Systems with many stable configurations abound in nature, both in living and inanimate matter, encoding a rich variety of behaviors. In equilibrium, a multistable system is more likely to be found in configurations with lower energy, but the presence of an external drive can alter the relative stability of different configurations in unexpected ways. Living systems are examples par excellence of metastable nonequilibrium attractors whose structure and stability are highly dependent on the specific form and pattern of the energy flow sustaining them. Taking this distinctively lifelike behavior as inspiration, we sought to investigate the more general physical phenomenon of drive-specific selection in nonequilibrium dynamics. To do so, we numerically studied driven disordered mechanical networks of bistable springs possessing a vast number of stable configurations arising from the two stable rest lengths of each spring, thereby capturing the essential physical properties of a broad class of multistable systems. We found that there exists a range of forcing amplitudes for which the attractor states of driven disordered multistable mechanical networks are fine-tuned with respect to the pattern of external forcing to have low energy absorption from it. Additionally, we found that these drive-specific attractor states are further stabilized by precise matching between the multidimensional shape of their orbit and that of the potential energy well they inhabit. Lastly, we showed evidence of drive-specific selection in an experimental system and proposed a general method to estimate the range of drive amplitudes for drive-specific selection.

1.
M.
Tsamados
, “
Plasticity and dynamical heterogeneity in driven glassy materials
,”
Eur. Phys. J. E
32
(
2
),
165
181
(
2010
).
2.
W.
Henderson
,
E. Y.
Andrei
,
M. J.
Higgins
, and
S.
Bhattacharya
, “
Metastability and glassy behavior of a driven flux-line lattice
,”
Phys. Rev. Lett.
77
(
10
),
2077
2080
(
1996
).
3.
L.
Berthier
and
J.
Kurchan
, “
Non-equilibrium glass transitions in driven and active matter
,”
Nat. Phys.
9
(
5
),
310
314
(
2013
).
4.
N. C.
Keim
and
S. R.
Nagel
, “
Generic transient memory formation in disordered systems with noise
,”
Phys. Rev. Lett.
107
(
1
),
010603
(
2011
).
5.
N. C.
Keim
,
J. D.
Paulsen
, and
S. R.
Nagel
, “
Multiple transient memories in sheared suspensions: Robustness, structure, and routes to plasticity
,”
Phys. Rev. E
88
(
3
),
032306
(
2013
).
6.
J. D.
Paulsen
,
N. C.
Keim
, and
S. R.
Nagel
, “
Multiple transient memories in experiments on sheared non-Brownian suspensions
,”
Phys. Rev. Lett.
113
(
6
),
068301
(
2014
).
7.
R. L.
Harne
,
Z.
Wu
, and
K. W.
Wang
, “
Designing and harnessing the metastable states of a modular metastructure for programmable mechanical properties adaptation
,”
J. Mech. Des.
138
(
2
),
021402
(
2015
).
8.
J.
Hertz
,
A.
Krogh
, and
R. G.
Palmer
,
Introduction to the Theory of Neural Computation
(
Addison-Wesley Longman Publishing Co., Inc.
,
Boston, MA
,
1991
).
9.
M.
Stern
,
H.
Sompolinsky
, and
L. F.
Abbott
, “
Dynamics of random neural networks with bistable units
,”
Phys. Rev. E
90
(
6
),
062710
(
2014
).
10.
C.-Y.
Cheng
,
K.-H.
Lin
, and
C.-W.
Shih
, “
Multistability in recurrent neural networks
,”
SIAM J. Appl. Math.
66
(
4
),
1301
1320
(
2006
).
11.
V.
Bacot
,
S.
Perrard
,
M.
Labousse
,
Y.
Couder
, and
E.
Fort
, “
Multistable free states of an active particle from a coherent memory dynamics
,”
Phys. Rev. Lett.
122
(
10
),
104303
(
2019
).
12.
A. N.
Pisarchik
,
Y. O.
Barmenkov
, and
A. V.
Kir’yanov
, “
Experimental demonstration of attractor annihilation in a multistable fiber laser
,”
Phys. Rev. E
68
(
6
),
066211
(
2003
).
13.
A. N.
Pisarchik
and
B. F.
Kuntsevich
, “
Control of multistability in a directly modulated diode laser
,”
IEEE J. Quantum Electron.
38
(
12
),
1594
1598
(
2002
).
14.
J.
Fern
,
J.
Lu
, and
R.
Schulman
, “
The energy landscape for the self-assembly of a two-dimensional DNA origami complex
,”
ACS Nano
10
(
2
),
1836
1844
(
2016
).
15.
F.
Ravelet
,
L.
Marié
,
A.
Chiffaudel
, and
F.
