EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 3, 2018, 628-644 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Spectra of Local Cluster Flows on Open Chain of Contours Alexander P. Buslaev1,2, Alexander G. Tatashev2,1 1 Department of Higher Mathematics, Automobile Faculty, Moscow Automobile and Road Construction State Technical University (MADI), Moscow, Russia 2 Department of Mathematical Cybernetics and IT, Faculty of Information Technology, Moscow Technical University of Communications and Informatics, Moscow, Russia Abstract. A dynamical system is considered. This dynamical system is a flow of clusters with the same length l on contours of unit length connected into open chain. A similar system, contours of which are connected into closed chain, was considered earlier. It has been found that, in the case of closed chain of contours, the dynamical system has a spectrum of velocity and mode periodicity consisted of more than one component. In this paper, it has been shown that, in the case of open chain, the spectrum of cluster velocity and mode periodicity contains only one component. The conditions of self-organization and the dependence of cluster velocity on load l is developed. 2010 Mathematics Subject Classifications: 47J10, 05C21, 11J06, 76B75 Key Words and Phrases: Dynamical System, Spectrum, Self-Organization, Contour Graph, Cluster Model 1. Introduction. Chains, clusters, local flows We consider a system of N contours C1, . . . , CN (N ≥ 1). Each contour is a circle of unit length. On each contour, from the Eastern pole counterclockwise, a coordinate system is given. On the contour Ci, coordinates of points are xi ∈ [0, 1), i = 1, . . . , N. Contours C1, . . . , CN form a graph (chain). The point of the contour Ci(0) with coordinate 0 is identified with the point Ci+1(1/2), i = 1, . . . , N − 1. These points are called nodes of the contour network. We denote by (Ci, Ci+1) the common node of contours Ci and Ci+1. A fragment of a contour, which conserves the length and can move, is called a cluster. We consider a system with one cluster of length l on each contour. A state of the system is admissible if, for this state, no node is covered by more than one cluster. In the general state, each cluster moves counterclockwise, uniformly with velocity equal to 1. We shall call such movement local. Simultaneous movement of more than one cluster through a node is forbidden. If cluster B comes to a node when the DOI: https://doi.org/10.29020/nybg.ejpam.v11i3.3292 Email addresses: apal2006@yandex.ru (A.P. Buslaev), a-tatashev@yandex.ru (A.G. Tatashev) http://www.ejpam.com 628 c© 2018 EJPAM All rights reserved. A.P Buslaev, A.G. Tatashev / Eur. J. Pure Appl. Math, 11 (3) (2018), 628-644 629 cluster A of neighboring contour moves through the node, then the cluster B stops, and its movement does not continue while the cluster A covers the node. If two clusters come to the node (Ci, Ci+1) simultaneously, then the cluster of contour Ci (cluster Cli) moves through the node, and the cluster Cli+1 stops (left-priority conflict resolution rule), i.e., the cluster Cli+1 moves through the node first. The delay of the cluster A is the duration of time interval such that, in this interval, the cluster A waits the node release. The state of the system at the time t is a vector α(t) = (α1(t), . . . , αN (t)), where αi(t) is the coordinate of the frontal point of cluster Cli, i = 1, . . . , N, Fig. 1. The back point of cluster Cli is located in the point with coordinate αi(t) − l (substraction modulo 1), i = 1, . . . , N. C 1 C 2 C 3 C 4 Figure 1: State (0, 0, 0.9, 0.8) 2. Formulation of problem. Spectra of flows on graphs 2.1. Wreaths of trajectories and spectra of velocities The considered system is deterministic. The system behavior in the future is de- termined fully by the state α(t0) at current time t0. Each admissible state generates a trajectory α(t) in the space of admissible states. Two trajectories αA(t) and αB(t) can coincide at the some moments tA and tB. A wreath is a batch of trajectories such that any two trajectories join after a finite time interval. A pair of states on the wreath can be classified as recurrent if the system comes after a finite time from any of these states to the other state. A pair of states is called dependent if only from one of the states it is possible to come to the other state. A pair of states is called independent if from none of these states it is possible to come to the other state. Any recurrent pair of states form a cycle. The