hyperscaling law is satisfied. Detailed discussion is given concerning the dependence of DS on initial conditions especially for the $A$ model case, where extremely long relaxation time was found.