Abstract
In this paper a discrete dynamical system is considered . There is a dial with $N$ positions (vertices) and $M$ particles. Particles are located in vertices. Each particle moves, at every time unit, in accordance with its plan. The plan is logistics, given through a real number which belongs to the segment $[0,1].$ The number is represented in positional numeral system with base $N$ equal to the number of vertices. A competition takes place if particles must move in opposite directions simultaneously. A rule of competition resolution is given. Systems characteristics are investigated for sets of rational and irrational plans. Some algebraic constructions are introduced for this purpose. Probabilistic analogues (random walks) are also considered.
Highlights
A dynamical system, named bipendulum, was introduced and investigated in (Kozlov, Buslaev, & Tatashev, 2015c)
The plan is logistics, given through a real number which belongs to the segment [0, 1]
A competition takes place if particles must move in opposite directions simultaneously
Summary
A dynamical system, named bipendulum, was introduced and investigated in (Kozlov, Buslaev, & Tatashev, 2015c). Digits of plans are shifted to the left onto a position, and the digit to the left of the point is excluded This transition is equivalent to multiplication of a number by 2 and exclusion of the integral part. The algebra WN, depending on N, is represented uniquely as a sum of connected components which are subalgebras WNi , where Ni are divisors of N In this case, dynamics of particles can be interpreted as motion of particles on the subalgebra graph, where weights of arcs generate logistics. The plan of each particle is given by a number which belongs to the segment [0, 1] This number is represented in the numeral positional system with the base equals to the number of vertices. Some discrete dynamical systems with symmetric periodic structures were introduced and investigated in (Kozlov, Buslaev, Tatashev & Yashina ,2014) (Kozlov, Buslaev & Tatashev, 2015b), (Kozlov, Buslaev & Tatashev, 2015a)
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