space of admissible states is divided into non-intersecting sets, which are trajectories, continuous modulo 1. These trajectories are piecewise linear functions with inclination 0 or 1. The system is in the the state of free movement at the time t0 if at any time t ≥ t0 all clusters move without delays. Self-organization is the property of the dynamical system such that the system comes to the state of free movement after a finite time interval from any admissible initial state. It is interesting to study the average velocity of cluster on each contour and the average velocity of all clusters of the system. The average velocity of the cluster Ci is defined as the limit, if this limit exists, vi = lim T→∞ Si(T ) T , A.P Buslaev, A.G. Tatashev / Eur. J. Pure Appl. Math, 11 (3) (2018), 628-644 630 where Si(T ) is the total distance such that the cluster passes this distance in the time interval (0, T ), i = 1, . . . , N. The average value of clusters velocities is called the average velocity of system clusters in time. In general case, partial self-organization takes place, [1]–[4]. The system comes to the state of free movement from some initial conditions. From the other admissible initial conditions, the free movement is not reached and the movement is characterized by the velocity less the 1. It is possible that the self-organization does not take place for any initial states. The set of admissible velocity values, for fixed values of system parameters and different initial states is called the spectrum of system velocities. The following problems are interesting for research. Is the spectrum of velocities a countable or continual set? Whether the trajectory of the process in the system state space is repeated cyclicly, and therefore the average distance covered by clusters per time unit, reaches a limit value, or the trajectory of the process in the system state space is an attractor and the average distance covered by clusters per time unit tends to a limit value? Whether the average velocity of each cluster is the same or the average velocities of different clusters can be different? The same value of the average velocity can correspond to the different cyclic trajectories in the system state space. How many possible cyclic trajectories correspond to the fixed value of velocity? Thus the study of systems, considered in [1]–[4], leads to the concept of the spectrum of limit cyclic trajectories in the state space and the spectrum of velocities corresponding to these trajectories. 2.2. Chain of discrete binary contours A closed chain of contours was studied in [4]. Each contour has common points (nodes) with two neighboring contours. There are two cells (the lower cell - cell 0, and the upper cell - cell 1) and a particle on each contour. In every discrete moment, the particle is located in upper (lower) cell and, if this is allowed, moves counterclockwise to the lower (upper) cell, Fig. 2. Figure 2: The binary vector 110010 Particles cannot move through the common node simultaneously. If the particle, lo- cated to the right of the node, tries to move from the upper cell to the lower cell, and the particle, located to the left of the node, tries to move from the lower cell to the upper cell, A.P Buslaev, A.G. Tatashev / Eur. J. Pure Appl. Math, 11 (3) (2018), 628-644 631 then there is a conflict. Assume that, in the case of conflict, the particle, located to the left of the node, moves, and the particle, located to the right of the node, does not move. 2.2.1. Open chain of contours with binary states It is obvious that, in the case of open chain of contours, Fig. 3, for any of 2N initial states (all possible states are admissible in this case), no more than after time 2N all particles will move without delays. Figure 3: The state 110010 on an open chain 2.2.2. Closed chain with binary states The following has been proved in [4]. (i) The dynamical system is equivalent to elementary cellular automaton CA 063 in terms of Wolfram classification, [5]. States of the system are cyclic vectors with N coordinates. The ith coordinate of vector equals 0 if the particle is in the cell 0, and equals 1 if the particle is in the cell 1. At any discrete moment, the value of each coordinate is changed except the case in which the value of this coordinate equals 1, and the value of the neighboring coordinate on the left equals 0. (ii) The space of states is divided into the set of recurrent states and the set of nonrecur- rent states. A state is non-recurrent if and only if the vector