EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 1, 2018, 260-283 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Flows Spectrum on Closed Trio of Contours with Uniform Load Alexander P.Buslaev1,2,∗, Alexander G.Tatashev2,1, Marina V.Yashina2,1 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. Considered dynamical system is a flow of clusters with the same length l on contours of unit length connected in polar-remote points into closed chain. When clusters move through common node, the left-priority rule of competition resolution works. In the paper it is shown that, in the case of chain containing three contours, the dynamical system has a spectrum of velocity and mode periodicity consisted of not more than two components. Spectrum distribution in dependence on load l is developed. Hypotheses on discrete spectrum in the case of arbitrary number of contours are formulated. 2010 Mathematics Subject Classifications: 47J10, 05C21, 11J06, 76B75 Key Words and Phrases: Dynamical System, Spectrum, Self-organization, Contour Graph, Collapse, Cluster Model 1. Introduction 1.1. System description We consider a closed chain of 3 contours - circles of unit length (C1, C2, C3). On each contour there is a standard coordinate system defined from 0 to 1 in counterclockwise direction. The coordinates of nodes - common points of neighboring contours (C1, C2), (C2, C3), (C3, C1) are equal to (0, 1/2), (0, 1/2), (0, 1/2) correspondingly. On all contours Ci, i = 1, 2, 3, clusters of length l move counterclockwise and with velocity 1 equal to the complete circle per time unit. A system state at time t is a vector (α1(t), α2(t), α3(t)), where α1(t), α2(t), α3(t) are coordinates of the leading points of clusters C1, C2, and C3 correspondingly. ∗Corresponding author. Email addresses: apal2006@yandex.ru (A.P. Buslaev), a-tatashev@yandex.ru (A.G. Tatashev ), yashinamv14@gmail.com (M.V. Yashina ), http://www.ejpam.com 260 c© 2018 EJPAM All rights reserved. A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 261 The trailing points of clusters C1, C2 and C3, at the moment t, are located in points with coordinates α1(t)− l, α2(t)− l, α3(t)− l respectively (subtraction by modulo 1). The system state is called admissible state, if no one node is covered by more than one cluster. The delay of cluster movement occurs when the cluster approaches the node at the time that cluster on adjacent contour covers this node. If two clusters approach the same node simultaneously, then there is a competition. For the dynamical system it needs to determinate a rule of competition resolution. After a result of conflict resolution one of these clusters is delayed the node, and another cluster begins to pass the node. Let us define the left-priority conflict resolution rule that means the following. If the conflict occurs between the clusters of contours Ci and Ci+1, (addition modulo 3), then the cluster on contour Ci, i = 1, 2, 3, moves though the common node (advantage over priority), and cluster on contour Ci+1 stops. Periodic spectral point is called the admissible state S0 = (α1(0), α2(0), α3(0)), for which there exist the minimal time values T ∗∗ ≥ 1, T ∗ ≥ 0, such that ∀ t ≥ T ∗ α1(t+ T ∗∗) ≡ α1(t), α2(t+ T ∗∗) ≡ α2(t), α3(t+ T ∗∗) ≡ α3(t). (1) Collapse (congestion) is a mode of dynamical system where all clusters do not have the capability of movement [2]. The dynamical system is in the free movement state at the time t0, if at any time t ≥ t0 all clusters move without delays, [4]. We say that self-organization of the dynamical system is a property of the system such that it should result in free movement state over finite time from any admissible initial state, [1]. 2. Movement and collapse 2.1. Collapse, l > 1 2 Proposition 1. If cluster length l > 1 2 , (2) then for any admissible initial state the dynamical system results in the collapse state no later than through 1/2 time units. Proof. From (2) we have that at each time unit any cluster covers at least one node. As the number of contours is equal to nodes number, then for any system state no cluster can cover two nodes simultaneously. And, therefore, the cluster approaching the node can not cross the node. Given that a moving cluster approaches one of the nodes no more than over 1/2 time units, the proposition is proved. 