Selective chaos of travelling waves in feedforward chains of bistable maps
SSelective chaos of travelling waves in feedforward chainsof bistable maps
Bastien Fernandez
Laboratoire de Probabilit´es, Statistique et Mod´elisation, CNRS - Univ. Paris Denis Diderot - SorbonneUniv., 75205 Paris CEDEX 13 France
Abstract. We study the chaos of travelling waves (TW) in unidirectional chains of bistablemaps. Previous numerical results suggested that this property is selective, viz. given theparameters, there is at most a single (non-trivial) velocity for which the corresponding set ofwave profiles has positive topological entropy. However, mathematical proofs have remainedelusive, in particular because the related symbolic dynamics involves entire past sequences.Here, we consider instead inite (short) rank approximations for which the symbolic dynamicshas finite memory. For every possible velocity, we compute the existence domains of all possiblefinite type subshifts of TW with positive entropy. In all examples, chaos of TW turns out tobe selective, indeed.
Dedicated to the memory of Valentin Afraimovich.
Valentin Afraimovich had a strong interest for lattices of coupled dynamical systems, the so-called LatticeDynamical Systems (LDS). He made a number of diverse and important theoretical contributions to thisfield; a summary of them can be found in the notes of his lecture at the CML2004 school in Paris [1]. Inshort terms, a LDS is a (continuous or discrete time) dynamical system whose phase space is M Z d (latticeconfigurations), where M is a subset of R or of a (compact) manifold, and d ∈ N is the lattice dimension.In this setting, Valentin introduced me to the problem of the detection of a preferred direction in space-time (from his own words, often accompanied with expressive hand motions!), presumably a measure ofthe velocity of information flow in the system. During several years, we frequently spent time together invarious places, San Luis Potosi, Marseille, etc. While he continuously showed receptive patience, Valentinwas moved by a strong will, which manifested itself as a vigorous stimulation. A young fellow at that time,I had no previous experience of interaction with a senior colleague who was both scientifically demandingand open to feedback and discussion. This collaborative experience has been truly beneficial to me, and alsoa particularly good time of a close relationship.Valentin’s incentive brought us (also with Antonio Morante) to extend Milnor’s notion of directional entropyin cellular automata [2, 3] to lattice dynamical systems [4]. Two context-dependent definitions emerged. Thefirst one involved space-time normalisation and was intended for systems with chaotic (temporal) dynamics.The second one used temporal normalisation only and was aimed at the case of regular dynamics.We proceeded to an extended investigation of basic characteristics, by analogy with the analysis of the topo-logical entropy in dynamical systems [5]. Furthermore, we obtained explicit estimates in simple examples,which unexpectedly showed that the space-time normalised quantity did not depend on the direction. Thisinvariance was later confirmed to hold in every translation invariant LDS, thanks to a decisive contribu-tion by Edgardo Ugalde [6]. Therefore, the directional entropy unfortunately appears to be insufficient todetect preferred space-time directions, at least for chaotic systems whose dynamics commutes with spatialtranslations.What about systems with regular dynamics? The space-time normalised directional entropy would haveto vanish but not necessarily the time normalised one. Moreover, the examples below [4] show non trivialdependence and call for further investigation. Translation invariant means that the dynamics commutes with the operator σ of spatial translations, which is definedbelow. More precisely, the entropy as defined in [4] depended on the direction. However, [6] showed that this dependence wastrivial, because it could be removed by suitable renormalisation. a r X i v : . [ n li n . C G ] N ov Unidirectional systems, chaos of travelling waves and symbolicdynamics
