Controlled pathways and sequential information processing in serially coupled mechanical hysterons

Significance Driven frustrated materials evolve via sequences of transitions between metastable states, associated with the flipping of one or more material bits called hysterons. Interactions between hysterons are key for obtaining and understanding complex pathways, yet their control has remained elusive. Here, we uncover a general geometric mechanism that yields controllable hysteron interactions. We characterize the resulting pathways and leverage our insights to fabricate metamaterials with targeted pathways that materialize counting, pattern generation, and pattern recognition. Our results open a route toward material-based sequential computing.


Supporting Information Text
Supplementary videos.Supplementary videos S1-S8.Front view of a three-hysteron metamaterial described by the graph 3A12 during stretch-release cycles.The traction speed is 0.05 mm/s, the movies were recorded at a frame rate of 0.41 Hz and accelerated 40 times.Model of hysterons in series.We consider n serially coupled hysterons under quasistatic, overdamped dynamics.The forcedisplacement relation of hysterons is modeled by a bilinear relation: fj = (uj − djsj) , [1] with fj the force, uj the displacement, dj the force discontinuity and sj the state.Sweeps of the global deformation U yields smooth episodes interspersed with discontinuous transitions.In each smooth episodes, {sj} is constant, and mechanical equilibrium yields relations between the global and local deformations: [2] F = fi = fj for all i, j, [3] The force balance equations can be used to link the extensions of pairs of hysterons as uj = (ui − disi) + djsj.By systematically eliminating all uj for j ̸ = i, we can then express, for fixed sj, the global deformation U as a function of any single ui, and vice-versa, Transitions.Consider sweeping U so that hysteron i flips its internal state; due to the mechanical balance equations, the force and the deformations of all hysterons change.To calculate these changes, we note that just before this transition ui approaches one of the local switching fields u ± i , and that the global driving U just before and just after a transition are equal.Using Eq. ( 4), we find: Denoting the states and deformations after the flipping event as {s ′ } and {u ′ j }, making use of the fact that s ′ j = sj if j ̸ = i, and using Eq. ( 5), we find the following explicit expressions for the discontinuous changes in the hysteron deformations and forces: [8] all hysterons k: We note that these expressions take the correct form for n = 1, that all forces jump by the same amount as required by force balance, that a 0 → 1 (1 → 0 transition) lowers (increases) the force, and that the sum of all deformations adds up to zero as required by length conservation (Eq.4).We note that as the relation between U and ui depends on the state of all hysterons (Eq.4,5), the expressions for the changes in ui are cumbersome.
Mapping serial hysterons to explicit model.To map the system of n serially coupled hysterons (Eqs.1-3) to an explicit model of interacting hysterons (1) , we focus on mapping the expression for U c i for serial hysterons (Eq.6) to the expression for V ± i for interacting hysterons: Importantly, the explicit model allows to gauge out all diagonal terms in cij ('self-coupling') by modifying the lower bare switching fields v − i , and also allows for a rescaling of cij, provided the gaps between the upper and lower bare switching fields are scaled accordingly (1).Hence, there is no unique mapping from the serial to the explicit model; for completeness we give three mappings here.In all cases, we only require that v This mapping simply defines

of 10
Note that the upper bare switching field can also be simplified as v + i = nu + i , as si = 0 for an up transition of hysteron i.
Substituting these definitions in Eq. ( 10) yields the desired form Notice that in this mapping, cii ̸ = 0 and the bare upper and lower switching fields scale with n.

Mapping II:
To gauge out the self-coupling term, we define For the upper global switching field we find U + i (S) = nu + i + Σ j̸ =i djsj.As si = 0, we can rewrite the first term as n(u and the second term as Σjdjsj, thus obtaining the desired form Eq. ( 6) for upper global switching field U + i (S).For the lower global switching field we find As si = 1 for a down transition, we can rewrite the last two terms as Σjdjsj, thus obtaining the desired form Eq. ( 6) for lower global switching field U − i (S).