2.2. Movement (life), l < 1 2 Proposition 2. If l < 1 2 , then the instantaneous velocity of each cluster and instantaneous average velocity of the system are strictly separated from zero. A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 262 Proof. If the condition is satisfied, then at any time unit at least one of three clusters is moving. Hence the instantaneous velocity of the system is not less than 1/3. On the other hand, each cluster either moves without conflicts, or after a delay equals not more than l time units at least 1/2 moves without delay. Thus each cluster makes a turn during a finite time. 2.3. System velocity The basic studied characteristic of the system is the average cluster velocity, determined by the limit lim T→∞ V (T ) 3T , where V (T ) is the total distance such that all clusters passed during the time interval (0, T ). There arises the problem of the existence of velocity and periodic spectra. Since the system is deterministic and, as it will be shown by direct verification, for any initial state of the system, the system states are periodically repeating, starting from some finite time unit, then the limit exists and thus the average system velocity is determined. Suppose that the initial state of the system is such that the average cluster velocity is less than 1. Cluster delays can occur at the node located on the left, at the point with the coordinate 1 2 , and at the node located on the right, at the point with coordinate 0. In the first case, the cluster delay is ending when the coordinate of the leading point of cluster on the left contour takes the value l. In the second case, the cluster delay is ending when the coordinate of the leading point of cluster on right contour takes the value 1 2 + l. Then we can reduce the study of the spectrum of possible values of velocity to the consideration of behavior systems for two one-parameter sets of initial states. Proposition 3. If the state( α1(0), 1 2 , α3(0) ) , 0 < α1(0) < l < 1 2 is periodic spectral point, then the state ( l, 1 2 , α3(0) + l − α1(0) = α30 ) is also periodic spectral point. If the state (α1(0), 0, α3(0)) , 1 2 < α3(0) < 1 2 + l is periodic spectral point, then the state ( α10 = α1(0) + 1 2 + l − α3(0), 0, 1 2 + l ) is also periodic spectral point. Proof. We suppose that at time t = 0 the system is in the state( α1(0), 1 2 , α3(0) ) , 0 < α1(0) < l < 1 2 . Then during the time interval t ∈ ( 0, 12 + l − α1(0) ) the cluster of contour C2 does not move, but clusters of contours C1 C3 are moving. A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 263 Thus, at the moment 1 2 + l − α1(0) the system is in the state( α1(t0) + 1 2 + l − γ0, 0, 1 2 + l ) . Similarly, we consider the case when at the moment 0 the system is in the state (α1(0), 0, α3(0)) , 1 2 < α3(0) < 1 2 + l. Proposition 3 has been proved. Thus, if the cluster velocity is not equal to 1, then the system approaches to one of the two states ( l, 12 , α30 ) , 0 ≤ α30 ≤ 1, Fig.1, or ( α10, 1 2 , 1 2 + l ) , 0 ≤ α10 ≤ 1, Fig.2, up to a shift, during a finite time 0,5 0,50,5 0 0 0 l 0,5 - l γ0γ -0 l Figure 1: System state l, 1/2, α30 0,5 0,50,5 0 0 0 α 0 1 - l 0,5 + l α 0 -l Figure 2: System state α10, 1/2, 1/2 + l 2.4. Potential of delay and properties Let us denote β+i (t) = (αi+1(t)− αi(t))mod(1), β−i (t) = (αi(t)− αi+1(t))mod(1), A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 264 where the indices are computed by modulo 3. Let θ(t) be Heaviside function, ψ(t) = (θ(t− 1/2)θ(l+ 1/2− t)) Suppose that the coordinates of the leading points of clusters on adjacent contours Ci Ci+1, i = 1, 2, 3, at time t0 satisfy the relation 1 2 ≤ β−i (t0) < 1 2 + l, (5) Then at the time moment t = 1 2 −αi+1(t0) the cluster delay of contour Ci+1 begins, if the delay does not occur earlier. We define potential delay at the moment t0 of cluster of contour Ci+1 relative of the cluster of contour Ci the value equal to hi+1,i(t0) = 1 2 + l − β−i (t0), if condition (3) fulfils, and hi+1,i(t0) = 0, if condition (5) does not fulfill, i = 1, 2, 3. If the coordinates of the leading points of the clusters on neighbor contours Ci and Ci−1, i = 1, 2, 3, at time unit t0 satisfy the relation 1 2 < β+i−1(t0) < 1 2 + l, (6) then at the moment t = 1 2 − αi−1(t0) the delay of the cluster on contour Ci+1 will begin, if the delay does not begin earlier. We shall say that potential delay at the time t0 of the cluster on Ci−1 with respect to cluster of the contour Ci, is the value equal to hi−1,i(t0) = 1 2 + l − β+i−1(t0), if condition (6) fulfils, and hi−1,i(t0) = 0, if condition (6) does not fulfill, i = 1, 2, 3. Thus, hi+1,i(t0) = ( 1 2 + l − β−i (t0) ) ψ(β−i (t0)), (7) hi−1,i(t0) = ( 1 2 + l − β+i−1(t0) ) ψ(β+i−1(t0)). (8) We say that delay potential is a function of time H(t) = h1,2(t) + h2,1(t) + h2,3(t) + h3,2(t) + h3,1(t) + h1,3(t) = = i=3∑ i=1 ( (1/2 + l − β−i (t))ψ(β−i (t)) + (1/2 + l − β+i (t0)ψ(β+i (t))) ) . (9) Proposition 4. Delay potential is non-negative piecewise linear function of time, and its derivative has one of the following values −2, −1, 0 or 1. A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 265 Proof. Proposition 4 follows from the fact that the velocity of changes of potential delay for any cluster relative to neighbor cluster at each time unit has one of values −1, 0 or 1. And, in a closed chain of 3 contours, a delay of not more than one cluster takes place simultaneously, and this cluster