of this state contains at least one coordinate such that the value of this coordinate is equal to 1, and the values of neighboring coordinates on the left and on the right are equal to 0. (iii) The system can be in a non-recurrent state only at the initial moment. (iv) Each recurrent state is repeated after no more than 2N steps, Fig. 4. (v) The vector of state is shifted onto one position to the right for every two steps. (vi) The quantity of sign changes in the state vector, divided by 2, is called the system variation. The variation decreases if the system moves through a non-recurrent state to a recurrent state, and does not change if the system is in the recurrent state. (vii) If the system is in a recurrent state, then the variation is not more than [N/3] - integer part of N/3. A.P Buslaev, A.G. Tatashev / Eur. J. Pure Appl. Math, 11 (3) (2018), 628-644 632 t = 0 t = 1 t = 3 t = 5 t = 7 t = 2 t = 4 t = 6 t = 8 0 1 0 0 1 1 1 1 0 1 1 1 0 0 0 1 1 1 1 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 0 1 Figure 4: A cyclic trajectory in the state space (viii) If the system is in a recurrent state at initial moment and the variation is equal to k, then the average velocity of particles is equal to (N − k)/N, k = 0, 1, . . . , [N/3]. (ix) If the initial state is non-recurrent, then the average velocity of particles is equal to (N−k)/N, where k is the value of variation at the next moment, k = 0, 1, . . . , [N/3]. (x) If k is an integer value and satisfies the condition 0 ≤ k ≤ [N/3], then there exists an initial state such that the average velocity equals (N − k)/N. (xi) For any initial condition, the value of average velocity satisfies the above conditions. 2.2.3. The open chain with two contours with discrete states A two contour system with a common point (node) is studied in [1], [2]. There are fixed quantities of cells and particles on each contour. Particles are located in cells and move in accordance with a given rule. In particular, this system can be an open chain with two A.P Buslaev, A.G. Tatashev / Eur. J. Pure Appl. Math, 11 (3) (2018), 628-644 633 contours and one cluster of particles on each contour. If the contour length (the quantity of cells on the contour) is the same for the contours, then the velocity of particles does not depend on the initial state, i.e., if the system parameters are fixed, the spectrum of possible states contains only one value. It is noted in [2], that the average velocity of clusters can depend on initial state if the lengths of clusters are different. 2.2.4. Closed chain of three contours with continual clusters A closed dynamical chain of contours is considered in [3]. A cluster of the same length moves on each contour. This system is characterized by continuous state space and time. Each contour has common nodes with two neighboring contours. It is found that, after a finite moment the system is in a set of recurring states, and the average velocity of clusters depends, in general, on the set in which the system is. In what set of recurring states system will be, depends on the initial state. It has been found that the spectrum of velocity values is finite. If the length of a cluster is more than a half, then, for any quantity of contours, all clusters stop (collapse) after a finite time. Therefore, in this case, the spectrum of velocities contains only the value 0. In the case of three clusters, it has been proved that, if the length of cluster is not more than 1/6, then the system comes to the state of free movement from any initial state, i.e., the spectrum of velocities contains only the value 1. If the cluster length l satisfies the condition 1/6 < l < 1/2, then the spectrum contains the value 1 and one value not equal to 1. Hypotheses, characterizing the motion on the closed chain of contours, have been formulated in [3]. If the quantity of contours is not less than 6, then the spectrum of velocities can contain more than one value. It has been proved that, for any arbitrarily small value, there exist values of the quantity of contours and the initial state such that the system does not come to the states of free movement. 