Phase space classification of an Ising Cellular Automaton: the Q2R model
aa r X i v : . [ n li n . C G ] M a r Phase space classification of an Ising Cellular Automaton: the Q2R model
Marco Montalva-Medel , Sergio Rica , , and Felipe Urbina Facultad de Ingenier´ıa y Ciencias, Universidad Adolfo Ib´a˜nez, Avda. Diagonal las Torres 2640, Pe˜nalol´en, Santiago, Chile. UAI Physics Center, Universidad Adolfo Ib´a˜nez, Santiago, Chile. Facultad de Ciencias, Universidad Mayor, Camino La Piramide, 5750, Huechuraba, Santiago, Chile
Abstract
An exact characterization of the different dynamical behavior that exhibit the space phase of a reversible and conservative cellularautomaton, the so called Q2R model, is shown in this paper. Q2R is a cellular automaton which is a dynamical variation of theIsing model in statistical physics and whose space of configurations grows exponentially with the system size. As a consequenceof the intrinsic reversibility of the model, the phase space is composed only by configurations that belong to a fixed point ora limit cycle. In this work we classify them in four types accordingly to well differentiated topological characteristics. Threeof them –which we call of type S-I, S-II and S-III– share a symmetry property, while the fourth, which we call of type AS,does not. Specifically, we prove that any configuration of Q2R belongs to one of the four previous limit cycles. Moreover, at acombinatorial level, we are able to determine the number of limit cycles for some small periods which are almost always presentin the Q2R. Finally, we provide a general overview of the resulting decomposition of the arbitrary size Q2R phase space, inaddition, we realize an exhaustive study of a small Ising system (4 ×
4) which is fully analyzed under this new framework.
1. Introduction
A central problem in statistical physics concerns the manifestation of irreversibility whenever the system is governed by alarge number of elements, or more precisely the number of degrees of freedom. Despite the reversible character of the equationof motions in mechanics, the nature does not allow to observe a reversible behavior of a macroscopical system. After Boltzmanntheory and the subsequent Loschmidt and Zermelo’s considerations this central question has been in the core of debates in basicphysics since the end of the 19th century. In particular, Zermelo argued that the Boltzmann H-theorem is in contradiction withPoincar´e’s recurrence theorem, however, as Boltzmann replied, the hypothetical recurrence time would be huge in comparisonwith all practical times in the usual thermodynamics.To model the recurrence time paradox, Paul and Tatiana Ehrenfest elaborated a particle exchange model [1, 2], the so-calledthe “dog-flea” model. This combinatorial model appears as an illustration of the irreversible exchange of heat between twodistinct reservoirs at different temperatures. Ehrenfest model consists of N particles that can be distributed in the left or rightside of a container, in such a way that N/ n balls are initially at the left container and N/ − n in the right one. As shownby M. Kac [3], the Ehrenfest model may be mapped into a random walk. Moreover, if initially the system is filling mostly onecontainer side n ≈ N/
2, then the average recurrence time is exponentially long, ∼ N . On the other hand, if initially the systemis almost equally distributed n ≈
0, then, the waiting time scales as a diffusion process √ N . Therefore, as one increases thetotal number of degrees of freedom N , there are some initial conditions with exponentially long recurrence times. The interestof the extremely simplified Ehrenfest model is that captures the essence of an exponentially long recurrence time showing thatsome initial configuration require exponentially long time to be back to the same state again.Generically, irreversibility arises as a consequence of systems that possess a large number of degrees of freedom. Moreover,even in moderate system size, say N = 64 for the Ehrenfest’s model, the recurrence time becomes of the order of 2 , i.e., essentially infinity. Therefore, although irreversibility appears to be a consequence of thermodynamic limit, N → ∞ , in practiceeven in moderate system size a thermodynamic statistical description appears to be the adequate one [4].In a recent article [5], two of us, developed a master equation approach to a reversible and conservative cellular automatonmodel: the Q2R model. Introduced in the 80 by Vichniac [6], Q2R is a dynamical variation of the Ising model for ferromagnetismthat possesses quite a rich and complex dynamics. Remarkably, the evolution of Q2R preserves an Ising-like energy [7], appealingthe analogy with the continuous dynamics of Hamiltonian systems . Because the Q2R model is a reversible cellular automaton More details in Section 2.3.3.
Preprint submitted to Elsevier March 29, 2019 ts phase space is finite and there are neither attractive nor repulsive attractors, all attractors must be fixed points or limitcycles.Q2R is a two variable automaton, i.e. , a state is defined through ( x t , y t ) in which each component x t and y t belong to agraph which is defined via a lattice and a neighbor (see Section 2). Although it can be defined in any kind of lattice, we restrictourselves to the particular case of a square grid with a von-Neuman four nearest neighbors. The size of the lattice will be N = L × L , thus the phase space is the set of the 2 N vertices of a 2 N -dimensional hypercube. However, as we show in thispaper, the phase space is partitioned in a large number of subspaces composed by periodic orbits or fixed points. A given initialcondition belongs to one of this limit cycles or is a fixed point.It has been reported numerically, that the phase space is composed of a huge number of limit cycles with probable exponen-tially long periods [8]. For small Ising systems, e.g. , for a 2 × = 256 states and the longest orbitis of period 4. In the case, of a 4 ×
4, the phase space has 2 ≈ . × elements, being T = 1080 the longest limit cycle.More important, this case can be scrutinized exactly, and we are able to conjecture that the number of states of a given periodis exponentially large with the number of sites N .In Ref. [5], following the Nicolis and Nicolis coarse-graining approach [9], we have applied it to the time series of the totalmagnetization, leading to a master equation that governs the macroscopic irreversible dynamics of the Q2R automata. Themethodology works out for various lattice sizes. Notably, in the case of small systems, we show that the master equation leadsto a tractable probability transfer matrix of moderate size, which provides a master equation for a coarse-grained probabilitydistribution. The success of a consistent thermodynamic description is based on the existence of rich nature of the phase space.Similarly, Lindgren and Olbrich [10] have recently considered the equilibrium properties of the Q2R model but with a differentapproach. Furthermore, for a large system size it has been established that the evolution presents an irreversible behaviortowards an equilibrium ruled by a micro-canonical ensemble [11, 12]. Moreover, in Ref. [12], it has been shown numericallythat for a set of random initial conditions with different energies one recovers statistically the Ising phase transition ruled bythe Onsager and Yang exact solutions [15, 16].The aim of the present article, is to study and classify the different possible attractors (fixed points and limit cycles) of thephase space of the Q2R cellular automaton in a square lattice of arbitrary size. The starting point is the reversibility propertyof the Q2R model and essentially all results of the current paper follow after the Lemma 3.8 (on Reversibility).Our main results are the following:1. A fully classification of all attractors in four types of limit cycles consisting of symmetric and asymmetric ones (Theorem4.11). More precisely this characterization is according to the specific topological features of the cycle. These limit cyclesmay be symmetric limit cycle of type S-I, S-II and S-III (See Sec. 4.4) and asymmetric limit cycle (AS).2. The fixed points are of type S-I, moreover with the aid of splitting the lattice in two sub lattices we are able to show thatthe total number of fixed points is β = 4 k , with k ∈ N (Theorems 4.1 and 4.29).3. The characterization and existence of β ( β −
1) period-two limit cycles. (Theorems 4.5 and 4.29).4. The characterization and existence of period-three limit cycles (Theorem 4.7 and Proposition 4.24).The paper is organized as follows: In Section 2, we define the Q2R model and its main properties. In Section 3, we establishthe formal definitions scheme, and we state the fundamental Lemma on reversibility (Lemma 3.8) which is the key propertyafter it all results in the paper follows. In Section 4, we prove the main results listed above. Next, in Section 5, we presenta general overview of the resulting decomposition of the phase space of Q2R. In Section 6 we conclude and discuss on furtherresults and conjectures. Finally, in the Appendix 7 we provide an exhaustive study of a small Ising system (4 ×
4) which is fullyanalyzed under this new framework with some specific examples of limit cycles.