can have non-zero potential delays relative to one or two clusters. Proposition 5. Delay potential is a nonincreasing function of time. Proof. Suppose at some time unit t the function H(t) increases. This is possible only in the case such that at least one term on the right-hand side of equation (3) is increasing function. Suppose the term hi0,i0+1(t) increases (index addition by modulo n). We note the term can increase only with velocity 1. This term can increase only if the cluster on contour Ci0 moves, and the cluster on contour Ci0+1 does not move. If cluster on contour Ci0+1 is near a node being common with contour Ci0 , then term hi0,i0+1(t) needs to decrease, but not increase. Hence cluster on contour Ci0+1 does not move near node being common for contours Ci0+1 and Ci0+2, and cluster on contour Ci0+2 passes the node. But the value hi0+1,i0+2(t) at time t decreases with velocity 1. Similarly, it is proved that if at a given time t0 the term hi0,i0−1(t0) increases, then at t0 term hi0−1,i0−2(t0) decreases. Thus each increasing term in the sum on the right-hand side of equality (3) corresponds to the decreasing term with the same velocity, and different increasing terms are associated with various decreasing terms. Thus Proposition 5 has been proved. Proposition 6. For any i the following equation holds hi+1,i(t)hi,i+1(t) = 0 (8) Proof. In accordance with definitions (5)-(6) the identity β−i (t) + β+i (t) = 1 (9) holds. Proposition 6 is proved. Proposition 7. The system at the time t0 is in the free movement state if and only if H(t0) = 0. Proof. If H(t0) = 0, then not for any i = 1, 2, 3 the condition (5) or (6) fulfils, and therefore, after the time t0 there are not delays. If H(t0) 6= 0, then for some i = 1, 2, 3 the condition (5) or (6) fulfils, and therefore, after the time t0 there is at least one delay in cluster movement. 2.5. Concept of closed clusters group We introduce the following definitions. Clusters at the time unit t form closed clusters group if for any i, i = 1, 2, 3 either hj,j+1(t) + hj+1,j(t) > 0 , or β−j (t) = 1 2 + l, or β+j (t) = 1 2 + l is fulfilled. A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 266 Note that if equality β−i (t0) = 1 2 + l is true, then if in interval of time t ∈ (t0, t0 + ε) cluster on contour Ci does not move, and cluster on contour Ci+1 moves, then for any arbitrarily small ε > 0 the value hi+1,i(t0 + ε) is positive. Analogically, equality β+i (t0) = 1 2 + l means that if in interval of time t ∈ (t0, t0 + ε) cluster on contour Ci moves, and cluster on contour Ci+1 does not move, then for any arbitrarily small ε > 0 the value hi,i+1(t0 + ε) is positive. Clusters Ci, . . . , Ci+k−1 form at time unit t closed group of left-type, if for any j, j = 1, . . . , i + k − 2 either hj+1,j(t) > 0 (addition modulo 3), nor β−j (t) = 1 2 + l. Clusters form at time unit t closed group of right-type, if for any j, j = 1, . . . , i + k − 2 either hj,j+1(t) > 0 , nor β+j (t) = 1 2 + l. We call closed group by closed of mixed type, if it is not neither an open left-group nor an open group of the right-type. Note that in the case of left-type group the cluster can not delay at node located right from it. And in the case of right-type group the cluster can not delay at node located left from it. Proposition 8. Suppose that from some time unit t0 the delay potential is positive and does not change H(t) ≡ H(t0) > 0, t ≥ t0. Then clusters form closed group left- or right-type for t > t0. Proof. As H(t0) > 0, then the system at time t0 is not in the free movement state. Suppose that in the time interval (t0, t1) all clusters move, and at the moment t1 the delay of cluster on contour Ci begins near the node common with contour Ci+1, then hi,i+1(t0) = hi,i+1(t1) > 0 and term hi,i+1 decreases in the time interval (t1, t2), at the same time the value hi+1,i is equal to 0 in this time interval. The values hi−1,i+1 and hi+1,i−1 in interval (t1, t2) don’t change, because in this interval the clusters on contours Ci−1 and Ci+1 move. If the function H(t) does not decrease in the interval (t1, t2), then one term of hi−1,i+1 needs to increase, at the same time another term is equal to 0. As cluster on contour Ci does not move, then term hi−1,i needs to increase. If in some time unit t3 (t1 ≤ t3 < t2) the equality β+i−1(t3) = 1 2 fulfils, then in time t3 the value of hi−1,i abruptly decreases from l to 0, and value hi,i−1 with jump increases from 0 to l. In time interval (t2, t3) there will decrease both term hi,i+1 and term hi−1,i. So the value H(t) will decrease. If the value β+i−1 does not achieve the value 1/2 in the interval (t1, t2) (it is equivalent that hi,i−1 does not achieve the value l), then the following relations hold hi,i+1(t2) = hi+1,i(t2) = hi,i−1(t2) = 0, hi−1,i(t2) > 0. If hi+1,i(t0) > 0, then at time t0 the clusters form closed group of right-type. If hi−1,i+1(t0) > 