The probably simplest (non trivial) example of LDS with regular dynamics is the direct product (over Z )of bistable one-dimensional maps ( ie. maps of the interval having two stable fixed points). Explicitly, thedynamics of configurations x ∈ [0 , Z is generated by the map F defined by( F ( x )) s = f ( x s ) , ∀ s ∈ Z . Assuming that f (0) = 0 and f (1) = 1 are the two stable fixed points of f , any configuration in { , } Z mustbe a (stable) fixed point of F . This property is an instance of spatial chaos in LDS, and it implies that the(time normalised) directional entropy vanishes in the direction of time and is maximal in the direction ofspace.The next example is when this uncoupled system is combined with the spatial transition operator σ , viz. ( σ ◦ F ( x )) s = f ( x s − ) , ∀ s ∈ Z . Under the same circumstances, the system σ ◦ F has a spatial chaos, now of (convectively stable) travellingwaves of velocity 1. Its directional entropy vanishes in the direction of wave propagation (corresponding tothe angle π in the space-time) and, as before, is maximal in the orthogonal direction.These examples are somewhat naive and rather irrelevant. As modelling of transport phenomena is con-cerned, it would be more interesting to have similar information for the following unidirectional chain ofcoupled (bistable) maps ( F (cid:15) ( x )) s = (1 − (cid:15) ) f ( x s ) + (cid:15)f ( x s − ) , ∀ s ∈ Z , (1)for (cid:15) ∈ [0 ,
1] (NB: this system reduces to the previous examples for the limit values (cid:15) = 0 and (cid:15) = 1respectively). So far, estimates of the directional entropy for this chain have remained elusive (expected inthe neighbourhood of the limit cases, using perturbative arguments). To address this issue, the considerationsin the limit cases above suggest to investigate the chaos of travelling waves (TW) for an arbitrary velocity.Travelling waves in (continuous or discrete time) LDS can be defined following the basic notion in physics,namely that they are trajectories given by x ts = u ( s − vt ) for some profile u : R → R and some velocity v ∈ R [9]. However, TW in discrete-time LDS can also be characterised using spatial translations [10], includingfor irrational velocities [11]. In particular, any solution of the equation G q ( x ) = σ p ( x ) , (2)where p ∈ Z and q ∈ N , defines a TW of velocity v = pq for the LDS generated by the map G . Of note, thevelocity is constrained by the coupling range, in particular we must have v ∈ [0 ,
1] for the LDS (1).Extending the notion above, the LDS generated by G is said to have (spatial) chaos of TW of velocity pq if theaction of σ on the set of solutions of (2) has positive topological entropy. Together with Vladimir Nekorkin,Valentin has asserted the existence of chaos of TW in some examples of LDS by constructing horseshoes ofthe profile generating dynamics associated with equation (2) [12]. This approach can be viewed as a nonlinearextension of the transfer matrix technique in theoretical solid-state physics. However, the construction isvelocity specific and makes it difficult to evaluate the velocity dependence on parameters, not to mention toportray a global description of the chaos of TW in parameter space.To address this issue, we considered a special case of the unidirectional chain (1) above, when the individualmap f is piecewise affine with two branches of unique slope, separated by a discontinuity. Formally speaking,the map writes f ( u ) = au + (1 − a ) H ( u − T ) , ∀ u ∈ [0 , , (3)where H is the (right continuous) Heaviside function, the slope a ∈ [0 ,
1) and the discontinuity T ∈ (0 , Another area of Valentin’s expertise, a LDS is said to have spatial chaos when the action of the spatial translations( σ ( x )) s = x s − on the set of LDS fixed points (viewed as a dynamical system whose time is given by the spatial variable s [7, 8]), is chaotic, typically with positive topological entropy [1]. T Figure 1: Entropy-velocity diagrams of TW of the LDS (1) with individual map (3) for a = 0 . Left:
Original diagram in the full square of the parameters ( (cid:15), T ). Right:
Zoom into the central region delimitedby the square). Painted points represent the velocity dependent quantity log P ,v (approximation of theTW entropy), based on color and intensity. The color indicates the velocity v ∈ V ∪ − V where V = { , , , , , , , } , 0 (resp. 1) is painted in gray (resp. black) and the other velocity colors monotonicallyrange from yellow to red. The intensity is proportional to the entropy, from 0 to log 2. Positive entropydomain of intermediate velocities v ∈ (0 ,
1) appear to be pair-wise disjoints. However, the domains withsmall velocities overlap with the one at v = 0 (and similarly in the right part of the pictures) indicating thata chaos of fixed points (resp. of TW with v = 1) may coexist with a chaos of patterns with intermediatevelocity.By using symbolic dynamics (see below), we numerically discovered [13, 14] that, given any values of theparameters, there is at most one (non-trivial) velocity v ∈ (0 ,
1) for which the corresponding TW set mayhave positive entropy, see Fig. 1 for an illustration for a = 0 .