2.2.5. Open chain of contours with uniform load In this paper, we consider a one-dimensional system of contours. This system differs from the closed chain in that the leftmost (rightmost) contour has common node only with one neighboring contour (open chain of contours). We have proved that for an open chain, as for a closed chain, after a time interval, the same set of states is repeated with a period. For the open chain, this set is determined by the quantity of contours and the cluster length and does not depend on the initial states. Hence the spectrum of velocities of clusters contains only one value if the quantity of contours and the cluster length are fixed. We have found a formula for the velocity of movement for fixed quantity of contours and cluster length. Though contours differ in their location on the open chain, they move with the same average velocity, i.e, the velocity of the cluster does not depend on the distances to the ends of the chains. The value v = v1 + · · ·+ vN N A.P Buslaev, A.G. Tatashev / Eur. J. Pure Appl. Math, 11 (3) (2018), 628-644 634 is called the average velocity of movement in the network. It is obvious that the lower bound of the average velocity on the time interval T is more than 0 for any l, 0 < l < 1. Proposition 1. At every moment t, at least one cluster moves with velocity 1. Proof. If all clusters do not move at the moment t, then coordinates of their frontal points are equal to 0 or 1/2, and, in this pair, the left cluster moves. The first coordinate of the vector of the initial state is equal 0, and the last coordinate of the vector of the initial state is equal 1/2. Then there exists a pair 0, 1/2. 3. Formulation of maim results Let us formulate main results that will be proved in this paper. Theorem 1. If l ≤ 1 2 , then there is one point of the spectrum of the system v = 1. The system reaches the state of free movement after a finite time interval. Theorem 2. If l ≥ 1 2 , then there is one point of the spectrum v = 1 2(N − 1)l −N + 2 , and, from a finite moment, a cyclic trajectory in the state space is repeated. The trajectory does not depend on the initial state. The point (0, . . . , 0) belongs to this trajectory. 4. Potential of delays and its properties 4.1. Definitions of delay potential and one-sided potential of delay By definition, put di(t) = { 1 2 + αi(t), 0 ≤ αi(t) < 1 2 , αi(t)− 1 2 , 1 2 ≤ αi(t) < 1. Now we introduce the following concept. The measure of time inside the segment t ∈ [0, 1] is called the potential of delays in the node (Ci, Ci+1) if, during this time, clusters Cli, Cli+1 move through this node, provided that clusters can move through the node simultaneously. A.P Buslaev, A.G. Tatashev / Eur. J. Pure Appl. Math, 11 (3) (2018), 628-644 635 Suppose that simultaneous movement of clusters through the node is possible. By Hi,i+1 we denote the potential of delays in the node (Ci, Ci+1) It is readily seen that, dependent on values l, αi, di+1, one of the following three available alternatives is realized. (i) Clusters Cli, Cli+1 move through the node simultaneously on one time segment belonging to the segment [0, 1]. (ii) Clusters Cli, Cli+1 move through the node simultaneously on two time segments belonging to [0, 1]. The first of these segments begins when one of these two clusters begins to move through the node. The second segment begins when the other cluster begins to move through the node. (iii) Clusters Cli, Cli+1 cannot move through the node simultaneously on the time seg- ment [0, 1]. We shall give the following definition. We assume again that simultaneous movement of clusters Cli and Cli+1 through the node on the time segment [0, 1] is possible. The duration of time segment the potential of the cluster Cli delay concerning the cluster Cli+1 if these clusters move through the node simultaneously during this segment such that, at the initial moment of this interval, the cluster Cli comes to the node, and the cluster Cli+1 moves through the node, if there exists such time segment of non-zero duration. We denote the potential of the cluster Cli delay concerning the cluster Cli+1 by hi,i+1. If such time interval does not exist, then we suppose hi,i+1 = 0. Similarly, the duration of time segment is called the potential of the cluster Cli delay concerning the cluster Cli+1 if these clusters move through the node simultaneously during this segment such that, at the initial moment of this interval, either the cluster Cli+1 comes to the node, and the cluster Cli+1 moves through the node, or clusters Cli, Cli+1 come to the node simultaneously, if there exists such time segment of non-zero duration. We denote the potential of the cluster Ci+1 delay concerning the cluster Ci by hi+1,i. If such time interval does not exist, then we suppose hi+1,i = 0. Now we shall give a formal definition of the delay potential and one-sided potentials. Suppose, at the moment t = 0, coordinates of frontal