0, then hi−1,i+1(t2) > 0, hi+1,i−1(t0) = hi+1,i−1(t0) > 0. Therefore, at some time unit t4 > t2, the delay of cluster on contour Ci−1 begins near node common with contour Ci, or near node common with contour Ci+1. In both cases after the time t2 both terms A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 267 hi−1,i+1 and hi+1,i−1 will decrease. Thus, the value H will decrease. If hi−1,i+1(t0) = hi−1,i+1(t2) = hi+1,i−1(t0) = hi+1,i−1(t2) = 0, then in some time unit a delay of cluster on contour Ci−1 will begin near node common with contour Ci. Thus immediately after time t5, the potential delay hi−1,i will decrease, and in order to the delay potential H does not decrease, it is necessary that the value hi+1,i−1 increases. As hi+1,i−1(t0) = hi+1,i−1(t5) = 0, then the condition β+i+1(t0) = β+i+1(t5) = 1 2 + l needs to hold, and so at time t0 the clusters form the group of right-type. Analogically it is proved that if at time t1 the delay of cluster on contour Ci begins near the common node with the contour Ci−1, then at time t0 the clusters form a group of left-type. The Proposition 8 has been proved. 3. Self-organization as a simple spectrum of a dynamical system Theorem 1. For a closed chain of 3 contours the sufficient condition for self- organization is the following condition l ≤ 1 6 . From Proposition 5 we have that delay potential is a nonincreasing function of time. From Proposition 8, it is true, that either clusters from some moment t0, or there are time intervals such that clusters are delayed and delay potential decrease, and, in this case, the delay potential will reach the value 0 for a finite time . Then consequently, the system results in free movement state. But in the case l < 1/6, clusters can not form a closed group. And in the case l = 1/6 clusters can form a closed group, only if the delay potential is 0. Hence Theorem 1 is true. 4. Case of a multiple spectrum ( 1 6 < l < 1 2 ) Proposition 9. Suppose 1 6 < l < 1 2 and at time t0 clusters form left- or right type. Then H(t0) = 3l − 1 2 . (12) Proof. Let it be left-type group. For right-type group the proof is analogically. We have H(t0) = h13(t0) + h21(t0) + h32(t0) = A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 268 = 3 2 + 3l − β−3 (t0)− β−2 (t0)− β−1 (t0). As 1 2 ≤ β−i (t0) ≤ 1 2 + l, i = 1, 2, 3, then β−1 (t0) + β−2 (t0) + β−3 (t0) = = (α2(t0)− α1(t0))mod(1) + (α3(t0)− α2(t0))mod(1)+ +(α2(t0)− α3(t0))mod(1) + (α3(t0)− α1(t0))mod(1) = 2. Proposition 10. Let 1/6 < l ≤ 1/2 and the system does not result in free movement state over finite time. Then from some time unit t0 clusters form a closed group of left- or right-type. And after time unit t0, the vector state of the system is cyclically shifted one position to the right over time interval 1 2 + l (in the case of left-type group), or one position to the left (in the case of right-type group). Also for this interval the total delay of clusters H(t) = 3l − 1/2 = H(t0), and the clusters average velocity equals 4/(3 + 6l). Proof. According to condition of Proposition 10 the system does not result in free movement state, then we consider with taking to account the Proposition 3 the system behavior with initial condition ( l, 1 2 , α3,0 ) , 0 ≤ α3,0 < 1. The case of initial condition( α1,0, 0, 1 2 + l ) , 0 ≤ α1,0 < 1 is considered analogically. At initial time unit one of the following conditions fulfils: 0 ≤ α,0 < l, (13) l ≤ α3,0 ≤ 1 2 , (14) 1 2 < α3,0 < 1 2 + l, l < 1 4 , (15) A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 269 1 2 + l ≤ α3,0 ≤ 1− l, l ≤ 1 4 , (16) 1− l < α3,0 < 1 2 + 2l, l < 1 4 , (17) 1 2 + 2l ≤ α3,0 < 1, l < 1 4 , (18) 1 2 < α3,0 ≤ 1− l, l ≥ 1 4 , (19) 1− l < α3,0 < 1 2 + l, l > 1 4 , (20) 1 2 + l < α30 < 1, l ≥ 1 4 . (21) For any initial state belonging the considered one-parameter family we have h1,2(0) = h2,1(0) = 0. Each of cases (13) – (21) is characterized by which values h1,3(0), h3,1(0) = 0, h2,3(0), h3,2(0) are nonzero. We describe the system behavior at the fulfilling of inequalities (13)-(21). We will show below that if condition (14) is satisfied, the system is in free movement state at the initial time. In this case, the delay potential H(t) is equal to 0 at any time. If conditions (13), (15), (19), (20) fulfill, the system results in free movement state over finite time. The delay potential H(t) is a piecewise-linear nonincreasing function equal to 0 over some finite time. If conditions (16), (17), (18), (21) hold, then beginning from finite time unit the system states periodically repeat, and average cluster velocity equals v = 4 3 + 6l . At fulfilling of conditions (16), (17), (21), cluster delays are arising cyclically in the following order: delay of cluster C1 with the duration 2l − α30 + 1 2 , delay of cluster C3 with the duration α3,0 + l− 1, delay of cluster C2 with the duration 2l−α3,0 + 1 2 , delay of cluster C1 with