6. Depending on the parameters, either there isno chaos of TW, or we have a chaos of TW with trivial velocity 0 or 1, or such chaos is combined with a chaosof TW with v ∈ (0 , the numerical results called for rigorous explanation. That a chaosof TW with v ∈ (0 ,
1) could coexist with one at v ∈ { , } , combined with proofs of parameter dependentcoexistence of TW with different velocities [13], suggested that the proof could not be elementary. In fact,excepted for the extreme velocities v = 0 and v = 1 for which the exact parameter domain for existence offull chaos could be determined [17], for arbitrary velocities, only estimates of pairwise disjoint domains ofexistence of chaotic sets of TW were available, based on shadowing arguments. In order to provide furtherinsights into analytic arguments, we need to introduce considerations about symbolic dynamics.A convenient aspect of the piecewise affine system (1) with individual map (3) is that its symbolic descriptionis easily accessible. Indeed, any trajectory { x t } (where x t +1 = F (cid:15) ( x t )) can be coded by a space-time symbolicsequence { θ ts } via the relation θ ts = H ( x ts − T ) ∈ { , } . Conversely, and more importantly, symbolic sequencesuniquely determine trajectories in the attractor of the LDS. Indeed, solving the iterations associated withtrajectories whose components exist and are bounded for all t ∈ Z yields the following expression [9, 13] x ts = (1 − a ) ∞ (cid:88) k =1 a k − k (cid:88) n =0 (cid:96) n,k θ t − ks − n (4) In symbolic systems, the topological entropy can be defined as the exponential growth rate of the number of admissibleblocks [15]. For numerical purposes, following [16], we used instead the number of periodic points and assumed that the entropyof the set of TW profiles with velocity v is given by lim sup L → + ∞ log P L,v L where P L,v is the number of L -periodic blocks ofvelocity v . The symmetric features of Fig. 1 are consequence of the following symmetries of the LDS: • If { x ts } is a trajectory for T (with all x ts (cid:54) = T ), then { − x ts } is a trajectory for 1 − T . • If F q(cid:15) ( x ) = σ p ( x ) then F q − (cid:15) ◦ R ( x ) = σ q − p ◦ R ( x ) where ( R ( x )) s = x − s for all s . (cid:96) n,k = (cid:0) kn (cid:1) (1 − (cid:15) ) k − n (cid:15) n ≥ k (cid:88) n =0 (cid:96) n,k = 1 , ∀ k ∈ N Not only the LDS attractor can be fully specified using symbolic dynamics, but so do the topologicalproperties of the dynamics in this set, when symbolic sequences are endowed with a suitable topology. Inparticular chaos of TW can be analyzed and quantified in the symbolic context.This topological equivalence between the dynamics in the original space and its symbolic representation isstandard in the theory of dynamical systems. However, what is specific to the current setting is the followingiterative process for symbolic codes θ t +1 s = H (cid:32) (1 − a ) ∞ (cid:88) k =0 a k k +1 (cid:88) n =0 (cid:96) n,k +1 θ t − ks − n − T (cid:33) , ∀ s, t ∈ Z (5)obtained by stipulating that the code of any trajectory given by (4) must coincide with the input symbolicsequence. In other words, the LDS attractor and its topological properties can be captured by an explicititeration scheme for symbolic codes. This formulation has proved useful in several cases [9, 17, 13], forinstance to obtain the estimates mentioned above about chaos of TW. A full mathematical proof of the numerical results of [13] remains elusive, especially because the iterations(5) involve the entire past sequence ( ie. the series in k is infinite when a >
0) and this makes it virtuallyimpossible to determine all solutions for arbitrary values of the parameters. Here, we suggest to investigateinstead the following finite rank iteration schemes, obtained by truncating the series in the expression (4)(and using a suitable normalisation) θ t +1 = F R ( θ t − R +1 , · · · , θ t ) where (cid:0) F R ( θ − R +1 , · · · , θ ) (cid:1) s = H (cid:32) − a − a R R − (cid:88) k =0 a k k +1 (cid:88) n =0 (cid:96) n,k +1 θ − ks − n − T (cid:33) . The iteration scheme F R can be viewed as a kind of cellular automaton (CA), which, thanks to the updatednormalisation, actually shows the same symmetries as the original LDS. Our goal is to systematically de-termine the existence domains of TW for increasing values of R . Exponential decay of the coefficients a k suggests that the F R should be good approximations of the LDS; the larger R , the better the approximation.The