Mapping III:
To gauge out the self coupling term and to have the bare switching fields of both models be similar, we define which by substitution yields the desired form Eq. (6).In this mapping, instead of the extensive quantity 'total deformation', we use the intensive strain-like quantity Vi, which is the deformation per hysteron.The upper bare switching fields in serial and explicit model are equal and system size independent.Our mapping also clarifies that the interactions come from a global coupling, and vanish in the limit of large n.Finally, the lower switching field v − i is a bit more complex.It contains both a system size independent shift (−di) and a system size dependent shift (+di/n) with respect to the bare switching fields u − iplease note that the size dependent shift is due to the convention that cii = 0.
T-graph construction.An important restriction for our t-graphs is that we require each elementary transition to only have one unstable hysteron (1).However, for some parameter values this requirement is violated.Consider hysteron i in state S becoming unstable at a certain driving value U c .This defines then a landing state S ′ .Due to interactions, it can not be ruled out that hysterons are unstable in state S ′ at U = U c .If one hysteron is unstable, this leads to an avalanche.However, if more than one hysteron is unstable (say hysteron l and k), the situation is more complex and features a race condition: flipping first hysteron l and then k may lead to a different scenario than first flipping k and then l -hysteron flips do not commute.
Alternatively, one may flip both hysterons simultaneously, which may lead to a different situation again.More work is needed to understand the physical reality of such situations, and whether a simple coupled hysteron model can consistently describe such situations.Hence, while these type of transitions may enlarge the range of t-graphs, in our studies here we have excluded these transitions and label the system then ill-defined (1).
Avalanches.In general, avalanches refer to transitions involving multiple hysteron flips; they are referred to as 'up' or 'down' depending on whether triggering them requires the increase or decrease of the global driving, and their length is defined by the number of elementary transitions separating the initial and final states (1).As an example of an avalanche, we consider graphs Scrambling.A scrambled t-graph features pairs of transitions that are not consistent with a global ordering of the switching fields.As an example of state dependent ordering of the switching fields, consider the pair of transitions {000} → {001} and {100} → {110}.The presence of the first transition indicates that in state {000} the up switching field for the third hysteron is lower than the up switching fields of the other hysterons, i.e., u + 3 (000) < u + 2 (000) and u + 3 (000) < u + 1 (000); while the second transition indicates that in state {100}, the up switching field of the second hysteron is lower than the up switching field of the third hysteron, i.e., u + 3 (100) > u + 2 (100).Such pairs of scrambled transitions cannot be observed in the absence of interactions, as in that case the upswitching fields are state independent (u + j (S) = u + j (000)), but are an important contributor to the wide variety of t-graphs observed for general interactions.We note that due to the specific nature of their coupling, scrambling is not possible in systems of serial hysterons (see section below).
General properties of simple SC-graphs.To discuss general features of the t-graphs for serially coupled hysterons, we use the mapping to the explicit model of interacting hysterons.While coupled hysterons with arbitrary interactions feature a very wide palette of t-graphs (1, 2), for the specific case of serially coupled hysterons the t-graphs are restricted.
First, simple SC-graphs do not feature scrambling, so that the sequence of hysteron switches on the main loop completely determines the elementary transitions in the rest of the graph.This follows directly from the definition of the global antiferromagnetic interactions cij = −dj.For a given state S, the effective switching fields can be written as u ± i + djsj (using the mapping to the explicit model).Hence, the shift of the switching fields for a given state is the same for each hysteron i, the ordering of U ± i and u ± i are the same, and the interactions do not impact the switching hysteron.Hence, a simple SC-graph without avalanches is a P-graph.