the duration α3,0+ l−1, delay of cluster C3 with the duration 2l−α3,0+ 1 2 , delay of cluster C2 with the duration α3,0 + l − 1. Then, after the end of the period, the system results in a state ( 1 2 , α′3,0, l ) such that its vector is obtained from initial state vector by cyclic shift of one position to the right and substitution α3,0 to α′3,0, α3,0 + α′3,0 = 3 2 + l. During the following part of the period A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 270 the clusters are delayed in the same order, while the delay times are equal to α′3,0 + l − 1 2l − α3,0 + 1 2 . At fulfilling of conditions (14), clusters are delayed in the period the following order: cluster delay C3, cluster delay C1, cluster delay C2, while the duration of each of these delays is equal to 3l − 1. Delay potential at fulfilling of conditions (16), (17), (21) has constant value equal to 3l− 1. At fulfilling of conditions (14) delay potential is piece-linear function such that its values do not change from some time and equal to 3l − 1, if 1 2 + 2l ≤ α3,0 < 1, and equal to 3l − 1 from initial time, if α3,0 = 1 2 + 2l < 1. As we will show below the behavior of the system with the initial state ( l, 12 , α30 ) , 0 ≤ α30 < 1 and various values of α30, it holds the following. If condition (14) is satisfied, the system is in free movement state from the initial time unit. And if conditions (15), (19), (20) fulfill, then the system results in free movement state over a finite time. At fulfilling of conditions (16), (17), (21) H(0) = 3l − 1 2 and over time interval with the duration 1 2 + l the state vector cyclicly moves to one position to the right, and total cluster delay over this interval equals to H(0). At fulfilling of conditions (18) clusters form a closed group from time unit t = 1/2− 2l (at condition (18) l < 1/4). Then at time unit t = 1/2 − 2l the delay potential H(t) has value 3l − 1 2 , and after this time the value delay potential of does not change. From time t = 1/2− 2l the state vector over time interval with duration 1 2 + l cyclicly moves to one position to the right, and total cluster delay over this interval equals to H(0). Thus at initial state ( l, 12 , α30 ) , with any α30 the state behavior satisfies the condition of Proposition 10. We have an average cluster velocity v = 1− H(t0) 1 2 + l = 1− 3l − 1 2 3 ( 1 2 + l ) = 4 3 + 6l . Let us prove the statement by direct verification of each conditions (13) - (21). a) Suppose the conditions (13) fulfils. Then in time interval t ∈ ( 0, 12 ) all clusters move. At time t ∈ [ 0, 12 ] h1,2(t) = h2,1(t) = h1,3(t) = h3,1(t) = h3,2(t) = 0, h2,3(t) = l − α3,0. Thus, H ( 1 2 + l − α3,0 ) = 0. A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 271 At moment t = 1 2 the system results in the state α1 ( 1 2 ) = 1 2 + l, α2 ( 1 2 ) = 0, α3 ( 1 2 ) = 1 2 + α3,0. Over time interval ( 1 2 , 1 2 + l − α3,0 ) cluster on contour C2 does not move. We have t ∈( 1 2 , 1 2 + l − α3,0 ) , h1,2(t) = h2,1(t) = h1,3(t) = h3,1(t) = h3,2(t) = 0, h2,3(t) = l − α3,0 − ( t− 1 2 ) , h1,2 ( 1 2 + l − α3,0 ) = h2,1 ( 1 2 + l − α3,0 ) = h1,3 ( 1 2 + l − α3,0 ) = = h3,1 ( 1 2 + l − α3,0 ) = h2,3 ( 1 2 + l − α3,0 ) = = h3,2 ( 1 2 + l − α3,0 ) = 0, H(t) = { l − α3,0, 0 < t < 1 2 , l − α3,0 − ( t− 1 2 ) , 1 2 < t < 1 2 + l − α3,0, H ( 1 2 + l − α3,0 ) = 0. Therefore, delay potential is constant and equals l−α3,0 from initial time unit to time t = 1 2 , then in time interval t ∈ ( 1 2 , 1 2 + l − α3,0 ) it linear decreases to 0 with velocity 1. After time unit 1 2 + l − α3,0 the delay potential equals to 0. At the time unit t = 1 2 + l − α3,0 the system results in the state α1 ( 1 2 + l − α3,0 ) = 1 2 + 2l − α3,0, α2 ( 1 2 + l − α3,0 ) = 0, α3 ( 1 2 + l − α3,0 ) = 1 2 + l, that is free movement state. b) Suppose the conditions (14) fulfils. Then t ∈ [0,+∞). h1,2(t) = h2,1(t) = h1,3(t) = h3,1(t) = h2,3(t) = h3,2(t) = 0, H(t) ≡ 0. A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 272 Thus, in this case the delay potential equals 0 as at initial time and at any time unit. The system is in free movement state from initial time unit. c) Suppose that there fulfils either condition (15) or (19). Then in time interval t ∈ (0, 1− α3,0) all clusters move. At time t ∈ [0, 1− α3,0] h1,2(t) = h2,1(t) = h1,3(t) = h2,3(t) = h3,2(t) = 0, h3,1(t) = α3,0 − 1 2 . At time t = 1− α3,0 the system results in the state α1 (1− α3,0) = 1 + l − α3,0, α2 (1− α3,0) = 3 2 − α3,0, α3 (1− α3,0) = 0. In time interval t ∈ ( 1− α3,0, 1 2 ) cluster on contour C3 does not move. We have t ∈ ( 1− α3,0, 1 2 ) h1,2(t) = h2,1(t) = h3,1(t) = h2,3(t) = h3,2(t) = 0, h1,3(t) = α30 − 1 2 − (t− 1 + α3,0) , H(t) = α30 − 1 2 − (t− 1 + α3,0) . At time unit t = 1 2 h1,2 ( 1 2 ) = h2,1 ( 1 2 ) = h1,3 ( 1 2 ) = = h3,1 ( 1 2 ) = h2,3 ( 1 2 ) = h3,2 ( 1 2 ) = 0, Thus at initial time unit the delay potential equals to α3,0 − 1 2 and preserves the value