Scaling theory and dimensional arguments for periodic solutions of spring-block model

Earlier, the authors have studied a one-dimensional version of the Burridge-Knopoff model [Burridge R & Knopff L, Bull Seismol Soc Am, 57 (1967) 341] of N-site chain of spring-blocks with stick-slip dynamics. Their numerical analysis and computer simulations lead to a set of different results corresponding to different boundary conditions [Gutenberg B & Richter C F, Ann Geofis, 9 (1956) 1]. The authors showed that, there are stable periodic solutions in a parameter-space. They have presented elsewhere, arguments to justify the occurrence of lower edges of the window [Carlson J M & Langer J S, Phys Rev Lett, 62 (1989) 2632]. The authors follow here, the same arguments and show that the lower and upper edges of the window can be understood in terms of simple scaling arguments and can be presented as a function of two dimensionless parameters. They try to understand the origin of the results and give a theory to explain them. An improved formula for lower threshold driving velocity will be presented, which-is in good agreement with their numerical experiments inside the parameters window of periodic solutions. Also, they discuss different types of instability occurring in the system.