2. The model
The Q2R model, introduced by Vichniac [6], is defined in a regular two dimensional toroidal lattice with even rank L × L ,being N = L the total number of nodes which have associated an index k ∈ { , . . . , N } , as well as a relative position in the We focus our work with periodic boundary conditions on the lattice, but other possibilities may be also considered. In particular, the lattice doesnot require a square lattice. It could be a rectangular one: L × L . k ∈ { , . . . , L } and k ∈ { , . . . , L } (the respective row and column indices). Further, a node k is characterized by two possible values x k = ±
1, conforming with the following two-step rule: x t +1 k = x t − k H X i ∈ V k x t i ! , where V k denotes the von Neuman neighborhood of the four closest neighbors with periodic boundary conditions. The function H is such a that H ( s = 0) = − H ( s ) = +1 in all other cases.The above two-step rule may be naturally re-written as a one step rule with the aid of a second dynamical variable [7]: y t +1 k = x t k x t +1 k = y t k H X i ∈ V k x t i ! . (2.1)Thus, the state x belongs to the discrete set Ω = {− , } N (of size 2 N ) and the set of configurations , denoted by Ω , it iscomposed by couples of states in Ω = Ω × Ω = { ( x, y ) /x ∈ Ω ∧ y ∈ Ω } (of size 2 N ). Definition 2.1.
We denote the symbol ⊙ by the Hadamard product, which is the multiplication component to component ofthe state x ∈ Ω and y ∈ Ω. Hence, x ⊙ y ∈ Ω represents that each component is defined by: [ x ⊙ y ] ij ≡ x ij y ij . This product iscommutative, associative, and it possesses a neutral element, that we denote by and corresponds to the state of Ω composedonly by 1s. Moreover, we also define − ∈ Ω by the states composed only by -1s. Given x ∈ Ω, we will write − x to refer to − x = [ − ] ⊙ x . Definition 2.2.
Let be the function φ : Ω → Ω such that, if x ∈ Ω then, the k -th component of [ φ ( x )] k = − k -th components is null, namely P i ∈ V k x i = 0. Notice that the neighborhood, V k , includes theperiodic boundary condition of the lattice. Otherwise, [ φ ( x )] k = +1. Therefore, the function φ ( x ) is a state in Ω that has a -1in the sites that x has a null neighborhood. Example.
Consider the state x ∈ Ω below. The node (3,2) has null neighborhood (its neighbors are marked by boxes), whilethe neighborhood of the node located at (1,4) is not null (its neighbors are marked by double boxes accordingly, to the toroidallattice). So, the state φ ( x ) will have a -1 value at position (3,2) and a 1 value at position (1,4) that are also marked, with a boxand a double box, respectively, in φ ( x ). In a similar way, all other values of φ ( x ) are obtained. x = φ ( x ) = − . (2.2) Definition 2.3.
The state x does not have any null-neighborhood iff φ ( x ) = . Notice that φ ( ) = φ ( − ) = . Given ( x t , y t ) ∈ Ω at time t , and according with the previous definitions we re-write the Q2R model (2.1) as the followingtwo step deterministic rule: y t +1 = x t x t +1 = y t ⊙ φ (cid:0) x t (cid:1) . (2.3)The evolution is dictated by the rule (2.3) and is complemented with an initial configuration ( x , y ) ∈ Ω . For instance, letus consider x = y = x , the example given in (2.2). The evolution of the initial configuration ( x , y ) are obtained as follows:3 − − | {z } x , − − | {z } y (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ − − − − | {z } x y ⊙ φ ( x , − − | {z } y x (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ ... ... that we schematize with the following abbreviated notation:( x , y ) → ( x , y ) → · · · . Remark . In general, we will write ( x t , y t ) → ( x t +1 , y t +1 ) for the one-step evolution from ( x t , y t ) to ( x t +1 , y t +1 ) =( y t ⊙ φ ( x t ) , x t ), according to rule (2.3). Definition 2.5.
The
Phase Space of the Q2R model it is composed by the set of configurations Ω and its one-step evolutions.Because is finite, the phase space has two types of attractors: limit cycles or fixed points . A limit cycle C of period T ∈ N isa sequence dictated by the evolution ( x , y ) → ( x , y ) → · · · → ( x T − , y T − ) → ( x T , y T ) such that all configurations ( x t , y t )are different, except ( x , y ) = ( x T , y T ). We will write ( x, y ) ∈ C if ( x, y ) is a configuration that is in C and, more general, thenotation [( x t , y t ) → ( x t +1 , y t +1 ) → · · · → ( x t + τ , y t + τ )] ∈ C will be used to refer to the subsequence of C that goes from ( x t , y t )to ( x t + τ , y t + τ ), τ ≤ T . A fixed point is a limit cycle of period T = 1, i.e., is a configuration ( x, y ) ∈ Ω such that ( x, y ) → ( x, y ). Observe that φ ( x t ) ⊙ φ ( x t ) = , ∀ x t ∈ Ω. Then Q2R rule may be inverted getting the backward evolution of the systembut for the couple ( y t , x t ), that reads: (cid:26) x t − = y t y t − = x t ⊙ φ ( y t ) , (2.4)which is exactly the same rule (2.3), displaying the remarkable property of reversibility. This property will be highlighted inthe “Reversibility Lemma” (Lemma 3.8). Since the Q2R rule is reversible and the phase space is finite, each configuration has two possibilities; to be a fixed point orto belong in a limit cycle.
Definition 2.6.
Let be the energy function E [ (cid:0) x t , y t (cid:1) ] = − X h i , k i x t i y t k , (2.5)where the symbol h· · · i stands for sum over the four near neighbors. The energy (2.5) is bounded by − N ≤ E ≤ N .4 emark . As shown by Pomeau [7], the energy function (2.5) is conserved, under the dynamics defined by the Q2R rule(2.3). That is E [( x t , y t )] = E [ (cid:0) x , y (cid:1) ] , ∀ t ∈ N . Remark . Other dynamical invariants are known in the literature [13].
The phase space of all configurations is defined through all possible values of the 2 N dimensional state ( x, y ) ∈ Ω . Theresulting phase space is composed by the 2 N vertices of a hypercube in dimension 2 N . This phase space is partitioned indifferent sub-spaces accordingly to its energy, E , and accordingly with its dynamical characteristic such that the period, andother unknown parameters.For instance, the constant energy subspace shares in principle many limit cycles of different periods, as well as, many differentfixed points. An arbitrary initial condition of energy E , falls into one of these limit cycles, and it runs until a time T , which couldbe exponentially long, and displaying a complex behavior (not chaotic stricto-sensu , see for instance [14]). More important,the probability that an initial condition belongs to an exponentially long period limit cycle and it exhibits a complex behavioris finite [8]. Moreover, Q2R manifests sensitivity to initial conditions, that is, if one starts with two distinct, but close, initialconditions, then, they will evolve into very different limit cycles as time evolves [12]. In some sense, any initial state exploresvastly the phase space justifying the grounds of statistical physics, as we shown in the Remark 3.2 and the main Theorem 4.11on the limit cycle general classification.
3. Preliminary Results
The core of this work aims to propose a new framework to study the dynamic of the Q2R model. It is based in particularproperties of its configurations that allow to partition Ω in order to characterize the full spectrum of fixed points and limitcycles, as well as delving into topological and combinatorial aspects. We begin by establishing the basic concepts that will beused along the text and the first results, necessary to understand the main results (Section 4). Ω . Observe that, given the states x, y ∈ Ω, there are two possibilities: [ x = y ] or [ x = y ]. Therefore a first partition of Ω arisesas follows: Definition 3.1.
