A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science Part Viii: More Isles of Eden

Abstract

This paper presents the basin tree diagrams of all hyper Bernoulli σ τ -shift rules for string lengths L = 3, 4, . . . , 8. These diagrams have revealed many global and time-asymptotic properties that we have subsequently proved to be true for all L < ∞. In particular, we have proved that local rule 60 has no Isles of Eden for all L, and that local rules 154 and 45 are inhabited by a dense set (continuum) of Isles of Eden if, and only if, L is an odd integer. A novel and powerful graph-theoretic tool, called Isles-of-Eden digraph, has been developed and can be used to test the existence of dense Isles of Eden of any local rule which satisfies certain constraints, such as rules 154 , 45 , 150 , 105 , as well as all invariant local rules, such as rules 170 , 240 , 15 and 85 , subject to no constraints.

The Topological Pressure of Linear Cellular Automata

“…They are indexed as 0, 60, 90, 102, 150, 170, 204, 240 respectively. When periodic boundary condition is considered, Chua et al [17] investigate the dynamical behavior of these rules such as Isles of Eden, period of attractors, and so on [17,18]. This demonstrates that additive CA can still propose rich dynamics.…”

Transitivity and Chaoticity in 1-D Cellular Automata

“…In fact, mathematical theory of CA was firstly developed by Hedlund about two decades after Neumann's seminal work [6]. Since 2002, Chua et al provided a rigorous nonlinear dynamical approach to Wolfram's empirical observations [7][8][9][10]. All elementary CA (ECA) rules are reorganized into four groups in terms of finite bit stings.…”

Chaos emerged on the ‘edge of chaos’

“…In 2002, Wolfram published his monumental work A New Kind of Science [38], which led to continuously rising interest in the subject later on. Since 2002, Chua et al have provided a nonlinear dynamics perspective to Wolfram's empirical observations from mathematical analysis, using concepts like the characteristic function, forward time-τ map, basin tree diagram and the Isle-of-Eden digraph [7][8][9][10]. Recently, the dynamical properties of Chua's periodic rules and the robust Bernoulli-shift rules with distinct parameters have also been investigated from the viewpoint of symbolic dynamics [2][3][4][5][6][22][23][24].…”