points of clusters Cli, Cli+1 are αi(0) = αi and di+1 = di+1(0). Assume that clusters move free and can pass through the node (Cli, Cli+1) simultaneously. If 0 ≤ αi < l, (1) then the cluster Cli covers the node (Ci, Ci+1) on time segments [0, l−αi] and [1−αi, 1]. If l ≤ αi < 1, (2) then the cluster Cli covers the node (Ci, Ci+1) on time segment [1− αi, 1− αi + l]. A.P Buslaev, A.G. Tatashev / Eur. J. Pure Appl. Math, 11 (3) (2018), 628-644 636 If 0 ≤ di+1 < l, (3) then the cluster Cli covers the node (Ci, Ci+1) on time segments [0, l−di+1] and [1−di+1, 1]. If l ≤ di+1 < 1, (4) then the cluster Cli covers the node (Ci, Ci+1) on the segments [1− di+1, 1− di+1 + l]. If (1) and (4) and conditions di+1 − αi > 1− l, (5) di+1 − αi ≥ l (6) are fulfilled, then clusters Cli and Cli+1 cover the node simultaneously on the segment [1− di+1, l − αi]. In this case, Hi,i+1 = l + di+1 − αi − 1. hi,i+1 = 0, hi+1,i = l + di+1 − αi − 1, i = 1, . . . , N − 1. If conditions (1), (4), (5) are fulfilled, and the condition (6) is not fulfilled (this case is possible only if l > 1 2), then clusters Cli, Cli+1 cover the node (Ci, Ci+1) simultaneously on time segment [1− di+1, l − αi] and [1− αi, 1− di+1 + l]. In this case, Hi,i+1 = 2l − 1, hi,i+1 = l − di+1 + αi, hi+1,i = l − αi + di+1 − 1. If conditions (1), (4), (6) are fulfilled, and the condition (5) is not fulfilled (this case is possible only if l ≤ 1 2), then clusters Cli, Cli+1 cannot move through the node simulta- neously. In this case, Hi,i+1 = 0, hi,i+1 = hi+1,i = 0. If conditions (1), (4) are fulfilled, and conditions (5), (6) are not fulfilled, then clusters Cli, Cli+1 move through the node (Ci, Ci+1) on time segment [1−αi, 1− di+1 + l]. In this case, Hi,i+1 = l − di+1 + αi, hi,i+1 = l − di+1 + αi, hi+1,i = 0. If conditions (2), (3) and αi − di+1 > 1− l (7) A.P Buslaev, A.G. Tatashev / Eur. J. Pure Appl. Math, 11 (3) (2018), 628-644 637 αi − di+1 ≥ l, (8) are fulfilled, then clusters Cli, Cli+1 cover the node on the segment [1 − αi, l − di+1]. In this case, Hi,i+1 = l + αi − di+1 − 1, hi+1,i = 0, hi,i+1 = l + αi − di+1 − 1. If conditions (2), (3), (7) are fulfilled, and the condition (8), is not fulfilled (this is possible only if l > 1 2), then clusters Cli, Cli+1 cover the node (Ci, Ci+1) on time segments [1− αi, l − di+1] [1− di+1, 1− αi + l]. In this case, Hi,i+1 = 2l − 1, hi,i+1 = l − di+1 + αi − 1, hi+1,i = l − αi + di+1. If conditions (2), (3), (8) are fulfilled, and condition (7) is not fulfilled (this is possible only if l ≤ 1 2), then clusters cannot cover the node simultaneously. In this case, Hi,i+1 = 0, hi,i+1 = hi+1,i = 0. If conditions (2), (3) are fulfilled, and conditions (7), (8) are not fulfilled, then clusters Cli, Cli+1 cover the node (Ci, Ci+1) on time segment [1− di+1, 1− αi + l]. In this case, Hi,i+1 = l − αi + di+1, hi,i+1 = 0, hi+1,i = l − αi + di+1. Suppose conditions (2), (4) and one of the equalities αi ≥ di+1 + l and di+1 ≥ αi + l are fulfilled. Then the clusters Cli and Cli+1 cannot move through the node simultane- ously. In this case, Hi,i+1 = 0, hi,i+1 = hi+1,i = 0. If conditions (2), (4) and αi < di+1 < αi + l are fulfilled, then clusters move through the node (Ci, Ci+1) simultaneously on the time segment (1− αi, 1− di+1 + l). In this case, Hi,i+1 = αi − di+1 − l, A.P Buslaev, A.G. Tatashev / Eur. J. Pure Appl. Math, 11 (3) (2018), 628-644 638 hi,i+1 = αi − di+1 − l, hi+1,i = 0. If conditions (2), (4) and di+1 ≤ αi < di+1 + l are fulfilled, then clusters Cli and Cli+1 move through the node simultaneously on time segment [1− di+1, 1− αi + l]. In this case, Hi,i+1 = di+1 − αi − l, hi,i+1 = 0, hi+1,i = di+1 − αi − l. If the conditions (1), (3) are fulfilled simultaneously, then the state is not admissible. 4.2. Definition of system potential of delays The sum of potentials of delays in the nodes H(t) = N−1∑ i=1 Hi,i+1(t) (9) is called the potential of delays of the system. 