dependence on a indicates that the approximation should be better for smaller values of this parameter.Since F R only involves input symbols in a time window of length R , the velocity of the TW F qR ( θ ) = σ p ( θ )must be of the form pq for q ∈ { , · · · , R } and p ∈ { , · · · , q } (and assuming wlog that p and q are co-prime).Similarly, the inputs are limited to the R + 1 neighbour sites, so the TW profile { θ s } , when read from leftto right, can be regarded as being generated by a topological Markov chains (subshift of finite type); givenan arbitrary site s and an admissible ( R + 1)-block θ s − R +1 · · · θ s , the subshift specifies those subsequentsadmissible blocks θ s − R +2 · · · θ s +1 at the next site. This viewpoint is convenient for the computation of theTW entropy, as the logarithm of the largest eigenvalue of the corresponding transition matrix (see [15] formore details on finite-type subshift and the computation of their entropy). The simplest approximation is when the summation on k only has one term ( R = 1) so that the iterativeprocess on symbolic sequences reduces to one for elements of { , } Z , viz. we have the following (genuine)CA ( F ( θ )) s = H ((1 − (cid:15) ) θ s + (cid:15)θ s − − T )4 T Figure 2: Entropy-velocity diagrams of TW of the cellular automaton F . The square of parameters ( (cid:15), T )decomposes into 4 regions. In the left (blue) triangle, every element θ ∈ { , } Z is invariant under thedynamics (full chaos of fixed points). In the lower white triangle, the only non-homogeneous TW profiles are { H ( s ) } mod σ ( v = 0) and { H ( − s ) } mod σ ( v = 1). The TW in the other domains follow from symmetries.The TW velocities of F are v ∈ { , } are trivial. By the symmetry v ↔ − v , it suffices to study the fixedpoints ( v = 0). To that goal, let X ( θ θ ) = (1 − (cid:15) ) θ + (cid:15)θ . It is simple to check that • In the interval 0 < T ≤ min { X (10) , X (01) } = min { (cid:15), − (cid:15) } , the only possible fixed points arethe homogeneous ones 0 Z and 1 Z , and { H ( s ) } mod σ . Indeed, we then have H ( X (01) − T ) = H ( X (10) − T ) = 1. Furthermore, using that X (01) = 1 − X (10), a symmetric conclusion holds for T ∈ ( X (01) , • In the interval (cid:15) = X (10) < T ≤ X (01) = 1 − (cid:15) , (which is non-empty obviously iff (cid:15) < ) everyelement of { , } Z is a fixed point of F (full chaos of fixed points, with entropy equal to log 2).The entropy-velocity diagram of TW for F is given in Fig. 2. The situation is pretty obvious in this caseas the two domains of existence of chaos of TW do not overlap. We now turn to the analysis of waves for F . The only possible TW velocities in this case are v ∈ { , , } .Let X (cid:18) θ θ θ θ θ (cid:19) = 11 + a (cid:88) k =0 a k k +1 (cid:88) n =0 (cid:96) n,k +1 θ kn , (where sub- and super-scripts are denoted with non-negative integers for simplicity) which obviously dependson a and (cid:15) . In order to determine the existence of fixed points ( v = 0), one needs to order the values of X for θ = θ .Using the symmetry X (1 − · ) = 1 − X ( · ), it suffices to consider those θ = θ with θ = θ = 0. Using forsimplicity, only the top row to identify the 3-blocks in this case, one can show that the corresponding valuescomply with the lexicographic order, ie. we have0 = X (000) < X (100) < X (010) < X (110) . All the LDS here are translation invariant; hence all their trajectories come as equivalence classes { x t } mod σ or { θ t } mod σ . T Figure 3: Entropy-velocity diagram of TW of the cellular automaton F (and zoom into the central region).Chaos of fixed points is full in the central blue region (all 3-blocks are admissible and entropy = log 2) andpartial in the two other blue domains (either 110 or 001 is forbidden, depending on the domain/entropy =log √ in both cases). In the blacks domains, the same comments apply to v = 1 TW. Orange domainscorrespond to chaos of TW with v = . In particular, in low intensity domains (left and right), both 010and 101 are forbidden (entropy = log √ ). In each of the high intensity domains (bottom and top), onlyone of these words is forbidden (entropy = log 1 . θ ∈ { , } Z is a fixed point of F iff the following constraint holds on its 3-blocks { θ s − θ s − θ s } s ∈ Z H ( X ( θ s − θ s − θ s ) − T ) = θ s , ∀ s ∈ Z . As mentioned above, this constraint is regarded as defining a subshift over 3-blocks, which depends onparameters, see Fig. 4 for the graphs obtained from the analysis to follow.