We denote by Ω xx the set of configurations with equal states , i.e., Ω xx = (cid:8) ( x, y ) ∈ Ω /x = y (cid:9) , whose size is | Ω xx | = 2 N . Similarly, the set of configurations with different states will be denoted by Ω xy and corresponds tothe complement set of Ω xx in Ω , i.e., Ω xy = Ω − Ω xx = (cid:8) ( x, y ) ∈ Ω /x = y (cid:9) whose size is | Ω xy | = 2 N (2 N − = Ω xx ∪ Ω xy . Remark . We underline that | Ω xx | < | Ω xy | . Further, the probability to have a configuration in Ω xx is p xx = | Ω xx | / | Ω | = 2 − N ,while the probability to have a configuration in Ω xy , is p xy = | Ω xy | / | Ω | = 1 − − N . Hence, in practice, p xx ≪ p xy . Moreover p xy → N → ∞ ).A second partition of Ω will allow us to know in detail the topology of the limit cycles: here, the sets Ω xx and Ω xy are alsopartitioned regarding the two possibilities of φ ( x ) in a configuration ( x, y ), that is [ φ ( x ) = ] or [ φ ( x ) = ], regardless the valueof φ ( y ), as in the following definition: Definition 3.3.
Let be the following sets: A = (cid:8) ( x, y ) ∈ Ω xx /φ ( x ) = (cid:9) B = (cid:8) ( x, y ) ∈ Ω xy /φ ( x ) = (cid:9) C = (cid:8) ( x, y ) ∈ Ω xx /φ ( x ) = (cid:9) D = (cid:8) ( x, y ) ∈ Ω xy /φ ( x ) = (cid:9) .
5e say that ( x, y ) ∈ Ω is a configuration of type A , B , C or D , if ( x, y ) belongs to one of the sets A , B , C or D , respectively.Later on, we refer to a evolution of type U → V to the one step evolution of a configuration ( x, y ) ∈ U up to ( w, x ) ∈ V with U, V ∈ {
A, B, C, D } . In such a case, we will say that U evolves to V , or V comes from U (see Figure 3 and Corollary 4.9 as agiven examples of this terminology). Definition 3.4.
We denote by P T ⊂ Ω the set of configurations belonging to a limit cycle of period T ∈ N . In particular, P correspond to the set of fixed points of Q2R. Moreover, we will denote by ν ( T ) the size of the set P T and by n ( T ) the numberof limit cycles of period T . That is: ν ( T ) ≡ | P T | and n ( T ) ≡ ν ( T ) T . Remark . From definition 3.3, Ω xx = A ∪ C and Ω xy = B ∪ D . Further, we show in Remarks 4.2 and 4.6 that P = A ⊂ Ω xx and P ⊂ B ⊂ Ω xy , respectively. Definition 3.6.
We say that, the symmetric configuration of ( x, y ) ∈ Ω is the configuration ( y, x ) ∈ Ω . In particular, thesymmetric configuration of ( x, x ) ∈ Ω is itself, ( x, x ), and we will call it as a self-symmetric configuration. We say that a limitcycle C is symmetric if satisfy: ( x, y ) ∈ C ⇒ ( y, x ) ∈ C . Otherwise, we say that C is non-symmetric .Naturally, the above definition allow us to separate all the attractors of Q2R into symmetric and non-symmetric, however,from our (main) Theorem 4.11, it will be shown that the symmetric ones are of 3 types, while that the non-symmetric onespossess the following (stronger) property defined below (see Figure 1). Definition 3.7.
A non-symmetric limit cycle C is said to be asymmetric if satisfy:( x, y ) ∈ C ⇒ ( y, x ) / ∈ C . Figure 1: Scheme of an asymmetric limit cycle C ; if a configuration ( x, y ) belongs to C , then ( y, x ) does not. Next, we continue with a key property of the Q2R model that will allow to have an easy understanding of the attractorclassification shown in this paper.
The following Reversibility Lemma shows a main characteristic of the Q2R system (2.3).
Lemma 3.8 (Reversibility) . Let x , y , z in Ω , then, [( x, y ) → ( z, x )] ⇔ [( x, z ) → ( y, x )] . Proof.
From (2.3) and because of φ ( x ) ⊙ φ ( x ) = :[( x, y ) → ( z, x )] ⇔ (cid:26) x = xz = y ⊙ φ ( x ) ⇔ (cid:26) x = xz ⊙ φ ( x ) = y ⇔ [( x, z ) → ( y, x )]6his Reversibility Lemma says that, if there is a one time step evolution between two configurations, then, there is also a onetime step evolution between their symmetric configuration, but, in the opposite sense. As a consequence, we have the followinggeneralization: Corollary 3.9.
Let x t , y t in Ω , t ∈ { , ..., p } , p ∈ N , then, (cid:0) x , y (cid:1) → · · · → ( x p , y p ) ⇔ ( y p , x p ) → · · · → (cid:0) y , x (cid:1) . Figure 2 illustrates the proof of this property.( x , y ) → (cid:0) x , y (cid:1)| {z } → · · · → ( x p , y p ) | {z } m · · · m z }| { ( y , x ) ← (cid:0) y , x (cid:1) ← · · · z }| { ← ( y p , x p ) Figure 2: Applying successively the Lemma 3.8 at each step-evolution ( x t , y t ) → ( x t +1 , y t +1 ), for t ∈ { , . . . , p } , p ∈ N , one constructs the backwardevolution between their symmetric configurations. Let us study the possible evolutions, according to the type of configurations involved. A , B , C and D Given a configuration of type U ∈ { A, B, C, D } , then the only possible evolutions are:(T1) Configurations of type A .Let ( x, y ) ∈ A , then [ x = y ] ∧ [ φ ( x ) = ]. Since φ ( x ) = , then ( x, y ) → ( y, x ), and because y = x , then φ ( y ) = .Therefore, ( x, x ) → ( x, x ) ∈ A . In fact, this is the characterization of the fixed points of Q2R (Theorem 4.1). Thus, A → A as shown in Figure 3-(T1).(T2) Configurations of type B .Let ( x, y ) ∈ B , then [ x = y ] ∧ [ φ ( x ) = ]. Because φ ( x ) = , then ( x, y ) → ( y, x ). Hence, there are two possibilities for φ ( y ):(i) If φ ( y ) = , then ( x, y ) → ( y, x ) ∈ B . In fact, this is the characterization of the limit cycles of period two of the Q2Rmodel (Theorem 4.5).(ii) If φ ( y ) = , then ( x, y ) → ( y, x ) ∈ D .Thus, [ B → B ] or [ B → D ], as shown in Figure 3-(T2).(T3) Configurations of type C .Let ( x, y ) ∈ C , then [ x = y ] ∧ [ φ ( x ) = ]. Hence, ( x, y ) → ( z = y ⊙ φ ( x ) , x ) with z = y (consequently, z = x ) and there aretwo possibilities for φ ( z ):(i) If φ ( z ) = , then ( x, y ) → ( z, x ) ∈ B .(ii) If φ ( z ) = , then ( x, y ) → ( z, x ) ∈ D .Thus, [ C → B ] or [ C → D ], as shown in Figure 3-(T3).(T4) Configurations of type D .Let ( x, y ) ∈ D , then [ x = y ] ∧ [ φ ( x ) = ]. Hence, ( x, y ) → ( z = y ⊙ φ ( x ) , x ) with z = y (but eventually, z = x ) and thereare two possibilities for z , which implies three possible evolutions for ( x, y ):(i) If z = x , then ( x, y ) → ( z, x ) = ( x, x ) ∈ C .(ii) If z = x , then we have the same two possibilities (T3)-(i) and (T3)-(ii) for φ ( z ).Therefore, [ D → B ], [ D → C ] or [ D → D ], as shown in Figure 3-(T4).The above analysis is summarized in Figure 3. 7T1) (T2) (T3) (T4) Figure 3: (T1) A type A configuration only can evolve to itself. (T2)-(T3) Configurations of type B and C can evolve to configurations of type B or D . (T4) A type D configuration can evolve to any configuration, excepting the configurations of type A .