4.3. Properties of delay potential in node and one-sided potentials Suppose, at time t0, the potential of delay of the cluster Cli concerning the cluster Cli+1 positive, hi,i+1 > 0. Then a delay of the cluster Cli at the node (Ci, Ci+1) begins at the time t = 1 − αi(t0) if there were no delays earlier, i = 2, . . . , N. The duration of this delay is hi,i+1. Similarly, if hi+1,i > 0, then a delay of the cluster Cli+1 begins at the time t = 1 − di+1(t0) if there were no delays earlier, i = 2, . . . , N. The duration of this delay is hi+1,i. Proposition 2. (i) It is true for any t ≥ t0 Hij(t) = hij(t) + hji(t), (10) H(t) = N−1∑ i=1 (hi,i+1(t) + hi+1,i(t)). (11) (ii) If l ≤ 1 2 , then at least one of the values hi,i+1(t) and hi+1,i(t) equals 0 for any t ≥ 0, i = 1, . . . , N − 1. Proof. Equation (10) follows from definitions of delay potential in a node and one-sided potential of delays. All versions of relations between system parameters are considered. Equation (11) follows from (9) and (10). The second statement of Proposition 2 is also proved on the basis of definition. Propo- sition 2 has been proved. A.P Buslaev, A.G. Tatashev / Eur. J. Pure Appl. Math, 11 (3) (2018), 628-644 639 4.4. Properties of the system delay potential Let us prove properties of the system delay potential. Proposition 3. If H(t) = 0, then the system is in the state of free movement at time t. Proof. If H(t) = 0, then Hi−1,i(t) = 0, i = 1, . . . , N. Proposition 3 follows from the definition and invariance of the clusters intersection measure with respect to the same shift. Proposition 4. If there is a delay in the node (Ci−1, Ci), then Hi−1,i(t) does not increase at t ≥ t0. If, in addition l ≤ 1 2 , then Hi−1,i(t) decreases strictly in a neighborhood of t ≥ t0. Proof. Proposition 4 follows from the definitions of delay potential in node and one- sided potentials. Proposition 5. The potential of delays is non-increasing function of time for any value of l, 0 < l < 1. Proof. Suppose l ≤ 1 2 , and the function H(t) increases at time t. It is possible only if at least one term on the right side of (11) increases. Assume that the term hi0,i0+1(t) increases. The term can increase only with velocity 1. This term can increase only if the cluster Cli0 moves and the cluster Cli0+1 does not move. If the cluster Cli0+1 is at the node (Ci0 , Ci0+1), then the term hi0,i0+1(t) equals 0 and does not increase. Therefore the cluster Cli0+1 does not move and is located at the node (Ci0+1, Ci0+2). The cluster Ci0+2 moves through the node. Hence, hi0+1,i0+2(t) decreases with velocity 1 at time t. Similarly, it is proved that, if, at the moment t0, hi0,i0−1(t0) increases, then hi0−1,i0−2(t0) decreases at this moment. Therefore each increasing term on the right side of (11) correspond to a term, decreasing with the same velocity, and different terms corresponds to different decreasing terms. Thus Proposition 5 is true in the case of l ≤ 1 2 . Assume that l > 1 2 . We shall prove that, in this case, the potential of delays does not increase in any node, and therefore the system potential H(t) does not also increase. If at the time t clusters Cli0 , Cli0+1 move or both the clusters do not move, then the delay potential in the node (Ci0 , Ci0+1) does not change at the time t, i0 = 1, . . . , N − 1. The potential of delays can change at moment t. The delay potential can change at the moment t only if one of two clusters move. Assume that the cluster Cli0 moves, and the cluster Cli0+1 does not move (the case in that only the cluster Cli0+1 moves can be considered similarly). Then the coordinate of the frontal coordinate of the cluster Ci0+1 is equal to 1 2 and 0. Suppose the coordinate of frontal point of the cluster Cli0+1 is equal to 1 2 . Since the cluster Cli0+1 does not move the value αi0(t0) satisfies the condition 0 ≤ αi0(t0) ≤ l. Without loss of generality we assume that t0 = 0 and αi0(t0) = αi0,0. If both the clusters move, then the cluster Cli0 move through the node (Ci0 , Ci0+1) on time segments (0, l − αi0,0) and (1− αi0,0, 1), and the cluster Cli0+1 move through the node on the segment (0, l). Hence the clusters Cli0 , Cli0+1 move through the common node simultaneously on the segment (0, l − αi0,0) and, A.P Buslaev, A.G. Tatashev / Eur. J. Pure Appl. Math, 11 (3) (2018), 628-644 640 if αi0,0 > 1− l, on the segment (1−αi0,0, l) too. From this follows that the delay potential does not increase with respect to αi0,0, and therefore the delay potential does not increase with respect to time. Proposition 6. Suppose l ≤ 1 2 . If a delay of cluster C1 or cluster ClN takes place in the time interval (t, t+ a), then H(t+ a) = H(t)− a. Proof. Assume that the cluster ClN does not move in the time interval (t, t+ a). The case in that the cluster Cl1 does not move can be considered similarly. We have hN,N−1(t+ a) = hN,N−1(t)− a, hN−1,N (t+ a) = hN−1,N (t) = 0. (12) In accordance with Proposition 5 the value of H does not increase if the system comes from the state (α1(t), . . . , αN (t)) to the state (α1(t+ a), . . . , αN (t+ a)). Suppose the value of H does not change, i.e., H(t+ a) = H(t). Then, in accordance with (12), N−2∑ i=1 (hi,i+1(t+ a) + hi+1,i(t+ a)) > N−2∑ i=1 (hi,i+1(t) + hi+1,i(t)). (13) Using (13), we obtain that, for a system with N − 1 contours such that it differs from the system under consideration by the absence of the contour CN , the potential of delays increases if this system comes from the state (α1(t), . . . , αN−1(t)) to the state (α1(t+ a), . . . , αN−1(t+ a)). However, in accordance with Proposition 5, the potential of delays cannot increase. The contradiction proves that, if the cluster CLN does not move in the time interval (t, t+ a), then the potential of delays increases in this interval with the unit velocity. In the case of the cluster Cl1 movement, Proposition 5 is proved similarly. Proposition 7. Suppose l < 1 2 , and, at time t0, a delay of the cluster ClN−1 at the node (CN−1, CN−2) ends (or a delay of the cluster Cl2 at the node (C2, C3) ends), and, at the moment t1 > t0, another delay of this cluster begins. The latter delay ends, at the moment t = t1 + a, and the total duration of the cluster ClN−1 (the delay of the cluster Cl2) in the time interval (t0, t1) (denote this duration by b) is not more than 1− 2l. Then H(t1 + a) ≤ H(t0)−min(a+ b, 1− 2l). A.P Buslaev, A.G. Tatashev / Eur. J. Pure Appl. Math, 11 (3) (2018), 628-644 641 Proof. We have αN (t0) = αN−1(t0) + l (modulo 1). While the total delay of the cluster ClN−1, after the moment t0, is not more than 1 − 2l, the cluster ClN−1 can be delayed at the node (CN−2, CN−1), and clusters Cl1, . . . , ClN−1 behave in such a way that if the contour CN is absent. Taking into account Proposition 6, we get Proposition 6 in the case of cluster ClN−1. Similarly, the proposition is proved in the case of the cluster Cl2. Remark 1. Although the potential of delays is non-increasing function in time, the quan- tity of non-moving particles is not in general non-increasing function in time. 5. Criterion for the system to enter the state of free movement Theorem 3. If l < 1 2 , (14) then the system comes to the state of free movement after a finite time interval from any initial state. If l > 1 2 , (15) then the system does not come to the state of free movement after a finite time interval from any initial state. Proof. If (17) is fulfilled, then, in accordance with Theorem 4 (Section 6), the average velocity of clusters is not equal to 1, and therefore the system does not come to the state of free movement. We shall prove by induction on N that (14) is sufficient for self-organization. The statement is true for N = 1. Suppose that the statement is true for N = K − 1, K ≥ 2. Consider the case l < 1 2 . The system comes to the state of free movement after a finite time. From Propositions 3–7 follows that either the potential of delays becomes equal to 0 and the system comes to the state of free movement or the total delay of the cluster ClK−1, on all infinite time interval from a moment, does not exceed 1 − 2l, and, from this moment, the presence of the cluster ClK does not affect behavior of the system. Hence, in accordance with induction statement, the system comes to the state of free movement after a finite time interval. Thus Theorem 3 is true in the case l < 1 2 . Theorem 3 has been proved. 6. Behavior of system in case l > 1 2 Theorem 4. Let (15) be fulfilled. Then, from a finite time, the system passes, in the system state space, the same cyclic trajectory, containing the state (0, 0, . . . , 0), and the A.P Buslaev, A.G. Tatashev / Eur. J. Pure Appl. Math, 11 (3) (2018), 628-644 642 system states are repeated with the period T = 2(N − 1)l −N + 2. (16) After a moment such that at this moment periodic movement begins, the potential of delays equals H(t) = 2(N − 1)l −N + 1. (17) The average velocity equals v1 = · · · = vN = 1 2(N − 1)l −N + 2 . (18) Proof. If (16) is fulfilled, the clusters Cl1 Cl2 cannot move through the node (C1, C2) without delays. Indeed, if both the clusters move through the node (C1, C2) without delays, then each of clusters Cl1, Cl2 covers the node during l time units. Therefore the inequality 2l ≤ 1 is fulfilled. However this contradicts (17). If, at the moment t0, a delay of the cluster Cl2, then there exists a moment t1 ≥ t0 such that α1(t1) = l, α2(t1) = 1 2 . (19) If a delay of the cluster Cl1 begins at the moment t0, then, at the moment of the end of this delay, the system comes to the state such that α1(t1) = 0, α2(t1) = 1 2 + l. After 1− l time units after the end of this delay, a delay of the cluster Cl2 begins. At the moment of the end of the latter delay, the system comes to the state such that (19) is