000 111 001 100 010 011 110 101
000 111 001 100 010 011 110 101
000 111 001 100 010 011 110 101
Figure 4: Graphs associated with the (nested) 3-blocks subshifts that generate the (parameter dependent)fixed point sets of the cellular automaton F . Left.
DeBruijn graph B (2 ,
3) (see Wikipedia or [18]) associatedwith the full shift.
Center.
All 3-blocks but 110 are allowed.
Right.
The four blocks 110, 010, 101 and100 are forbidden. Light red color corresponds to blocks/transitions that are forbidden. Dark red colorcorresponds to transient blocks/transitions that do not contribute to the subshift entropy.By considering the relative position of T with respect to the ordered values of X above, the following claimsresult about the existence of fixed point subshifts. Needless to say that similar conclusions immediatelyfollow for the TW with v = 1, from the symmetry (cid:15) ↔ − (cid:15) . • The full shift is admissible iff X (110) < T ≤ − X (110). • The subshift for which all 3-blocks but 110 are allowed is admissible iff X (010) < T ≤ min { X (110) , X (001) } . This means that every element in this set if a fixed point. To see this, note that we have H ( X ( θ θ , θ − T ) = 0 except when θ = θ = 1 and H ( X ( θ θ , θ − T ) = 1 forevery θ , θ ∈ { , } .
6y symmetry T ↔ − T , the subshift for which all 3-blocks but 011 are allowed is admissible iff 1 − T satisfies the same condition. • There are no other possibly admissible subshifts of positive entropy. In particular, the only non-homogeneous fixed points which exist for 0 < T ≤ min { X (010) , X (001) } , write { H ( s ) } mod σ . By symmetry T ↔ − T , the only non-trivial fixed points for max { X (001) , X (110) } < T ≤ { − H ( s ) } mod σ .As before, the intervals in either of the first two items are non-empty iff (cid:15) is not too large (and smaller thansome threshold smaller than that depends on a ). Moreover, they are adjacent when all non-empty. Theintervals in the third claim are never empty, see Fig. 3.As entropy is concerned, it is obviously equal to log 2 in the first case. To compute it in the second case(and also for any other subshift in the sequel), we consider that this quantity is not affected by passing tonon-wandering sets, so that the transitions painted in red in the central picture of Fig. 4 can be ignored.The remaining subshift turns out to be identical to the golden shift , Fig. 6 (a), whose entropy can be easilycomputed as the logarithm of the largest eigenvalue √ (cid:39) .
618 of the matrix (cid:18) (cid:19) . v = A configuration θ ∈ { , } Z of a TW with velocity v = is defined by θ = F ( σ − ( θ ) , θ ) (or equivalentlyby σ ( θ ) = F ( θ, σ ( θ ))). In this case, the analysis of admissible 3-blocks then amounts to consider bothquantities X (cid:18) θ θ θ θ θ (cid:19) and X (cid:18) θ θ θ θ θ (cid:19) and to obtain conditions so that its output is equal to θ in bothcases. The results are given below and the corresponding transition graphs are presented in Fig. 5.
