A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science Part VII: ISLES of Eden

نویسندگان

  • Leon O. Chua
  • Junbiao Guan
  • Valery I. Sbitnev
  • Jinwook Shin
چکیده

This paper continues our quest to develop a rigorous analytical theory of 1-D cellular automata via a nonlinear dynamics perspective. The 18 yet uncharacterized local rules are henceforth partitioned into ten complex Bernoulli στ -shift rules and eight hyper Bernoulli στ -shift rules, the latter including such famous rules 30 and 110 . All exhibit a bizarre composite wave dynamics with arbitrarily large Bernoulli velocity σ and Bernoulli return time τ as the length L → ∞. Basin tree diagrams of all ten complex Bernoulli στ -shift rules are exhibited for lengths L = 3, 4, . . . , 8. Superficial as it may seem, these basin tree diagrams suggest general qualitative properties which have since been proved to be true in general. Two such properties form the main results of this paper; namely,

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science Part VIII: More ISLES of Eden

This paper presents the basin tree diagrams of all hyper Bernoulli στ -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 (...

متن کامل

A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science Part I: Threshold of Complexity

This tutorial provides a nonlinear dynamics perspective to Wolfram’s monumental work on A New Kind of Science. By mapping a Boolean local Rule, or truth table, onto the point attractors of a specially tailored nonlinear dynamical system, we show how some of Wolfram’s empirical observations can be justified on firm ground. The advantage of this new approach for studying Cellular Automata phenome...

متن کامل

A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science Part XII: Period-3, Period-6, and permutive Rules

This 12th part of our Nonlinear Dynamics Perspective of Cellular Automata concludes a series of three articles devoted to CA local rules having robust ω-limit orbits. Here we consider only the two rules, 131 and 133 , constituting the third of the six groups in which we classified the 1D binary Cellular Automata. Among the numerous theoretical results contained in this article, we emphasize the...

متن کامل

A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science Part V: Fractals Everywhere

This fifth installment is devoted to an in-depth study of CA Characteristic Functions, a unified global representation for all 256 one-dimensional Cellular Automata local rules. Except for eight rather special local rules whose global dynamics are described by an affine (mod 1 ) function of only one binary cell state variable, all characteristic functions exhibit a fractal geometry where self-s...

متن کامل

A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science Part III: Predicting the Unpredictable

We prove rigorously the four cellular automata local rules 110, 124, 137 and 193 have identical dynamic behaviors capable of universal computations. We exploit Felix Klein’s remarkable Vierergruppe to partition the 256 local rules studied empirically by Wolfram into 89 global equivalence classes of which only 50 may exhibit complex dynamics. We define a 24-element rotation group which induces 3...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • I. J. Bifurcation and Chaos

دوره 17  شماره 

صفحات  -

تاریخ انتشار 2007