fulfilled. Hence, from any initial state, the system comes to a state such that (19) is fulfilled. Let the system is at the time t1 in the state( l, 1 2 , α3(t1), . . . , αN (t1) ) . Since α2(t1) = l, i.e., the cluster Cl2 is at the node (C2, C3) at the time t1, then, taking into account that l > 1 2 , we have that, at this moment, the cluster Cl2 covers the node (C2, C3), the cluster Cl3 covers the node (C3, C4), etc. Similarly, the cluster Cli covers the node (Ci, Ci+1) i = 4, . . . , N − 1, at time t1. For state to be admissible, it is necessary that l − 1 2 ≤ αi(t1) ≤ 1 2 , i = 3, . . . , N. (20) In accordance with (20), the frontal point of the cluster Cli comes to the node (Ci, Ci+1), not earlier than at the time t1 + 1 2 , and the cluster Cli+1 comes to the node (Ci, Ci+1) not later than at the time T1 + 1 2 − l. On the other hand, the cluster Cli+1, beginning to pass through the node (Ci, Ci+1) at the moment t2(i) such that αi+1 = 1 2 + l, releases the node (Ci, Ci+1) later than the cluster Cli comes to this node, i = 1, . . . , N − 1. Stopping at the node (Ci, Ci+1), the cluster Cli continues to hamper the movement of the cluster Cli−1. Therefore there exists a moment t0 such that α1(t) = · · · = αN−1 = 0, αN ≥ 1 2 , and only the cluster ClN moves at the time t0. Thus there exists a moment t = a such that A.P Buslaev, A.G. Tatashev / Eur. J. Pure Appl. Math, 11 (3) (2018), 628-644 643 α1(a) = · · · = αN−1(a) = 0. (21) Since, only on the contour ClN , there is no node at the point 0, only the ClN moves in the time interval ( a, a+ l − 1 2 ) . Movement of the cluster ClN−1 resumes at the moment t = a+ l− 1 2 . Movement of the cluster Cli resumes at the moment t2(i) = a+(N−i)(l− 1 2), i = 1, 2, . . . , N − 1. All clusters, except the cluster Cl1, coming to the point 1 2 , stops and waits for the node release. The cluster Cl1 comes to the point 1 2 last at the time t = a+ 1 2 + (N − 1) ( l − 1 2 ) . At this moment, the system is in the state α0 ( a+ 1 2 + (N − 1) ( l − 1 2 )) = α1 ( a+ 1 2 + (N − 1) ( l − 1 2 )) = · · · = = αN−1 ( a+ 1 2 + (N − 1) ( l − 1 2 )) = 1 2 . Hence, after the time t = a, all clusters pass half of the circle in time interval of dura- tion 1 2 + (N − 1) ( l − 1 2 ) . After new interval of the same duration, the system returns to the state (21), at which the system was at the time t = a, and, in this interval the movement of the clusters resumes in inverse order. Therefore the period is equal to 2 ( 1 2 + (N − 1) ( l − 1 2 )) = 2(N − 1)l − N + 2, i.e., (16) is fulfilled. Duration of the cluster Cli delay at the point 0 equals (N − 1 − i) ( l − 1 2 ) , i = 1, . . . , N − 1, and total delay of each cluster during the period is equal to (N − 1)(2l − 1) = 2(N − 1)l −N + 1. Thus the average velocity of each cluster equals v1 = · · · = vN = 1− 2(N − 1)l −N + 1 2(N − 1)l −N + 2 = 1 2(N − 1)l −N + 2 , i.e., (18) is fulfilled. It is proved by direct consideration that at the time t = a, and therefore, in accordance with Proposition 5, at any time t ≥ a, the potential of delays is calculated by (17). This completes the proof of Theorem 4. Remark 2. In accordance with (18), if N is fixed, the average velocity of the cluster is a continuous function on l, and this velocity tends to 1 N as l → 1. If l is fixed, then the average velocity of clusters tends to 0 as N →∞. 7. Conclusion A deterministic dynamical system is considered. This system is an open chain of N contours, on which clusters of length l move in accordance with specified rules. In [3], a similar system was considered. The supporter of the system is a closed chain of contours. It has been found in [3] that the dynamical system has a spectrum of velocity and mode periodicity consisted of more than one component. In this paper, it has been shown that, in the case of open chain, the spectrum of cluster velocity and mode periodicity contains only one component. If l < 1/2, then the system A.P Buslaev, A.G. Tatashev / Eur. J. Pure Appl. Math, 11 (3) (2018), 628-644 644 comes to the state of free movement after a finite time interval from any initial state. If l < 1/2, then the average velocity of clusters is less than 1. The dependence of this velocity on N and l has been found. Properties of delay potential function are studied. These properties are used in proof of self-organization conditions. Acknowledgements This work has been supported by the Russian Foundation for Basic Research. Grant No. 17-01-00821-a and No. 17-07-01358-a. References 1 A. P. 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