Daviaud
, “
Multistability and memory effect in a highly turbulent flow: Experimental evidence for a global bifurcation
,”
Phys. Rev. Lett.
93
(
16
),
164501
(
2004
).
16.
S. B.
Power
and
R.
Kleeman
, “
Multiple equilibria in a global ocean general circulation model
,”
J. Phys. Oceanogr.
23
(
8
),
1670
1681
(
1993
).
17.
S.
Rahmstorf
, “
Multiple convection patterns and thermohaline flow in an idealized OGCM
,”
J. Clim.
8
(
12
),
3028
3039
(
1995
).
18.
S.
Waitukaitis
,
R.
Menaut
,
B. G.-g.
Chen
, and
M.
van Hecke
, “
Origami multistability: From single vertices to metasheets
,”
Phys. Rev. Lett.
114
(
5
),
055503
(
2015
).
19.
N.
Singh
and
M.
van Hecke
, “
Design of pseudo-mechanisms and multistable units for mechanical metamaterials
,”
Phys. Rev. Lett.
126
(
24
),
248002
(
2021
).
20.
J.
Braun
and
M.
Mattia
, “
Attractors and noise: Twin drivers of decisions and multistability
,”
NeuroImage
52
(
3
),
740
751
(
2010
).
21.
B.
Marin
,
W. H.
Barnett
,
A.
Doloc-Mihu
,
R. L.
Calabrese
, and
G. S.
Cymbalyuk
, “
High prevalence of multistability of rest states and bursting in a database of a model neuron
,”
PLoS Comput. Biol.
9
(
3
),
e1002930
(
2013
).
22.
A. D.
Goldberg
,
C. D.
Allis
, and
E.
Bernstein
, “
Epigenetics: A landscape takes shape
,”
Cell
128
(
4
),
635
638
(
2007
).
23.
M.
Laurent
and
N.
Kellershohn
, “
Multistability: A major means of differentiation and evolution in biological systems
,”
Trends Biochem. Sci.
24
(
11
),
418
422
(
1999
).
24.
J.
Wang
,
K.
Zhang
,
L.
Xu
, and
E.
Wang
, “
Quantifying the Waddington landscape and biological paths for development and differentiation
,”
Proc. Natl. Acad. Sci. U. S. A.
108
(
20
),
8257
8262
(
2011
).
25.
F.
Sittel
and
G.
Stock
, “
Perspective: Identification of collective variables and metastable states of protein dynamics
,”
J. Chem. Phys.
149
(
15
),
150901
(
2018
).
26.
N.
Knowlton
, “
Thresholds and multiple stable states in coral reef community dynamics
,”
Am. Zool.
32
(
6
),
674
682
(
1992
).
27.
M.
Scheffer
,
S.
Carpenter
,
J. A.
Foley
,
C.
Folke
, and
B.
Walker
, “
Catastrophic shifts in ecosystems
,”
Nature
413
(
6856
),
591
596
(
2001
).
28.
D.
Angeli
,
J. E.
Ferrell
, and
E. D.
Sontag
, “
Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems
,”
Proc. Natl. Acad. Sci. U. S. A.
101
(
7
),
1822
1827
(
2004
).
29.
E. M.
Ozbudak
,
M.
Thattai
,
H. N.
Lim
,
B. I.
Shraiman
, and
A.
van Oudenaarden
, “
Multistability in the lactose utilization network of Escherichia coli
,”
Nature
427
(
6976
),
737
740
(
2004
).
30.
H. H.
Chang
,
P. Y.
Oh
,
D. E.
Ingber
, and
S.
Huang
, “
Multistable and multistep dynamics in neutrophil differentiation
,”
BMC Cell Biol.
7
(
1
),
11
(
2006
).
31.
M.
Frenkel-Pinter
,
M.
Samanta
,
G.
Ashkenasy
, and
L. J.
Leman
, “
Prebiotic peptides: Molecular hubs in the origin of life
,”
Chem. Rev.
120
(
11
),
4707
4765
(
2020
).
32.
M.
Frenkel-Pinter
,
M.
Bouza
,
F. M.
Fernández
,
L. J.
Leman
,
L. D.
Williams
,
N. V.
Hud
, and
A.
Guzman-Martinez
, “
Thioesters provide a plausible prebiotic path to proto-peptides
,”
Nat. Commun.
13
(
1
),
2569
(
2022
).
33.
K.
Matange
,
V.
Rajaei
,
G.
Schuster
,
N.
Hud
,
C.
Menor-Salvan
,
P.
Capera-Aragonès
,
M.
Frenkel-Pinter
, and
L.
Williams
, “
Origins of life: Chemistry and evolution
,” chemRxiv:2023-1jrcq-v2 (
2023
).
34.