4. Main Results
Theorem 4.1 (Characterization of fixed points) . Let ( x, y ) ∈ Ω be a configuration of the Q2R model. Then, ( x, y ) ∈ P ⇔ [ x = y ] ∧ [ φ ( x ) = ] . Proof. ( x, y ) ∈ P ⇔ [( x, y ) → ( x, y )], by definition of fixed point. ⇔ (cid:26) x = yx = y ⊙ φ ( x ) , by Remark 2.4. ⇔ [ x = y ] ∧ [ φ ( x ) = ] . Remark . The above result states that fixed points of the Q2R model are always configurations of type A (see Figure 4-a), i.e. , self-symmetric (( x, x ) ∈ Ω xx ) and without null neighborhoods ( φ ( x ) = ). Hence, P = A .a) b) Figure 4: a) Scheme for a fixed point (or limit cycle of period 1). b) Scheme for a period-2 limit cycle.
Since Q2R is a reversible system, if a configuration ( x, y ) ∈ Ω is not a fixed point, then necessarily it belongs to a limitcycle of period 2 or higher. In this context, as a characterization of such a limit cycles, we consider convenient to explicit thenext Corollary, which is the negation of Theorem 4.1. Corollary 4.3.
Let ( x, y ) ∈ Ω be a configuration of Q2R. Then, ( x, y ) belongs in a limit cycle of period 2 or higher ⇔ [ x = y ] ∨ [ φ ( x ) = ] . Remark . The possible evolutions analyzed before implies that the configurations involved in any limit cycle of period 2 orhigher are of type B , C or D (not A). Theorem 4.5 (Characterization of limit cycles of period 2) . Let ( x, y ) ∈ Ω be a configuration of Q2R. Then, ( x, y ) ∈ P ⇔ [ x = y ] ∧ [ φ ( x ) = ] ∧ [ φ ( y ) = ] . roof. ( x, y ) ∈ P ⇔ [( x, y ) → ( y ⊙ φ ( x ) , x ) → ( x, y )], by definition of P . ⇔ [( x, y ) → ( y ⊙ φ ( x ) , x )] ∧ (cid:26) y = y ⊙ φ ( x ) x = x ⊙ φ ( y ⊙ φ ( x ))(by Remark 2.4) ⇔ [( x, y ) → ( y ⊙ φ ( x ) , x )] ∧ (cid:26) φ ( x ) = φ ( y ⊙ φ ( x )) = ⇔ [( x, y ) → ( y, x )] ∧ (cid:26) φ ( x ) = φ ( y ) = ⇔ [ x = y ] ∧ [ φ ( x ) = ] ∧ [ φ ( y ) = ] . Remark . The above result says that the limit cycles of period 2 consists of configurations ( x, y ), of type B and with bothstates, x and y , without null neighborhoods (see Figure 4b). Hence, P ⊂ B .Observe that there are elements in B which do not belong to P . For instance, take the configuration ( , x ) ∈ B where thestate x ∈ Ω is composed by a 2 × i.e., x = · · · · · · · · · · · · · · · · · · · · · · · · ... 11 . . . 1 1 1 1 · · · · · · − − · · · ...... · · · − − · · · ...1 · · · · · · · · · · · · · · · · · · . . . 11 1 · · · · · · · · · · · · L × L , This configuration has only 4 null neighborhoods, located just in the nodes of the inner block of -1s. In other words, φ ( x ) = x .Thus, by applying the Q2R rule we get : • ( , x ) → ( φ ( ) ⊙ x, ) = ( x, ) • ( x, ) → ( φ ( x ) ⊙ , x ) = ( x ⊙ , x ) = ( x, x ) • ( x, x ) → ( x ⊙ φ ( x ) , x ) = ( x ⊙ x, x ) = ( , x )Therefore, the sequence of evolutions ( , x ) → ( x, ) → ( x, x ) → ( , x ) is a period-3 limit cycle and, consequently, ( , x ) ∈ P ,hence, ( , x ) / ∈ P . Theorem 4.7 (Characterization of limit cycles of period 3) . Let { ( x, y ) , ( z, x ) , ( y, z ) } ⊂ Ω such that ( x, y ) → ( z, x ) → ( y, z ) .Then, { ( x, y ) , ( z, x ) , ( y, z ) } ⊆ P ⇔ φ ( x ) ⊙ φ ( y ) ⊙ φ ( z ) = . Proof.
By Remark 2.4, ( x, y ) → ( z, x ) → ( y, z ) means that:[ y = x ⊙ φ ( z )] | {z } ( a ) ∧ [ z = y ⊙ φ ( x )] | {z } ( b ) ⇒ ) { ( x, y ) , ( z, x ) , ( y, z ) } ⊆ P ⇒ [( y, z ) → ( x, y )] ⇒ x = z ⊙ φ ( y ), by Remark 2.4. (4.1)9eplacing (a) in (b) and, after that, (b) in (4.1) we have that x ⊙ φ ( x ) ⊙ φ ( y ) ⊙ φ ( z ) = x , i.e., φ ( x ) ⊙ φ ( y ) ⊙ φ ( z ) = . ⇐ ) φ ( x ) ⊙ φ ( y ) ⊙ φ ( z ) = ⇒ x ⊙ φ ( x ) ⊙ φ ( y ) ⊙ φ ( z ) = ⊙ x = x ⇒ [ x ⊙ φ ( z )] ⊙ φ ( x ) ⊙ φ ( y ) = x ⇒ y ⊙ φ ( x ) ⊙ φ ( y ) = x , by (a). ⇒ z ⊙ φ ( y ) = x , by (b). ⇒ [( y, z ) → ( x, y )]Thus, [( x, y ) → ( z, x ) → ( y, z )] ∧ [( y, z ) → ( x, y )], i.e., { ( x, y ) , ( z, x ) , ( y, z ) } ⊆ P Remark . Contrary with the statements of the Remarks 4.2 and 4.6, in which fixed points and period-2 limit cycles are shownto have a unique topology, the condition φ ( x ) ⊙ φ ( y ) ⊙ φ ( z ) = for period-3 limit cycles allows different topologies (see Figure5). This fact occurs for all limit cycles of period 3 and higher and it will be explained later in Remark 4.12.a) b) Figure 5: The two different topologies for a period-3 limit cycle: a) A symmetric limit cycle, with one self-symmetric configuration ( x, x ). b) Anasymmetric limit cycle, hence, without any self-symmetric configuration.
The following result will be useful for the proof of our main Theorem 4.11 (on the attractors classification in Q2R) anddictates direct consequences that are easily deduced from Theorems 4.1 and 4.5 and the possible evolutions shown in Figure 3.
Corollary 4.9.
Let C be a limit cycle of Q2R with period 3 or higher. Then:(i) C has at least one configuration of type D .(ii) C does not have evolutions of type A → A , nor B → B (notice that C → C does not exist) but it could have evolutions oftype D → D .(iii) If C has a type D configuration, then D comes from a type V configuration with V ∈ { B, C, D } .(iv) If C has a configuration ( x, y ) ∈ B , then ( x, y ) → ( y, x ) ∈ D .4.5. Topological classification of limit cycles in Q2R Definition 4.10.
Let C be a limit cycle of the Q2R model with period T ∈ N . We say that C is: • A symmetric limit cycle of type I (S-I). If T = 1 or if there exists p ∈ N such that C has the topology of Figure 6-a, i.e. ,is symmetric with: – An odd period T = 2( p + 1) + 1. – Only one configuration of type C , only one configuration of type B and (2 p + 1) configurations of type D .10 A symmetric limit cycle of type II (S-II). If there exists p ∈ N such that C has the topology of Figure 6-b, i.e. , is symmetricwith: – An even period T = 2( p + 2). – Only two configurations of type C and 2( p + 1) configurations of type D . • A symmetric limit cycle of type III (S-III). If T = 2 or if there exists p ∈ N such that C has the topology of Figure 6-c, i.e. , is symmetric with: – An even period T = 2( p + 2). – Only two type B configurations and 2( p + 1) type D configurations. • An asymmetric limit cycle (AS). If there exists p ∈ N \ { } such that C has the topology of one of the two limit cycles ofFigure 6-d, i.e. , is an asymmetric limit cycle with: – Period T = p + 1 (it can be even or odd, depending on the value of p ). – All its configurations are of type D .a) b)c) d) Figure 6: Topology of different type of limit cycles. a) Symmetric limit cycle of type I. b) Symmetric limit cycle of type II. c) Symmetric limit cycleof type III. d) Two asymmetric limit cycles.