until the time unit 1 − α3,0. Then in time interval t ∈ ( 1− α3,0, 1 2 ) delay potential linear decreases with velocity 1 and equals to 0 from time t = 1 2 . At time t = 1 2 the system results in the state α1 ( 1 2 ) = 1 2 + l, α2 ( 1 2 ) = 0, α3 ( 1 2 ) = 0, that is free movement state. d) Suppose condition (20) fulfils. Then in time interval t ∈ (0, 1− α3,0) all clusters move. At t ∈ [0, 1− α3,0] h1,2(t) = h2,1(t) = h3,1(t) = h2,3(t) = 0, h3,1(t) = α3,0 − 1 2 , A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 273 h3,2(t) = α3,0 + l − 1, H(t) = 2α3,0 + l − 3 2 . At time unit t = 1− α3,0 the system results in the state α1 (1− α3,0) = 1 + l − α3,0, α2 (1− α3,0) = 3 2 − α3,0, α3 (1− α3,0) = 0. In time interval t ∈ ( 1− α3,0, 1 2 ) cluster on contour C3 does not move. We have t ∈[ 1− α3,0, 1 2 ] h1,2(t) = h2,1(t) = h3,1(t) = h2,3(t) = 0, h3,1(t) = α3,0 − 1 2 − (t− 1 + α3,0). For potential delay h3,2(t) we have h3,2(t) = l + α3,0 − 1− (t− 1 + α3,0) ∈ [1− α3,0, l], h3,2(t) = 0 ∈ ( l, 1 2 ] , H(t) = l + α3,0 − 1− (t− 1 + α3,0) ∈ [1− α3,0, l], h3,2(t) = 0 ∈ ( l, 1 2 ] . Therefore H(t) = 2α3,0 + l − 3 2 − 2(t− 1 + α3,0), 1− α3,0 ≤ t ≤ l, H(l) = 1 2 − l, H(t) = 1 2 − l − (t− l), l ≤ t ≤ 1 2 . At time t = 1 2 we have h1,2 ( 1 2 ) = h2,1 ( 1 2 ) = h1,3 ( 1 2 ) = = h3,1 ( 1 2 ) = h2,3 ( 1 2 ) = h3,2 ( 1 2 ) = 0, H ( 1 2 ) ≡ 0. Thus delay potential equals 2α3,0 + l− 3 2 at initial time and preserves this value until time unit t = 1 − γ. After this time the delay potential decreases with velocity 2 to the value A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 274 1 2 − l at time unit t = l, and in time interval t ∈ ( l, 12 ) it decreases to value 0, that is preserved at any next time. At time unit t = 1 2 the system is in the state α1 (1− α3,0) = 1 2 + l, α2 (1− α3,0) = 0, α3 (1− α3,0) = 0, that is free movement state. e) Suppose there fulfils condition either (16), or (17), or (21). Then in time interval t ∈ ( 0, 12 − l ) all clusters move. And at time t = 1 2 − l the system results in the state α1 ( 1 2 − l ) = 1 2 , α2 ( 1 2 − l ) = 1− l, α3 ( 1 2 − l ) = α3,0 − 1 2 − l. At time t ∈ [ 0, 12 − l ] h1,2(t) = h2,1(t) = h3,1(t) = h2,3(t) = 0, h1,3(t) = 2l − α3,0 + 1 2 , h3,2(t) = α3,0 + l − 1, H(t) ≡ 3l − 1 2 . In time interval ( 1 2 − l, 1 + l − α3,0 ) cluster on contour C1 does not move. At time t = 1 + l − α3,0 the system results in the state α1 (1 + l − α3,0) = 1 2 , α2 (1 + l − α3,0) = 3 2 + l − α3,0, α3 (1 + l − α3,0) = l. In time interval ( 1 2 − l, 1 + l − α3,0 ) cluster on contour C1 does not move. In time interval t ∈ [ 1 2 − l, 1 + l − α3,0 ] h1,2(t) = h3,1(t) = h2,3(t) = 0, h1,3(t) = 2l − α30 + 1 2 − ( t− 1 2 + l ) , h2,1(t) = t− 1 2 + l, h3,2(t) = α3,0 + l − 1, H(t) ≡ 3l − 1 2 . Suppose α′3,0 = 3 2 + l − α3,0. Then α3,0 = 3 2 + l − α′3,0. A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 275 We obtain 1 2 + l < α′3,0 < 1 2 + 2l, i. e. α′30 satisfies the condition analogous to the condition for α30. We have α3,0 + α′3,0 = 3 2 + l. (22) Vector of system state at time t = 1 + l−α3,0 is obtained from vector of initial system state by shifting on one position to the left and substitution of α3,0 to α′3,0 Delay potential equals to H(t) = 3l − 1 2 at initial time unit and reserves constant on any time. At any time unit the value H(t) does not change H(t) ≡ 3l − 1 2 . Thus at time t = 6(1 + l)− 3(α3,0 + α′3,0) = 3 2 + 3l the system returns to initial state. Over this time duration each cluster delays twice: first time it delays for (3l − α3,0) and second time it delays for (3l − α′3,0). According to (22), we obtain that total delay of cluster over period equals to 3l − 1 2 time units. f) Suppose condition (18) fulfils. In time interval t ∈ ( 0, 12 − l ) all clusters move. In time interval t ∈ ( 0, 32 − α3,0 ) all clusters move. We have h1,2(t) = h2,1(t) = h3,1(t) = h1,3(t) = h2,3(t) = 0, h3,2(t) = α3,0 + l − 1, H(t) ≡ α3,0 + l − 1, 0 ≤ t ≤ 1− α3,0. At time t = 3 2 − α3,0 the system results in the state α1 ( 3 2 − α3,0 ) = 3 2 − α3,0 + l, α2 ( 3 2 − α3,0 ) = 1− α3,0, α3 ( 3 2 − α3,0 ) = 1 2 . In time interval t ∈ ( 3 2 − γ0, 1 2 + l ) cluster on contour C3 does not move, while other clusters move. The function h3,2 decreases in this interval, and beginning from time t = 1 − 2l, function h1,3 increases (note that at α3,0 = 1 + 2l). The equality 3 2 − α3,0 = 1 + 2l holds, thus, in this case function h1,3 increases when time t = 3 2 − α3,0 comes. At t ∈ [ 3 2 − α30, 1 2 + l ] we have h1,2(t) = h2,1(t) = h2,3(t) = h3,1(t) = 0, A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 276 h3,2(t) = α30 + l − 1− ( t− 3 2 − α3,0) ) , h1,3(t) = 0, 3 2 − α3,0 ≤ t ≤ 1− 2l, h1,3(t) = 0, 1− 2l ≤ t ≤ 1 2 + 2l, H(t) = α3,0 + l − 1− ( t− 3 2 − α3,0 ) , 3 2 − α3,0 ≤ t ≤ 1 2 + l. At time t = 1 2 + l the system results in the state α1 ( 1 2 + l ) = 1 2 + 2l, α2 ( 1 2 + l ) = l, α3 ( 1 2 + l ) = 1 2 , that differs from