000 111 001 100 010 011 110 101
000 111 001 100 010 011 110 101
Figure 5: Graphs associated with subshifts that generate TW of velocity of the CA F . Left.
The block101 is the only forbidden one.
Right.
Both 101 and 010 are forbidden. Same color codes as in Fig. 4. • The subshift for which all 3-blocks but 101 are allowed (Fig. 5 left) is admissible iff max (cid:26) X (001) , X (cid:18) (cid:19)(cid:27) < T ≤ min (cid:26) X (010) , X (cid:18) (cid:19)(cid:27) . By the symmetry T ↔ − T , the subshit in which no 3-block is equal to 010 is admissible iff 1 − T satisfies the same condition (which amounts to reflect the interval wrt to ). Indeed, we then have H (cid:18)(cid:18) θ (cid:19) − T (cid:19) = 1 for θ ∈ { , } . Alternatively, one first observes that the new constraint T ≤ X (010) immediately implies that 010 is forbidden. The transition graph on Fig. 4 then implies that the block 101, andthen also 100, cannot be accessed either. The only non-trivial admissible path is the one that joins 000 to 111. Notice also thatthese fixed points exist in the larger interval 0 < T ≤ min { X (110) , X (001) } . Indeed, we then have H (cid:18) X (cid:18) θ θ θ θ θ (cid:19) − T (cid:19) = H (cid:18) X (cid:18) θ θ θ θ θ (cid:19) − T (cid:19) = θ for all θ θ θ (cid:54) = 101. Conversely, that theblock 101 is not admissible is obvious from the condition on T . (NB: When the space-time symbol block corresponds to a fixedpoint, we use the simplified notation.) The subshift for which all 3-blocks but 101 and 010 are allowed (Fig. 5 right) is admissible iffmax (cid:26) X (001) , X (cid:18) (cid:19)(cid:27) < T ≤ min (cid:26) X (110) , X (cid:18) (cid:19)(cid:27) . • There are no other possibly admissible subshifts of positive entropy. In particular, no non-homogeneousTW with v = exists when the previous condition fails.As before, the intervals for T depend on (cid:15) , but this time they are non-empty only when (cid:15) is close enough to (again depending on a ), and complement existence domains of chaos of fixed points and TW with v = 1,see Fig. 3. Moreover, the interval in the second item contains the two ones in the first item. These twointervals never intersect, which means that it is impossible to have full chaos of TW with v = .As entropy is concerned, using a similar non-wandering argument as above, we obtain that the entropy ofthe subshift(s) in the first item is obtained from the largest eigenvalue λ ∼ .
755 of the transition matrix associated with the graph in Fig. 6 (b). The entropy of the subshift in the second item isequal to that of the one depicted in Fig. 6 (c), which turns out to be equal to that of the golden shiftlog √ ( < log 1 .
0 1 (a) 0 1 (b) 0 0 1 (c) 1 0 0 1 (d) 1 0 1 (f) 0 0 0 1 (e) 0
Figure 6: Transitions graphs of some simple subshifts, which capture the entropy of TW profile subshifts inthe CA F R , see text more details. In order to obtain further refinement of the entropy-velocity diagram, the next logical step is to consider therank 3 approximation F . The analysis is similar to as before and we only provide the results here, which areillustrated by the entropy-velocity diagram in Fig. 7. To that goal, we shall need to investigate the valuesof the function X θ θ θ θ θ θ θ θ θ = 1 − a − a (cid:88) k =0 a k k +1 (cid:88) n =0 (cid:96) n,k +1 θ kn , Fixed points.