Second, all avalanches are of length two, are obtained by merging an up and a down transition and thus connect two states with the same magnetization m := Σisi, and for a given monotonic sweep of U , no two avalanches can occur in succession.To understand this, we note that avalanches must be composed of alternating up and down (elementary) transitions when the interactions are purely antiferromagnetic (1,4); hence during such an avalanche the magnetization M (S) must alternatingly increase and decrease by one.Moreover, all simple sc-graphs can be constructed by starting from avalanche-free and scrambling-free t-graphs-they are precisely the Preisach graphs-and then merging pairs of elementary transitions to obtain length two avalanches.The Preisach graphs satisfy l-RPM, which restrains the structure of transitions within a loop.
Suppose now that one has a series of elementary transitions S (0) → S (1) → S (2) → S (3) which form an avalanche.Let the first transition (S (0) → S (1) ) be an up transition (the same argument can be constructed if it is a down transition).Then, the second (S (1) → S (2) ) must be a down transition, so that S (1) must be the endpoint of a loop of which S (0) and S (2) lie on the up and down boundaries, respectively.Due to l-RPM, the up transition S (2) → S (3) cannot lead to any other state than S (1) .To see this, consider the up-orbit from S (2) , which under l-RPM must go through the endpoint S (1) .However, since S (2) → S (3) is an up transition, M (S (3) ) = M (S (1) ) and subsequent up transitions can never lead back to the state S (1) .Thus, the up transition S (2) → S (3) is only allowed under l-RPM if S (3) = S (1) .An avalanche cannot have any repeating nodes, as this would lead to a self-loop.Therefore, one cannot have avalanches of l = 3 and higher when there are only anti-ferromagnetic interactions and the base graphs obey l-RPM.
The argument above can be extended to show that no up avalanche can start from the final state of another up avalanche (similar for down avalanches).To see this, we consider an up avalanche composed from the elementary transitions S (0) up −→ S (1) down −−−→ S (2) .As we showed before, S (2) up −→ S (1) , and since S (1) down −−−→ S (2) the chain of states is (2) .The first two transitions can merge under an avalanche-S (0) up,ava −−−−→ S (2) , but the latter two transitions would form a loop, which is forbidden Return Point Memory and l-RPM.Return point memory (RPM) is the widespread ability of complex systems to 'remember their extremal driving', i.e., to return to a previously visited state when the driving revisits an extremum value (5)(6)(7)(8).Antiferromagnetic interactions can break RPM (4).The presence of strict RPM cannot be determined from the topology of a t-graph as it may depend on the precise values of the switching fields (9).We therefore use a recent definition of so-called loop-RPM (l-RPM) which can be determined from the topology of the t-graph (10,11).l-RPM requires that all loops within a t-graph are absorbing.In essence, a loop is defined by a pair of nodes Sm and SM , where the system evolves from Sm to SM and vice-versa by a series of up and down transitions.The intermediate states are defined as the up (down) boundaries of this loop.
l-RPM then requires states Sm (SM ) to be accessible by a sequence of down (up) transitions from any up (down) boundary state.Informally, l-RPM requires that any orbit starting from a state in a given loop escapes this loop by going through either the top or bottom state.We finally note that broken l-RPM implies broken RPM, that breaking l-RPM necessitates either scrambling or avalanches, and that t-graphs with broken l-RPM allow for more complex behavior than t-graphs that satisfy l-RPM (1).
Full hierarchy of t-graphs.We present in Fig. S1 the 44 simple sc-graphs found for n = 3 serially coupled hysterons hierarchically organized following the rules exposed above and in the main text.Our numerical sampling is exhaustive and we obtain precisely 44 simple SC-graphs, with six P-graphs (A − F ), and respectively 12, 11, 8, 4, 2 and 1 simple sc-graphs with 1, 2, 3, 4, 5 and 6 avalanches.As explained above, all avalanches connect two states with the same magnetization m := Σisi and are of length two.
To understand how these 44 simple sc-graphs emerge, we organize them hierarchically.We start from the six Preisach graphs (3A-3F ), and consider all possible length-two avalanches obtained by merging transition pairs.Collecting all possible avalanches, we find that for for m = 2, all six potential avalanches that satisfy these conditions occur.For m = 1, only three of these occur, with three others forbidden due to the ordering of the upper switching fields.Specifically, {100} we can determine the potential avalanches for each 'base' graph (3A-3F ) and obtain the remaining 38 simple sc-graphs by combining these, leading to a natural hierarchy (Fig. S1).