initial system state by shifting on one position to the right and substitu- tion of α3,0 on 1 + 2l, while h1,2(t) = h2,1(t) = h3,1(t) = h1,3(t) = h2,3(t) = 0, h3,2(t) = 3l − 1 2 , H(t) = 3l − 1 2 . Thus, in time interval t ∈ ( 0, 32 − α3,0 ) , the function H(t) has value α3,0+l−1, then in the time interval t ∈ ( 3 2 − α3,0, 1− 2l ) function H(t) linear decreases to value 3l− 1 2 , and beginning from time 1 2 − 2l value of H(t) preserves constant. Next system states return with period T = 1 2 + l. Over this time period one of clusters has delay with duration 3l − 1 2 time units. All possible cases are considered. Proposition 10 has been proved. Theorem 2. At value l satisfying the inequality 1 6 < l ≤ 1 2 , any admissible initial system state generates a spectral function with one from the two following spectral values 1 and 4/(3 + 6l). Proof. At any value l < 1/2 there exist initial states from which the system results to free movement state after finite time (or the system state is free movement state already). Thus the state is any state of type (α1,0(0), α2,0(0), α3,0(0)) such that α1,0(0) = α2,0(0) = α3,0(0). Hence and using Proposition 10 the Theorem 2 is proved. A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 277 5. 5. Remarks, generalizations, hypotheses 5.1. Behavior of contours trio at l = 1 2 In [3] a discrete analog of considered system is studied. There are 2 cells and one particle on each contour of closed chain consisting of n contours. Particle moves at discrete instants of time. If n = 3 and rule of conflict resolution is conflict left-hand, this system is equivalent to the following case of continuous considered system. The cluster length on each contour is 1 2 . In the initial state, each cluster is located at a point with coordinate 0 or 1 2 . Example 1. Suppose that at initial time the system was in the state α1(0) = l = 1 2 , α2(0) = l = 1 2 , α3(0) = α3,0. If α3,0 = 1 2 , then H(0) = 0, and the system is in free movement state. If α30 = 0, i. e. initial state is α1(0) = l = 1 2 , α2(0) = l = 1 2 , α3(0) = 0, then h1,2(0) = h2,1(0) = h3,1(0) = h2,3(0) = h3,2(0) = 0, h1,3(0) = 1 2 , H(0) = 1 2 and at initial time there is conflict of clusters C1 and C3. In according of left-priority rule of conflict resolution in time interval t ∈ ( 0, 12 ) cluster on contour C1 does not move, while clusters on contours C2 and C3 move. So at time t = 1 2 the system results in the state α1 ( 1 2 ) = 1 2 , α2 ( 1 2 ) = 0, α3 ( 1 2 ) = 1 2 , that is obtained from initial state vector by cyclic shift on one position to the right. At t ∈ [0, 12 ] we have h1,2(t) = h2,1(t) = h3,1(t) = h2,3(t) = 0, h1,3(t) = 1 2 − t, h3,2(t) = t, H(t) ≡ 1 2 A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 278 and, thus, h1,2 ( 1 2 ) = h2,1 ( 1 2 ) = h(1, 3) ( 1 2 ) = h3,1 ( 1 2 ) = h2,3 ( 1 2 ) = 0, h3,2 ( 1 2 ) = 1 2 , H ( 1 2 ) = 1 2 , i. e. the potential preserves constant value from initial time to any time unit. Initial system state repeats over time interval with duration T = 3 2 . Over period the delay of each clusters comes with duration equal to 1 2 . Average velocity of each clusters equals to v = 2 3 , that corresponds to formula v = 4 3 + 6l at l = 1 2 and results of [1]. 5.2. Behavior of contours trio at l = 1 6 In Sect. 2 it was proved that at l ≤ 1 6 the system results to self-organization over finite time . Example 2. Let cluster length be l = 1 6 and at time t = 0 system be in the state α1(0) = 1 6 , α2(0) = 1 2 , α3(0) = 3 4 . Then at t ∈ ( 0, 13 ) h1,2(t) = h2,1(t) = h3,1(t) = h2,3(t) = h3,2(t) = 0, h1,3(t) = 1 12 − ( t− 1 3 ) , H(t) = 1 12 . In time interval ( 0, 13 ) all clusters move and in t = 1 3 the system results in state α1 ( 1 3 ) = 1 2 , α2 ( 1 3 ) = 5 6 , α3 ( 1 3 ) = 1 12 . A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 279 In ( 1 3 , 5 12 ) cluster on contour C1 does not move, and other clusters move. While t ∈ ( 1 3 , 5 12 ) h1,2(t) = h2,1(t) = h3,1(t) = h2,3(t) = h3,2(t) = 0, h1,3(t) = 1 12 − ( t− 1 3 ) , h2,1(t) = t− 1 3 , H(t) = 1 12 . At time t = 5 12 the system is in state α1 ( 5 12 ) = 1 2 , α2 ( 5 12 ) = 11 12 , α3 ( 5 12 ) = 1 6 , h1,2 ( 5 12 ) = h2,1 ( 5 12 ) = h1,3 ( 5 12 ) = h3,1 ( 5 12 ) = h2,3 ( 5 12 ) = = h3,2 ( 5 12 ) = 0, h2,1 ( 5 12 ) = 1 12 , H(t) = 1 12 . In time interval ( 5 12 , 1 2 ) the system is at time t = 1 2 in state α1 ( 5 12 ) = 7 12 , α2 ( 5 12 ) = 0, α3 ( 5 12 ) = 1 4 . In time interval ( 1 2 , 7 12 ) cluster on contour C1 does not move, while at t ∈ ( 1 2 , 7 12 ) h1,2(t) = h1,3(t) = h3,1(t) = h2,3(t) = h3,2(t) = 0, h2,1(t) = 1 12 − ( t− 1 2 ) . We have h1,2 ( 7 12 ) = h2,1 ( 7 12 ) = h1,3 ( 7 12 ) = h3,1 ( 7 12 ) = h2,3 ( 7 12 ) = = h3,2 ( 7 12 ) = 0, H ( 7 12 ) = 0. A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 280 Thus, delay potential value equals 1 12 from initial time to time t = 1 2 , and delay potential linear decreases with velocity 1 in time interval t ∈ ( 1 2 , 7 12 ) to 0, and it reserved value 0 in all next time. At time t = 7 12 the system is in the state α1 ( 7 12 ) = 7 12 , α2 ( 7 12 ) = 0, α3 ( 7 12 ) = 1 4 that is free movement state. Thus, according to the above, following is true. Depending on the value of cluster length l we have 1) If l ≤ 1 6 , then cluster velocity equals to 1. 