As before, one can check that the values of this function, for configurations θ = θ = θ that are associated with fixed points, are ordered according the lexicographic order. We have for (cid:15) ∈ (0 , ),and using similar notation as for rank-2 fixed points, X (1000) < X (0100) < X (1100) < X (0010) < X (1010) < X (0110) < X (1110) . The admissible 4-blocks subshifts depend on the relative location of the threshold T with respect to thisordering. The possibly admissible subshifts are those listed in Table 1 - there are no other possibly admissiblesubshifts of positive entropy, excepted, evidently those obtained by the exchanging 0 and 1 - and illustratedin Fig. 8.As for the rank-2 fixed points, the intervals in T are only non-empty when (cid:15) is not too large. When non-empty, these intervals are adjacent to each other. The subshift are nested, with decreasing entropy, when T moves away from . Indeed, we have min (cid:26) X (cid:18) (cid:19) , X (cid:18) (cid:19)(cid:27) < min (cid:26) X (cid:18) (cid:19) , X (cid:18) (cid:19)(cid:27) = 1 − max (cid:26) X (cid:18) (cid:19) , X (cid:18) (cid:19)(cid:27) . because min (cid:26) X (cid:18) (cid:19) , X (cid:18) (cid:19)(cid:27) ≤ T Figure 7: Entropy-velocity diagram of TW of the cellular automaton F (and zoom into the central region).As before, chaos of fixed points is full in the central blue region and the successive lower blue domainscorrespond to successive subshifts in Table 1/Fig. 8. The central orange domains is where we have chaos ofTW with v = (Table 2 and Fig. 9). The intermediate yellow and red domains respectively correspond to v = and v = , see Table 3 and Fig. 9 for a description of these domains, together with the correspondingsubshifts.Forbidden 4-blocks Graphs Entropy Existence conditionNone Fig. 8 (a) log 2 X (1110) < T ≤ X (0001) = 1 − X (1110)1110 Fig. 8 (b)/6 (d) log 1 . X (0110) < T ≤ min { X (1110) , X (0001) } + 0110, 1100, 1101 Fig. 8 (c)/6 (a) log √ X (1010) < T ≤ min { X (0110) , X (0001) } + 1010 Fig. 8 (d)/6 (e) log 1 . X (0010) < T ≤ min { X (1010) , X (0001) } all non-homogenous Fig. 8 (e) 0 0 < T ≤ min { X (0010) , X (0001) } Table 1: Summary of possibly admissible (nested) subshifts of fixed points in the CA F , together withreferences to the graphs in Fig. 8, to their simplification in Fig. 6 (where it applies), to entropy estimatesand to the intervals of the parameter T where they are admissible. (NB: The constraint in the first columnmeans that words that are not forbidden are all admissible. The symbol + means in addition to forbiddenblocks in the previous row .) TW with velocity v = . For v = TW, one needs to check that we simultaneously have X θ θ θ θ θ θ θ θ θ = θ and X θ θ θ θ θ θ θ θ θ = θ , for every 4-block in the presumed subshifts. Candidate subshifts can be obtained from the 4-block graphassociated with the full shift on 2 symbols, de Bruijn graph B (2 , evidently, those obtained by the exchanging 0 and 1 - and illustrated in Fig. 9.As for the rank-2 TW v = , the intervals in T are non-empty when (cid:15) is close to . However, when bothnon-empty, the intervals in the first and second rows overlap and enhancement of the entropy results wheneither T or 1 − T lies in their intersection (Notice also, that the interval in the first row never intersects itsreflected image wrt ). TW with velocity v = . In addition to fixed points and v = TW, the rank 3 approximation may alsohave v = (and v = ) TW. We focus on v = since the other one can be deduced from the symmetry In particular, the analysis shows that no subshift can be admissible without 1001 and 0110 being admissible. Similarly, thenumerical computations of the existence conditions show that no admissible subshift can allow for 0101 or 1010.
000 1111 1001 0110 0001 1000 1110 0111 0011 1100 1010 0101 1011 0010 0100 1101 (a) (b) (c) (d) (e)
Figure 8: Graphs associated with the (nested) 4-blocks subshifts in Table 1. Details as in Fig. 4.Graphs Entropy Existence conditionFig. 9 (b)/6 (e) log 1 .
446 max X , X < T ≤ min X , X Fig. 9 (d)/6 (c) log √ max X , X < T ≤ min X , X Fig. 9 (e) log 1 .