Thus combining avalanches, we find a total of 45 simple sc-graphs.However, there is no guarantee that all combinations are realizable, and indeed we find one example of a putative graph-3B23-that cannot be realized.We now ask why this graph is not realizable.A graph is realizable if the underlying set of design inequalities relating the switching fields has a solution.We reason that all four transitions up −→ {001} must be involved in the combination of inequalities that is incompatible, since more than one of the candidate graphs would otherwise be impossible.
To know whether a set of inequalities is solvable, one can use the Fourier-Motzkin method for elimination of variables.In the Fourier-Motzkin method, a variable x is eliminated by combining the inequalities to create a new set that does not contain x.
A set of inequalities does not have a solution if, after all variables are eliminated, a contradictory statement remains such as 0 > 0. By working backwards, we can identify which combinations of inequalities lead to a contradictory statement and are therefore incompatible.
We find that for the putative graph 3B23, there is a single combination of six inequalities that is incompatible, namely: We further note that the six inequalities arise from the four transitions{001} Experimental realizations.The geometrical parameters of the different elements constituting a hysteron play a crucial and complex role on its mechanical response, and thus on the subsequent pathways.We used our numerical simulations to investigate the mechanical response of a single hysteron.The stiffness of the spring governs the hysteretic behavior of the beam, and we find that for a given beam design, the spring has to be soft enough for the system to be bistable (Fig. S2b).The stiffness of the spring, however, does not influence the force threshold.Fig. S2c illustrates how the force threshold, as defined in Fig. S2b evolves with the geometry of the pre-curved beam.
We where L = 20 mm, b = 1 mm.We provide in Fig. S3 the t-graph for sample A with all the switching fields.
Fig. S4 presents the full experimental response of sample A subjected to driving protocol 'yxxxyyxy' described in the main text (see Fig. 6 of the main text).Each state transition is denoted by a vertical grey line, and the state reached is indicated above the plot.
Transients.A popular characterization of the transient is the number of driving cycles τ before the system falls into a periodic orbit of period T (1,9).To connect the transient to i1-the index of the first repeating state-we define t1 := i1/2; note that t1 can take on half-integer values, and τ can be obtained by rounding up t1 to the nearest integer.Similarly, rounding up (i2 − i1)/2 yields the integer period T .For collections of n = 3 non-interacting hysterons, we found that (i1, i2) can only take on four values (0, 1), (0, 2), (1, 2) and (1, 3), which follows from the properties of strict RPM, and this remains true in the limit of large n.In contrast, sample A can feature longer transients: (i1, i2) can take on values (2, 3), (2,4), (3,4)   In each daughter graph, we use a colored arrow to indicate the avalanches that replace the direct transitions in the mother graphs -the remaining direct transitions are as in the mother graph and represented in grey and without arrowheads.3A has four groups of avalanches (1-4) that occur in any combination.Note that the individual avalanches in group 3 and 4 cannot occur independently.3B has four avalanches (1-4) that can occur in almost any combination -with the exception of t-graph 3B23 which cannot be realized with serial coupling.Notice that avalanches 2 and 3 can occur together when either avalanche 1 or 4 is present.3C, 3D and 3E all allow two avalanches that can occur together.3F has no daughter graphs, because all its states have different values of m.
Grey lines indicate daugther relations in graphs 3C, 3D and 3E and in the first two levels of 3A and 3B.
(Fig. S5).We note that we can use the plots of i1 and i2 to calculate statistical measures.For example, to calculate the probability of finding a certain transient (e.g., i2 = 2) for randomly sampled values of U m and U M , one can add the volumes in U p , U q space for each matching block (Fig. S5).