2) If 1 6 < l < 1 2 , then, depending on the initial state, the average clusters velocity is equal to 1 or 4 3+6l , Fig.3, 3) If l > 1 2 , then cluster velocity equals to 0. 0 1 ! 6 2 3 1 4 1 6 1 2 l α3,0 l=α -3,0 1 2 l= 1 4 α3,0 2 - 1 2 3 Figure 3: (1) v = 1, (2) v = 4/(3 + 6l), two delays of each cluster per period, (3) v = 4/(3 + 6l), one delay A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 281 5.3. Hypotheses about velocity spectrum for closed chain consisting of arbitrary number of contours We formulate hypotheses about the behavior of system that is analogous to con- sidered system, but contour number is arbitrary and equals n. Hypothesis 1. 1) The system results in free movement state over finite time, if a number n is odd, and the condition l < 1 n fulfils, or a number n is even and the condition l < 1 2n fulfils. 2) For given n ≥ 3 and l that satisfy the condition 1 n < l ≤ 1 2 , if n is even, or condition 1 2n < l ≤ 1 2 , if n is odd, in the spectrum of possible values of average velocity for various initial states of the system, except for value 1, there are no more than [ n 3 ] values such that less than 1 (square brackets denote the integer part of a number). These values are calculated as follows. Let n be even number; k0 be maximal natural number k such that k < nl. (23) Then all possible values of average velocity v = n 2 + k n 2 + nl , where k is natural number satisfying the inequalities (23) and k ≥ k0 − [n 3 ] + 1. Let n be odd; k0 be maximal natural number k satisfying the following condition k < nl + 1 2 . (24) A.P.Buslaev, A.G.Tatashev, M.V.Yashina / Eur. J. Pure Appl. Math, 11 (1) (2018), 260-283 282 Then all possible values of average velocity v = n 2 + k − 1 2 n 2 + nl , where k is natural number satisfying the inequalities (24) and k ≥ k0 − [n 3 ] + 1. 6. The behavior of closed chain with 4 contours In the case of closed chain with 4 contours, we have results of computer simulation modeling and formulate the following. 1) If l ≤ 1 4 , then the system results in a state of self-organization over finite time. 2) If 1 4 < l ≤ 1 2 , then, depending on the initial state, the average cluster velocity is equal to 1 or 3 2+4l , 3) If l > 1 2 . then the system results in a state of collapse. Thus hypothesis 1 is confirmed. 7. Conclusion A deterministic dynamical system is introduced and considered. It is a closed chain of contours, on which there are movement of clusters with length l in accordance with the specified rules. It is found that, if l ≤ 1/6, then self-organization takes place. i.e., the system results in free movement state over finite time from any initial state. For l > 1/2, the system results in in collapse state over a finite time. And for any value of l, satisfying the condition 1/6 < l ≤ 1/2, the spectrum of possible values of average velocity contains two possible values: 1 and 4/(3 + 6l). While what value will average velocity take, depends on the initial system state. We have developed a method to analyze the system behavior. The concept of delay potential is introduced. The properties of delay potential function are studied. In particular, it is proved that the delay potential is a nonincreasing function of time and has a value 0, when the system results in free movement state. REFERENCES 283 Possible generalizations of the results to a closed chain with arbitrary number of con- tours are considered. Hypothesis about the average clusters velocity and condition of free movement state of the system is discussed. 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. Buslaev and A. G. Tatashev. Flows on discrete traffic flower. Journal of Mathe- matics Research, 9(1):98–108, 2017. [2] A. P. Buslaev, A. G. Tatashev, and M. V. Yashina. Qualitative properties of dynam- ical system on toroidal chainmail. In T. Simos, G. Psihoyios, and Ch. Tsitouras, edi- tors, 11th International Conference of Numerical Analysis Mathematics.AIP Confer- ence Proceedings 1558., pages 1144–1147, Rhodes, 2013. American Institute of Physics (AIP). [3] V. V. Kozlov, A. P. Buslaev, and A. G. Tatashev. On synergy of totally connected flows on chainmails. In I.P.Hamilton and J.Vigo-Aquiar, editors, 13th International Conference on Computational and Mathematical Methods in Science and Engineering., pages 861–873, Spain, 2013. University of Almeria. [4] V. V. Kozlov, A. P. Buslaev, and A. G. Tatashev. Monotonic walks on a necklace and coloured dynamic vector. International Journal of Computer Mathematics, 92(9):1910– 1920, 2014.