674 intersection of previous conditionsTable 2: Summary of possibly admissible subshifts of TW v = in the CA F , together with references tothe graphs in Fig. 9, to their simplification in Fig. 6 (where it applies), to entropy estimates and to existencedomains, expressed in terms of intervals for the parameter T . In particular, the intervals in the middle roware symmetric wrt . (cid:15) ↔ − (cid:15) . In this case, a 4-block θ θ θ θ is admissible iff the following three conditions simultaneously hold X θ θ θ θ θ θ θ θ θ = X θ θ θ θ θ θ θ θ θ = X θ θ θ θ θ θ θ θ θ = θ , As before, candidate subshifts can be obtained by pruning the de Bruijn graph B (2 , ↔
1, the resulting possibly admissible subshifts are listed in Table 3 (again, no other subshift of positiveentropy can be admissible), where the existence conditions have been simplified by assuming (cid:15) ≤ . Thecorresponding existence domains do not intersect those of previous velocities, except for some boundaries(complementary domains). The corresponding intervals in T are only non-empty when (cid:15) lies in a intermediaterange, between existence domains of velocities 0 and . The subshifts are similar to those associated with v = TW (NB: in particular, no admissible subshift can allow for 0101 or 1010); however, there are twosignificant differences • The admissibility of the blocks 1001 and 0110 is not a consequence of that of other blocks; hence thereare additional (nested) existence domains (see 2nd and 4th rows in Table 3).10
000 1111 1001 0110 0001 1000 1110 0111 0011 1100 1010 0101 1011 0010 0100 1101 (a) (b) (c) (d) (e)
Figure 9: Graphs associated with the subshifts in Table 2 and 3. • The existence domains do not overlap as they did; so that there no enhancement effect on the entropyexists in this case.Graphs Entropy Existence conditionFig. 9 (a)/6 (f) log 1 .
380 max X (0001) , X < T ≤ min X (0010) , X Fig. 9 (b)/6 (e) log 1 .
446 max X (1001) , X < T ≤ min X (0010) , X Fig. 9 (c) log 1 .
466 max X (0001) , X < T ≤ min X (1110) , X Fig. 9 (d)/6 (c) log √ max X (1001) , X < T ≤ min X (0110) , X Table 3: Summary of possibly admissible subshifts of TW v = in the CA F , together with referencesto the graphs in Fig. 9, their simplification in Fig. 6 (where it applies), entropy estimates and existencedomains. The intervals in the last two rows are symmetric wrt to . The interval in the 2nd (resp. 4th)row is contained in the one of the 1st (resp. 3rd) row. The entropy of the subshift in rows 2 and 3 areunexpectedly equal. The analysis above has revealed that chaos of TW is selective in basic finite rank approximations F R ofthe LDS F (cid:15) , with unique velocity depending on parameters. More than in F (cid:15) , in the F R , this uniquenessapplies to all velocities and not only to v ∈ (0 , F , some boundaries curves in the ( (cid:15), T ) square, of the11omains associated with v = , coincide with some boundaries associated with v = 0 (Fig. 7) (and a similarcoincidence holds for v = and v = 0 in Fig. 3). This suggests that some overlap between the domains for v = 0 and v = R might exist for R large enough.Moreover, even though this analysis is too preliminary to anticipate full results for larger values of R , somesystematic features have emerged. In particular, the fact that for any R , the following extreme subshifts canbe certainly be admissible, depending on the parameters: • B R : every block of consecutive 0’s (or consecutive 1’s) must be of length R or longer; a generalisationof the graphs in Fig. 5 right and Fig. 9 (c). • A R : every block of consecutive 0’s must be of length R or larger, and every 1 must be isolated (or thesubshift resulting from exchanging 0’s and 1’s); a generalisation of the graphs in Fig. 9 (a).Sufficient existence conditions for these subshifts have been provided in [13], in term of existence regions offronts and solitary waves respectively. Namely, domains in the ( (cid:15), T ) square have been given, so that thesubshift B R (resp. A R ) of TW with v = pq given and R sufficiently large, is admissible. However, to computeexistence conditions of A R and B R for any R , is not totally obvious, even though there are some similaritiesbetween the case R = 2 and R = 3. Those computations could be part of a continuation to this paper. Acknowledgements.
I am grateful to Stanislav M. Mintchev for careful reading of the manuscript, com-ments and suggestions.
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