The following (main) Theorem shows that the only possible limit cycles existing in Q2R are the four ones defined above.
Theorem 4.11 (Attractors classification of Q2R) . Let C be a limit cycle of Q2R with period T ∈ N . Then C is of type S-I,S-II, S-III or AS. roof. Let C be a limit cycle of Q2R with period T ∈ N .If T = 1 or T = 2 then, by definition, C is of type S-I or S-III, respectively.Let T ≥
3, then, by Corollary 4.9-(i), C has at least one configuration of the type D , ( x , y ), which, according with Corollary4.9-(iii), it may come from a configuration of the type B , C or D . Previous statement allows us to consider only three casesfor the limit cycle C :(C1) C has only configurations of type D . In this case, C has the form ( x , y ) → ( x , y ) → · · · → ( x p , y p ) → ( x , y ) with p = T − i.e. , C is of type AS. (C2) C has a configuration ( x , y ) ∈ D coming from a configuration ( y , x ) ∈ B , as in Figures 6-a) or 6c). Because of Corollary4.9-(ii), a configuration of type D could (eventually) evolve to another configuration of type D , so, we can consider p ∈ N as the maximum time step such that [( x , y ) → ( x , y ) → · · · → ( x p , y p )] ∈ C and ( x t , y t ) ∈ D , ∀ i ∈ { , ..., p } . Hence, wehave only two possible evolutions for the configuration ( x p , y p ) ∈ D :(i) ( x p , y p ) → ( x p +1 , y p +1 ) ∈ B .In this case, by Corollary 4.9-(iv), ( x p +1 , y p +1 ) → ( y p +1 , x p +1 ) ∈ D and ( y p +1 , x p +1 ) → ( y p , x p ) → · · · → ( y , x ) → ( y , x ) (by Corollary 3.9), completing C with an even period T = 2( p + 2) as shown in Figure 6c), i.e., C is of typeS-III.(ii) ( x p , y p ) → ( x p +1 , y p +1 ) ∈ C .In this case, ( x p +1 , y p +1 ) = ( x p , x p ) and, again, the Corollary 3.9 justifies both: ( x p , x p ) → ( y p , x p ) and ( y p , x p ) → ( y p − , x p − ) → · · · → ( y , x ) → ( y , x ), completing C with an odd period T = 2( p + 1) + 1 as shown in Figure 6a),i.e., C is of type S-I.(C3) C has a configuration ( x , y ) ∈ D coming from a configuration ( y , y ) ∈ C . By considering p ∈ N the same as (C2),again we have only two possible evolutions for ( x p , y p ) ∈ D : ( x p +1 , y p +1 ) ∈ B or ( x p +1 , y p +1 ) ∈ C , but the analysis inboth cases are similar to (i) and (ii) of the previous case (C2), respectively. In the case (i), C ends being of type S-I withan odd period T = 2( p + 1) + 1 as shown in Figure 6a) while that, for the case (ii), C ends up being of type S-II with aneven period T = 2( p + 2) as shown in Figure 6b). Remark . Let C be a limit cycle of Q2R with period T ∈ N and ( x, y ) ∈ C . Some particular information about T and theconfiguration ( x, y ) can help us to deduce the particular type of C . In fact: • If T is odd, then C could be of type S-I or AS only. If T is even, then C could be of type S-II, S-III or AS. • If ( x, y ) ∈ B then, if there exists ( x ′ , y ′ ) ∈ C such that [( x ′ , y ′ ) = ( x, y )] ∧ [( x ′ , y ′ ) ∈ B ], then C is of type S-III, otherwise, C is of type S-I. • If x = y ( i.e. , ( x, y ) ∈ { A, C } ), then, if there exists ( x ′ , y ′ ) ∈ C such that [( x ′ , y ′ ) = ( x, y )] ∧ [ x ′ = y ′ ], then C is of typeS-II, otherwise, C is of type S-I. Remark . The asymmetric limit cycles always appear in pairs (in the sense that all the symmetric configurations of onelimit cycle belongs into the other limit cycle). Furthermore, we know that ( x, y ) ∈ D means [ x = y ] ∧ [ φ ( x ) = ] (regardless thevalue of φ ( y )) but, if ( x, y ) belongs to an asymmetric limit cycle, then ( x, y ) ∈ D and, necessarily, φ ( y ) = , i.e.,[ x = y ] ∧ [ φ ( x ) = ] ∧ [ φ ( y ) = ] . (4.2)The converse relation is not true, i.e., if ( x, y ) satisfy (4.2) then not necessarily ( x, y ) belongs to an asymmetric limit cycle (seethe limit cycle (7.2) in Appendix 7.3 as a counterexample). Note that, by Corollary 3.9, the symmetric configurations of the previous limit cycle produces a different limit cycle C ′ of the form ( y , x ) → ( y p , x p ) → · · · → ( y , x ) → ( y , x ) also composed by configurations of type D , as shown in Figure 6-d (left), i.e. , C ′ is also of type AS. efinition 4.14. We denote by ν SI ( T ), ν SII ( T ), ν SIII ( T ) and ν AS ( T ) as the number of configurations belonging to a limit cycleof period T and of type S-I, S-II, S-III and AS, respectively. Similarly, n SI ( T ), n SII ( T ), n SIII ( T ) and n AS ( T ) denote the numberof limit cycles of period T and type S-I, S-II, S-III and AS, respectively. Notice that: ν ( T ) = ν SI ( T ) + ν SII ( T ) + ν SIII ( T ) + ν AS ( T ) n ( T ) = n SI ( T ) + n SII ( T ) + n SIII ( T ) + n AS ( T ) n q ( T ) = ν q ( T ) T , with q ∈ { SI , SII , SIII , AS } . The following result shows that the sets S-I and S-II are naturally rare in the whole phase space.
Theorem 4.15. X T ≥ ( n SI ( T ) + 2 n SII ( T )) = | Ω xx | = 2 N .Proof. Because of the third point of Remark 4.12, one has that if ( x, x ) ∈ Ω xx , then, ( x, x ) belongs only to a cycle of type S-Ior S-II. Therefore, this Theorem is true because the limit cycles of type S-I include one configuration in Ω xx while that thosecycles of type S-II include two configurations in Ω xx . In this section we show that the number of fixed points, | P | , will be the fundamental quantity that determine the size ofthe different sub-spaces of Ω . Our starting point relates the neighborhoods of x ∈ Ω with those of − x ∈ Ω. Proposition 4.16.
Let x ∈ Ω . Then: φ ( x ) = ⇔ φ ( − x ) = .Proof. The null neighborhoods of x ∈ Ω have two -1’s and two 1’s. This is kept for − x ∈ Ω, so: φ ( x ) = ⇔ φ ( − x ) = .The second result relates fixed points with limit cycles of period 2. Theorem 4.17. { ( x, x ) , ( y, y ) } ⊆ P ⇔ { ( x, y ) , ( y, x ) } ⊆ P Proof. { ( x, x ) , ( y, y ) } ⊆ P ⇔ [ x = y ] ∧ [ φ ( x ) = ] ∧ [ φ ( y ) = ], by Theorem 4.1. ⇔ { ( x, y ) , ( y, x ) } ⊆ P , by Theorem 4.5.As a first consequence of the previous Theorem, we have the following result: Corollary 4.18.