Determining classes of FSMs.
To determine all FSMs for a given t-graph and alphabet we use a simple recursive algorithm.
We determine all states {S0} that are stable at U = U p , and build candidate FSMs by iteratively determining a(S0), b(S0), a.  a(a(S0)) etc (where a, b, . . .are characters in the alphabet), until no new states are found, for each S0.For the specific case of two pulses driving, this yields transition tables as function of p, qa, q b and S0.To turn these into candidate FSMs, we take S0 as initial state, label it grey, and select one or more accepting states -we do not consider each of these latter choices separately.
Many of the candidate FSMs thus obtained are equivalent under permutations of the characters a ↔ b or permutations of the (non-initial) states, and such permutations do not alter the computational complexity.We therefore determine the irreducible FSMs which are distinct under such permutations.
We finally note that there are two classes of such graphs: each either has one or no states without incoming edges.If there is one such state, it must be S0 (all the other nodes can be reached starting from S0, so they have incoming edges).However, some graphs have no nodes without incoming edges.In that case, all nodes can be chosen as initial nodes -the candidate graphs contain graphs with each of these nodes as initial node.

FSMs for non-interacting hysterons.
To understand the emergence of the limited set of eight FSMs for Preisach systems, we note that for independent hysterons, positive or negative pulses only affect individual hysterons.Hence, the operators a q 1 p and a q 2 p now act on individual hysterons, so that a q 1 p (si) is well defined.
We first consider the case where the operators a q 1 p and a q 2 p represent positive pulses (the case of two negative pulses follows by symmetry).We then order these two pulses such that |q2 − p| > |q1 − p|; in other words, a q 1 p represents a weak pulse and a q 2 p a strong pulse.We can distinguish three groups of hysterons, by how they transform their phase si under action of a q 1 p and a q 2 p .
Group one contains the hysterons that do not change under the action of either pulses (a q 1 p (si) = si and a q 2 p (si) = si); group two contains the hysterons that are set to one by the strong pulse but not by the weak pulse (a q 1 p (si) = si and a q 2 p (si) = 1); group three contains the hysterons that are set to one by either pulse (a q 1 p (si) = 1 and a q 2 p (si) = 1).
By considering different combinations of groups of hysterons (i.e., only group one, group one and group two, etc), we find four distinct FSMs.Labeling the eight FSMs in Fig. 6d of the main text that arise for no interactions as FSM 1 -FSM 8 (top to bottom), these correspond to FSM 1 -FSM 4 (where weak and strong pulses correspond to yellow and green transitions respectively).These four FSMs are the only ones possible for pulses of equal sign, as can also be understood from strict RPM.
When a q 1 p (si) and a q 2 p (si) represent positive and negative pulses, the situation is slightly more complex.We assume that q1 > p and q2 < p (the opposite case follows by symmetry).Each hysteron can now either be set to 1 by the action of a q 1 p or left invariant; and each hysteron can now either be reset to 0 by the action of a q 2 p or left invariant.Together, this yields four groups of hysterons.Again considering all permutations yields the remaining FSM types 5-8, as well as FSM 1-4 (Fig. 6d).As we will discuss elsewhere, there are two subtleties for pulses of opposite sign: First, if one includes so-called Garden of Eden states, there is a nineth FSM that can be created; second, one can get additional FSMs when one considers three distinct pulses.
From FSMs to t-graphs.Here we show how we can realize any FSM by a t-graph.Consider an FSM with N states and L characters, and a transition table ai(Sj) = S k (Fig. S6a).First, we encode the alphabet {ai} for i = 1, • • • as pulses a i 0 , so that an input string such as ′ a3a1a ′ 5 is translated to a sequence of driving pulses maxing out at U = 3, 1 and 5, respectively.For convenience, we split the driving pulses in positive ramps, i.e., use a i 0 (Sj) = r 0 i (r i 0 (Sj)) ((Fig.S6b).).Second, we define N × L states S i j , such that S 0 j are all stable at U = 0 -these correspond to the N states SJ of the FSM -and all other states are intermediate states reached during the upsweep of each pulse: r i 0 (S 0 j ) = S i j .The transition table ai(Sj) = S k can be realized by a t-graph where a down sweep from U = i to U = 0 leads to a transition from S i j to S 0 k , in other words, r 0 i (S i j ) = S 0 k (Fig. S6c).We finally give an example of how such t-graphs can be realized with interacting hysterons.We note that our construction requires a large number of hysterons, and for most FSMs, more effective implementations can likely be found.Moreover, our construction is for a specific FSM with N = 3 states and a single (L = 1) character denoted a1; while it can easily be extended to N states, we have not yet extended this construction to more characters.Although very technical, this construction is a Movies S1 and S2 demonstrate the breaking of l-RPM shown in, respectively, Figs.4a and 4b of the main text.Movies S3 and S4 demonstrate the transient response (τ = 2) shown in, respectively, Figs.4c and 4d of the main text.Movies S5, S6 and S7 demonstrate respectively FSM 1, FMS 2 and FMS 3 shown in Figs.5a-c, 5d-f and 5g-h of the main text.Movies S8 demonstrates FSM 4 shown in Figs.6e-f of the main text.
, in agreement with our prediction.Graph 3B23 thus illustrates that it is not trivial to see whether a combination of transitions of compatible, and that it therefore remains necessary to explicitly check whether candidate graphs are solvable by generating the design inequalities.