Let x ∈ Ω such that φ ( x ) = . Then, in the Q2R dynamics: • ( x, x ) and ( − x, − x ) are two different fixed points, as well as; • ( x, − x ) → ( − x, x ) → ( x, − x ) is a limit cycle of period 2.Proof. Let x ∈ Ω such that φ ( x ) = , then, by Theorem 4.1 and Proposition 4.16, { ( x, x ) , ( − x, − x ) } ⊆ P . Therefore, byTheorem 4.17, { ( x, − x ) , ( − x, x ) } ⊆ P . Finally, applying the Q2R rule, we have that ( x, − x ) → ( − x, x ) → ( x, − x ) is, in fact, alimit cycle of period 2. P , P and P . A second consequence of Theorem 4.17 shows that the number of limit cycles of period 2 is larger than the number of fixedpoints, moreover:
Corollary 4.19. | P | = | P | ( | P | − . roof. | P | = 2 · (number of limit cycles of period 2), by definition of P = 2 · (cid:18) | P | (cid:19) , by Theorem 4.17= | P | ( | P | − . Moreover, because of Remark 4.2, P = A = (cid:8) ( x, y ) ∈ Ω xx /φ ( x ) = (cid:9) . In other words, P = { ( x, x ) ∈ Ω : φ ( x ) = } . Thus, | P | = | Φ | where Φ = { x ∈ Ω /φ ( x ) = } , i.e. , Φ is the set of states without null neighborhood. Definition 4.20.
Let x ∈ Ω. We define the staggered-state [13] x B as the state x restricted to the nodes k ∈ { , ..., N } suchthat k + k is even. Analogously is defined the staggered-state x W but for the nodes k ∈ { , ..., N } such that k + k is odd.In other words, x B and x W can be seen as the black and white fields in the “chessboard x ∈ Ω” respectively, and, such that itssuperposition – denoted with the ⊎ symbol – is x , i.e.: x = [ x B ⊎ x W ] ∈ Ω . We will use the subindices ( · ) B and ( · ) W to refer to the corresponding restriction of the element ( · ) which we are working inorder that: ( · ) = [( · ) B ⊎ ( · ) W ] ∈ Ω . In particular, we define the chessboard states BW and W B of Ω as follows: BW ≡ [ − B ] ⊎ W and W B ≡ − BW = B ⊎ [ − W ] . Remark . This construction using the staggered-states is particularly useful in the case of the von Neuman neighborhood.Moreover, in the particular case of L being an odd number, the staggered-states loss its utility (see the first fact of Remark4.26). Remark . According with the above definition, notice the following statements:1. Ω = Ω B ⊎ Ω W .2. | Ω | = 2 N ⇒ | Ω B | = | Ω W | = 2 N/ .3. x ∈ Ω ⇔ [ x B ∈ Ω B ] ∧ [ x W ∈ Ω W ].4. φ ( BW ) = φ ( W B ) = .The following two propositions establish conditions for the existence of fixed points and limit cycles of period 2 and 3. Proposition 4.23. ∀ L even, ≤ | P | < | P | .Proof. Because of definition 2.3 and remark 4.22: φ ( − ) = φ ( ) = φ ( BW ) = φ ( W B ) = . Hence, by Theorem 4.1: { ( , ) , ( − , − ) , ( BW , BW ) , ( W B , W B ) } ⊆ P , therefore, 4 ≤ | P | < | P | ( | P | −
1) = | P | (where the last equality is by Corollary 4.19). Proposition 4.24. ∀ L ≥ , | P | ≥ N = 6 L .Proof. Let L ≥
4, and consider an state x ∈ Ω of Remark 4.6 where we have proved that { ( , x ) , ( x, ) , ( x, x ) } ⊆ P , i.e. , | P | ≥ x , there are N equivalent configurations, moreover, because theinverse configuration ( − , − x ) ∈ P , there are at least 2 N limit cycles of period 3. Therefore, | P | ≥ N . Definition 4.25.
We denote by B and W as the sets of staggered-states x B ∈ Ω B and x W ∈ Ω W without null neighborhoods,respectively. That is: B = { u ∈ Ω B : ∃ x ∈ Ω , [ x B = u ] ∧ [ φ ( x ) W = W ] } W = { v ∈ Ω W : ∃ x ∈ Ω , [ x W = v ] ∧ [ φ ( x ) B = B ] } xample. Consider the following state x ∈ {− , } and its corresponding state φ ( x ) ∈ {− , } : x = and φ ( x ) = The values in the boxes of x correspond to the staggered-state x B ∈ {− , } while the other values that are not in boxescorrespond to x W ∈ {− , } . Similarly, the values in the boxes of φ ( x ) correspond to φ ( x ) W and were obtained with the valuesof x B while the other values, that are not in the boxes of φ ( x ), correspond to φ ( x ) B and were obtained with the values of x W . Remark . The previous example allows us to understand the following facts, that are direct consequences of the abovedefinitions:1. The neighborhoods in x B are independent of those in x W , when L is even.2. φ ( x ) = ⇔ [ φ ( x ) B = B ] ∧ [ φ ( x ) W = W ]. In other words, studying the set Φ is equivalent to study the sets B and W .3. B ⊎ W = Φ .4. | B | = | W | . Definition 4.27.
We denote β ≡ | B | = | W | .The following corollary will be useful to prove, in Theorem 4.29, that β is an even number. Corollary 4.28.
Let x ∈ Ω . Then: x B ∈ B ⇔ − x B ∈ B x W ∈ W ⇔ − x W ∈ W Proof.
Let x ∈ Ω. (1) x B ∈ B ⇔ φ ( x ) W = W ⇔ φ ( − x ) W = W (by Proposition 4.16 and fact (2) of Remark 4.26) ⇔ − x B ∈ B (2) Is analogous to (1).The following Theorem shows that the number of fixed points of Q2R is always an square number and a multiple of 4. Theorem 4.29.
The following statements are true: β = 2 k , for some k ∈ N . | P | = | Φ | = β = 4 k . | P | = 4 k (4 k − .Proof. β ≥ { B , − B } ⊆ B and { W , − W } ⊆ W . The fact that β it is even is direct from Corollary 4.28.2. As already said: | P | = |{ ( x, x ) ∈ Ω : φ ( x ) = }| = |{ x ∈ Ω /φ ( x ) = }| = | Φ | . Since β = 2 k and because of the facts (3) and (4) ofRemark 4.26: | Φ | = | B | · | W | = β = 4 k . Therefore: | P | = | Φ | = β = 4 k , for some k ∈ N .3. This proof is a direct from Corollary 4.19 and statement (2) previously proved.15 -IIIS-IS-II AS cycles Figure 7: The phase space Ω decomposed according to the results of the previous sections.
5. A general overview of the phase space
Before concluding, we will explain the Figure 7 that summarizes briefly the phase space partitions in limit cycles accordinglywith the main results obtained in the previous sections. • The whole figure represents Ω that is partitioned according to definition 3.1: Ω = Ω xx ∪ Ω xy . • Ω xx (colored by yellow and green) is partitioned in A = P (yellow) and C =Ω xx − P (green). • Ω xy (colored by orange and blue) is partitioned in P (orange) and Ω xy − P (blue). • The figure tries to reflect, though not in the real scale, that | Ω xx | ≪ | Ω xy | and | P | ≪ | P | . • P ∪ P (yellow and orange) represents the set of configurations without null neighborhoods. For the complementaryconfigurations (those of green and blue regions), at least one of its states has a null neighborhood. • The dashed line, that limits the left region of Ω with colors orange, yellow, green and a part of the blue, corresponds toconfigurations that are symmetric limit cycles ( i.e. , limit cycles of type S-I, S-II or S-III only). The remaining region ofΩ (only in blue) correspond to configurations belonging to asymmetric limit cycles (i.e., of type AS). • Configurations belonging in C = Ω xx − P (green) are in limit cycles of type S-I or S-II only. Those of type S-I arerepresented only with one configuration in the green region (the remaining configurations lying in the blue region limitedby the dashed line). Those of type S-II are represented only with two configurations in the green region (the remainingconfigurations lying in the blue region enclosed by the dashed line). • The limit cycles with all its configurations belonging to the leftmost blue region limited by the dashed line are exclusivelyof type S-III. • While P and P are exactly the yellow and orange regions, respectively, the other sets P j ( j ≥
3) have configurations inthe green or blue regions (but not in the yellow nor the orange) as exemplified in the figure.The following Table 1 complements Figure 7 by showing the sizes of the different regions of Ω above mentioned for smallsystem ( N = 16 , , and 64).Since the sizes of the different partitions shown in Figure 7 essentially depends on β , this value was computationally calculatedin Table 1 by generating all the staggered-states x B ∈ Ω B without null neighborhoods ( i.e. , φ ( x ) W = ) in order to constructthe set B . This procedure gave us a computable size of | Ω B | = 2 N/ , for N = 16 , , and 64.16 abel variable size L = 4 L = 6 L = 8[b] N L
16 36 64[c] | Ω | N ∼ · ∼ · ∼ · [d] | Ω xx | N = 65536 2 ∼ · ∼ · [e] | Ω xy | [c]-[d] ∼ · < [c] ∼ · < [c] ∼ · < [c][f] —Yellow— | P | = β [g] — Orange— | P | = [f]([f]-1) 1335180 ∼ ∼ · [h] —Green— [d]-[f] 64380 ∼ · < [d] ∼ · < [d][i] — Blue— [c]-([f]+[g]+[h]) ∼ · < [e] ∼ · < [e] ∼ · < [e] Table 1: Summary of the sizes of the main regions discussed in the Figure 7. The 1st column labels the values of the “size” column. The 2nd columnhas the variables and the main regions explained in Figure 7. In the 3rd column are the size formulas for each “variable” of the 2nd column. In the4th, 5th and 6th columns are the calculations done for L ∈ { , , } , respectively.
6. Discussion
Because of the relevance of an accurate knowledge of the phase space in complex dynamical systems with many degrees offreedom, we attempted a classification of the phase space of the Q2R cellular automaton which is in close connection with theIsing model and its statistical properties. The Q2R model is reversible and essentially all results of the present paper follow afterthe Lemma 3.8 (on Reversibility). The main results in the present paper are: Theorem 4.11 that shows a fully classification ofall Q2R attractors in four types of limit cycles consisting of symmetric and asymmetric ones. Moreover, a general overview ofthe phase space decomposition has been provided. Besides, some specific results for small period limit cycles are the following:Theorem 4.29 that shows both, that the total number of fixed points is of the form | P | = β = 4 k , with k ∈ N and that thenumber of configurations belonging in a period-2 limit cycle is | P | = β ( β − Theorem 6.1.
If the neighborhood considered for the definition of Q2R is composed by an odd number of nodes, then theQ2R dynamic has only fixed points and period-2 limit cycles.Proof.
If the neighborhood has an odd number of nodes, then, given x ∈ Ω, all its neighborhoods will have more 1s than-1s or vice-versa. In both cases, its sum will never be 0. In other words, φ ( x ) = , ∀ x ∈ Ω. Thus, by Theorems 4.1 and4.5, the Q2R dynamic will have only fixed points and period-2 limit cycles.3. Notice that we do not have a mathematical expression for β . In this context, all numerical values involving β in this paper,such as those of Table 1, were obtained checking state by state if they have or not a null-neighborhood.4. At a computational level, it is important to note that, to make a correct dynamic study of Q2R, initial configurations ofboth Ω xx and Ω xy must be considered because if only the first one is considered, then it will never be possible to obtain,for instance, period-2 limit cycles nor asymmetric limit cycles. On the other hand, if only Ω xy is considered, then it willnever be possible to obtain fixed points.5. For the dynamical characterization of limit cycles of period-4 and higher, we have not proven general conditions of existenceneither we are not able to compute its cardinality, however, a period T limit cycle is characterized by( x , y ) → ( x , y ) → ( x , y ) → · · ·· · · → ( x T − , y T − ) → ( x T , y T ) = ( x , y ) , with x = y ⊙ φ ( x ), y = x , x = y ⊙ φ ( x ), . . . , x T = y T − ⊙ φ ( x T − ) = x , y T = x T − = y . Therefore, oneconcludes the following necessary (but not sufficient) conditions:17a) For T even, we have two, a priori , independent conditions: φ ( x ) ⊙ φ ( x ) ⊙ φ ( x ) ⊙ · · · ⊙ φ ( x T − ) ⊙ φ ( x T − ) = φ ( x ) ⊙ φ ( x ) ⊙ φ ( x ) ⊙ · · · ⊙ φ ( x T − ) ⊙ φ ( x T − ) = . (6.1)(b) For an odd period, one has the necessary condition φ ( x ) ⊙ φ ( x ) ⊙ φ ( x ) ⊙ · · · ⊙ φ ( x T − ) ⊙ φ ( x T − ) = . (6.2)As an example the period four limit cycles posses three different types of limit cycles, the type S-II, the type S-III andthe type AS (see Figure 8).a) b) c) Figure 8: a) A period-4 limit cycle ( x, x ) → ( y, x ) → ( y, y ) → ( y, x ) → ( x, x ) (type S-II). b) A period-4 limit cycle ( x, y ) → ( z, x ) → ( x, z ) → ( y, x ) → ( x, y ) (type S-III). c) A period-4 limit cycle ( x, y ) → ( z, x ) → ( u, z ) → ( y, u ) → ( x, y ) (type AS). The period five limit cycles posses two different types of limit cycles, the type S-I and the type AS (see Figure 9).a) b)
Figure 9: (a) A period 5 limit cycle ( x, x ) → ( y, x ) → ( z, y ) → ( y, z ) → ( x, y ) → ( x, x ). (b) Example of a limit cycle ( x, y ) → ( z, x ) → ( u, z ) → ( v, u ) → ( y, v ) → ( x, y ) this limit cycle does not exist in the case of a 4 ×
6. Because of the special topology of the limit cycles of type S-I and S-II (see Figure 6-a & 6-b) these limit cycles are fullycharacterized by a simpler set of conditions.Let be the sequence: x = x ⊙ φ ( x ) for the first step x t +1 = x t − ⊙ φ ( x t ) for t = { , , , . . . , T − } , (6.3)then, the closing conditions for limit cycle for an even periodic limit cycle (6.1) imposes φ ( x ) ⊙ φ ( x ) ⊙ φ ( x ) ⊙ φ ( x ) ⊙ · · · ⊙ φ ( x T − ) ⊙ φ ( x T − ) = , (6.4)while, for an odd period, the condition (6.2) simplifies to φ ( x ( T − / ) = . (6.5)Because the even and odd cases follow quite different conditions we conjecture that: Conjecture . Let be odd period T = 2 q + 1 with q ∈ N , and let x = x ⊙ φ ( x ), together with x t +1 = x t − ⊙ φ ( x t ) for t ∈ { , , , . . . , q − } , then, the pair ( x , x ) belongs to a type S-I limit cycle of period T = 2 q + 1, iff φ ( x ) = ∧ φ ( x ) = ∧ · · · ∧ φ ( x q − ) = ∧ φ ( x q ) = . Conjecture . Let be the even period T = 2 p , with p a prime number and let x = x ⊙ φ ( x ), and x t +1 = x t − ⊙ φ ( x t )for t = { , , , . . . , p − } , then the pair ( x , x ) belongs to a type S-II limit cycle of a period T = 2 p iff φ ( x ) = ∧ φ ( x ) = ∧ · · · ∧ φ ( x p − ) = ∧ φ ( x ) ⊙ φ ( x ) ⊙ · · · ⊙ φ ( x p − ) = . (6.6)As before, the cardinality of the sets S-II with even periods T = 2 p ( p a prime number), nevertheless, the generaleven period case requires more careful considerations. Essentially, there is a double counting, e.g. the case of period T = 2 ×