Fig. S1 .
Fig. S1.44 tgraphs.The P-graphs 3A-3F have a varying number of daughter graphs.In each daughter graph, we use a colored arrow to indicate the avalanches that replace the direct transitions in the mother graphs -the remaining direct transitions are as in the mother graph and represented in grey and without arrowheads.3A has four groups of avalanches (1-4) that occur in any combination.Note that the individual avalanches in group 3 and 4 cannot occur independently.3B has four avalanches (1-4) that can occur in almost any combination -with the exception of t-graph 3B23 which cannot be realized with serial coupling.Notice that avalanches 2 and 3 can occur together when either avalanche 1 or 4 is present.3C, 3D and 3E all allow two avalanches that can occur together.3F has no daughter graphs, because all its states have different values of m.Grey lines indicate daugther relations in graphs 3C, 3D and 3E and in the first two levels of 3A and 3B.
Fig. S2.Properties of a single hysteron (a) Schematic of a single mechanical hysteron, represented by a precurved beam of Young's modulus E, in series with a spring.(b) Numerical results.Normalized force as a function of the normalized displacement of a single hysteron of fixed geometry for different values of the spring stiffness k.We note that bistability appears for k < 10 mN/m.L = 20 mm, b = 1 mm, a = 0.15, t = 0.1.(c) Numerical results.Normalized threshold force (as defined in Fig. S2b ) as a function of the normalized amplitude of a single hysteron in series with an infinitely rigid spring for different thicknesses.•: F − , ▽: F + .

Fig. S3 .
Fig. S3.Full t-graph of sample A Sample A is described by t-graph 3A12, next to each arrow is indicated the related critical switching field in mm.

Fig. S4 .
Fig. S4.Experimental response of sample A to driving protocol 'yxxxyyxy'.Global force F (grey line) and deformation U (black line) as a function of time corresponding to the full FSM shown in Fig.6 of the main text.Each state reached is labeled on top of the plot.

Table S1 . Switching fields for four different three-hysterons samples described by
To avoid double counting, we represent such a group of graphs with a single graph without a (grey) labeled initial state -in such cases, each state can act as initial state (See Fig.6dof the main text).