7. Appendix: Exact results for the case 4 × Ω The 4 × configurations, that can be fully scanned numerically. Thenext 6 × configurations, making impossible this task.In Tables 2 and 3, we provide the exact distributions for the number of configurations and for the number of limit cyclesaccordingly with its period and the cycle type, respectively. One observes that the number of odd period limit cycles are rare.As a general rule, the number of configurations (and the number of limit cycles) of even period limit cycles are much larger thanthe odd ones limit cycles. We also notices that the periods 7, 11, 13, etc. are missing. Moreover, the largest odd period is 27.The reason why some periods exist and other does not is still an open problem. T ν SI ( T ) ν SII ( T ) ν SIII ( T ) ν AS ( T ) | P T | Table 2: The distribution of the configurations of Ω according with the type of limit cycle that belongs, for a 4 × T , the second (resp. third, fourth and fifth) one, the total number of configurations belonging in a period- T limit cycle of typeS-I (resp. S-II, S-III and AS). Finally, the column | P T | indicates the size of the set P T . The largest number of limit cycles for a given period is obtained for T = 12 where n (12) = 65 , , ν (12) ≈ × , which is about an 18% of the totalphase space. On the contrary, the smallest set is P with just 1,156 configurations ( ≈ − %). However, Table 3 shows thatthe smallest number of limit cycles are those of period 27 with 256 limit cycles.19 n SI ( T ) n SII ( T ) n SIII ( T ) n AS ( T ) n ( T )1 1,156 0 0 0 1,1562 0 0 667,590 0 667,5903 1,376 0 0 256 1,6324 0 3614 5,139,064 12,096,102 17,238,7805 384 0 0 0 3846 0 1,760 2,536,656 3,342,496 5 ,880 ,9128 0 5,344 7,299,968 29,375,904 36,681,2169 384 0 0 512 89610 0 768 517,440 294,144 812,35212 0 8,304 11,024,512 54,609,744 65,642,56018 0 3,840 1,045,824 7,985,152 9,034,81620 0 960 864,768 2,684,736 3,550,46424 0 1,152 4,820,992 22,342,528 27,164,67227 0 0 0 256 25630 0 0 528,384 98,304 626,68836 0 0 1,417,728 9,260,800 10 ,678 ,52840 0 1,920 657,408 6,160,512 6,819,84054 0 3,456 0 4,494,848 4,498,30460 0 0 552,960 1,886,208 2,439,16872 0 0 657,408 2,252,800 2,910,20890 0 0 147,456 196,608 344,064108 0 0 0 3,051,008 3,051,008120 0 0 0 1,671,168 1,671,168180 0 0 0 172,032 172,032216 0 0 0 831,488 831,488270 0 0 0 98,304 98,304360 0 0 0 172,032 172,032540 0 0 0 49,152 49,1521080 0 0 0 49,152 49,152Total 3,300 31,118 37,878,158 163,176,246 201,088,822 Table 3: The distribution of the number of limit cycles for a 4 × T ,the second (resp. third, fourth and fifth) one, the total number of period- T limit cycles of type S-I (resp. S-II, S-III and AS). Finally, the column n ( T ) indicates the total number of period- T limit cycles. More important, from the total 2 configurations, a fraction of 13 .
33 % are symmetric limit cycles (10 − % of type S-I,10 − % of type S-II and 13 .
31 % of type S-III) and 86 .
67 % are asymmetric (AS) limit cycles.The values showed in the above tables are also summarized in Figure 10 where it can be observed that, apparently, thequantities ν SI ( T ), ν SII ( T ), ν SIII ( T ) and ν AS ( T ) (see Figure 10-a) are upper bounded by K T − / , with K a constant. Onthe contrary the quantities n SI ( T ), n SII ( T ), n SIII ( T ) and n AS ( T ) (see Figure 10-b) are upper bounded by K T − / with K aconstant. We do not know the reasons for the bound.Moreover, because of Theorem 4.15 we are able to upper bound the total number of S-I and S-II limit cycles as follows: X T ≥ ( n SI ( T ) + n SII ( T )) < | Ω xx | = 2 N ≪ | Ω xy | = 2 N (2 N − . This can be noticed in both Figures 10.a) b) Figure 10: a) Number of configurations, ν q ( T ), and (b) number of limit cycles, n q ( T ), per period, for q ∈ { SI , SII , SIII , AS } . Finally, the following Figure 11 plots the normalized density of limit cycles for each topology. To do that, we normalize n SI ( T ) by Ω xx = 2 N , that is ̺ SI ( T ) = n SI ( T )2 N .
20n the same way we normalize ̺ SII ( T ) = 2 n SII ( T )2 N , ̺ SIII ( T ) = n SII ( T )2 N (2 N − , ̺ AS ( T ) = n AS ( T )2 N (2 N − . Figure 11: Normalized density of limit cycles per period for different topology.
The interest of this plot is that it shows that relative to its set, namely Ω xx , the S-I and S-II are of the same order ofmagnitude, than S-III and AS relative to Ω xy . An example of an odd period limit cycle (different of a fixed point) is the following period-3 limit cycle of the form ( x, x ) → ( y, x ) → ( x, y ) → ( x, x ) (type S-I), like those of Figures 5a) or 6a), where x is in red and y is in blue:( x, x ) = − − − − − − − − − − , − − − − − − − − − − ∈ C ( y, x ) = − − − − − − − − − − − − − − − − , − − − − − − − − − − ∈ B ( x, y ) = − − − − − − − − − − , − − − − − − − − − − − − − − − − ∈ D (7.1)Observe that, by Remark 4.12, there not exists limit cycles of type S-I with an even period. The following example is a period-4 limit cycle (type S-II) of the form ( x, x ) → ( y, x ) → ( y, y ) → ( x, y ) → ( x, x ), like thoseof Figures 6b) or 8a), where x is in red and y is in blue: 21 x, x ) = − − − , − − − ∈ C ( y, x ) = − − − , − − − ∈ D ( y, y ) = − − − , − − − ∈ C ( x, y ) = − − − , − − − ∈ D (7.2)Observe that, by Remark 4.12, there not exists limit cycles of type S-II with an odd period. An example of an even period limit cycle (different of a period-2 limit cycle) is the following period-4 limit cycle of the form( x, y ) → ( z, x ) → ( x, z ) → ( y, x ) → ( x, y ) (type S-III), like those of the Figures 6c) or 8b), where x is in red, y is in blue and z is in green: ( x, y ) = − − −
11 1 − − , − − − − ∈ D ( z, x ) = − − − − , − − −
11 1 − − ∈ B ( x, z ) = − − −
11 1 − − , − − − − ∈ D ( y, x ) = − − − − , − − −
11 1 − − ∈ B Observe that, by Remark 4.12, there not exists limit cycles of type S-III with an odd period.
The following is a period-3 limit cycle of the form ( x, y ) → ( z, x ) → ( y, z ) → ( x, y ) (type AS), like those of the Figures 5b)or 6d), where x is in red, y is in blue and z is in green: 22 x, y ) = − − − − − − − − − − − − , − − − − − − − − − − − − ∈ D ( z, x ) = − − − − − − − − − − − − , − − − − − − − − − − − − ∈ D ( y, z ) = − − − − − − − − − − − − , − − − − − − − − − − − − ∈ D The following ia a period-4 limit cycle of the form ( x, y ) → ( z, x ) → ( u, z ) → ( y, u ) → ( x, y ) (type AS), like those of theFigures 6d) or 8c), where x is in red, y is in blue, z is in green and u is in black:( x, y ) = − − − − − , − − ∈ D ( z, x ) = − − − − − − − − , − − − − − ∈ D ( u, z ) = − − − − − , − − − − − − − − ∈ D ( y, u ) = − − , − − − − − ∈ D Acknowledgment
Work partially supported by FONDECYT Iniciaci´on 11150827 (M.M-M.). S.R. thanks the Gaspard Monge Visiting ProfessorProgram of ´Ecole Polytechnique (France). F.U. thanks FONDECYT (Chile) for financial support through Postdoctoral N ◦ References [1] P. Ehrenfest and T. Ehrenfest,
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