P.
Marmillot
,
M.
Kaufman
, and
J. F.
Hervagault
, “
Multiple steady states and dissipative structures in a circular and linear array of three cells: Numerical and experimental approaches
,”
J. Chem. Phys.
95
(
2
),
1206
1214
(
1991
).
35.
R.
Aguado
and
G.
Platero
, “
Photoinduced multistable phenomena in the tunneling current through doped superlattices
,”
Phys. Rev. Lett.
81
(
22
),
4971
4974
(
1998
).
36.
L. K.
Madan
,
C. L.
Welsh
,
A. P.
Kornev
, and
S. S.
Taylor
, “
The ‘violin model’: Looking at community networks for dynamic allostery
,”
J. Chem. Phys.
158
(
8
),
081001
(
2023
).
37.
L. G.
Ahuja
,
S. S.
Taylor
, and
A. P.
Kornev
, “
Tuning the ‘violin’ of protein kinases: The role of dynamics-based allostery
,”
IUBMB Life
71
(
6
),
685
696
(
2019
).
38.
J. J.
Hopfield
, “
Neural networks and physical systems with emergent collective computational abilities
,”
Proc. Natl. Acad. Sci. U. S. A.
79
(
8
),
2554
2558
(
1982
).
39.
C. C.
Canavier
,
D. A.
Baxter
,
J. W.
Clark
, and
J. H.
Byrne
, “
Nonlinear dynamics in a model neuron provide a novel mechanism for transient synaptic inputs to produce long-term alterations of postsynaptic activity
,”
J. Neurophysiol.
69
(
6
),
2252
2257
(
1993
).
40.
N. C.
Keim
,
J. D.
Paulsen
,
Z.
Zeravcic
,
S.
Sastry
, and
S. R.
Nagel
, “
Memory formation in matter
,”
Rev. Mod. Phys.
91
(
3
),
035002
(
2019
).
41.
C.
Merrigan
,
D.
Shohat
,
C.
Sirote
,
Y.
Lahini
,
C.
Nisoli
, and
Y.
Shokef
, “
Emergent disorder and mechanical memory in periodic metamaterials
,” arXiv:2204.04000 [cond-mat] (
2022
).
42.
D.
Shohat
,
D.
Hexner
, and
Y.
Lahini
, “
Memory from coupled instabilities in unfolded crumpled sheets
,”
Proc. Natl. Acad. Sci. U. S. A.
119
(
28
),
e2200028119
(
2022
).
43.
A.
Kashiwagi
,
I.
Urabe
,
K.
Kaneko
, and
T.
Yomo
, “
Adaptive response of a gene network to environmental changes by fitness-induced attractor selection
,”
PLoS One
1
(
1
),
e49
(
2006
).
44.
P.
Bieling
,
T.-D.
Li
,
J.
Weichsel
,
R.
McGorty
,
P.
Jreij
,
B.
Huang
,
D. A.
Fletcher
, and
R.
Mullins
, “
Force feedback controls motor activity and mechanical properties of self-assembling branched actin networks
,”
Cell
164
(
1–2
),
115
127
(
2016
).
45.
S.
Majumdar
,
L. C.
Foucard
,
A. J.
Levine
, and
M. L.
Gardel
, “
Mechanical hysteresis in actin networks
,”
Soft Matter
14
(
11
),
2052
2058
(
2018
).
46.
N.
Pashine
,
D.
Hexner
,
A. J.
Liu
, and
S. R.
Nagel
, “
Directed aging, memory, and nature’s greed
,”
Sci. Adv.
5
(
12
),
eaax4215
(
2019
).
47.
D.
Hexner
,
N.
Pashine
,
A. J.
Liu
, and
S. R.
Nagel
, “
Effect of directed aging on nonlinear elasticity and memory formation in a material
,”
Phys. Rev. Res.
2
(
4
),
043231
(
2020
).
48.
D.
Hexner
,
A. J.
Liu
, and
S. R.
Nagel
, “
Periodic training of creeping solids
,”
Proc. Natl. Acad. Sci. U. S. A.
117
(
50
),
31690
31695
(
2020
).
49.
M.
Stern
,
M. B.
Pinson
, and
A.
Murugan
, “
Continual learning of multiple memories in mechanical networks
,”
Phys. Rev. X
10
(
3
),
031044
(
2020
).
50.
G.
Librandi
,
E.
Tubaldi
, and
K.
Bertoldi
, “
Programming nonreciprocity and reversibility in multistable mechanical metamaterials
,”
Nat. Commun.
12
(
1
),
3454
(
2021
).
51.
K.
Che
,
C.
Yuan
,
J.
Wu
,
H. J.
Qi
, and
J.
Meaud
, “
Three-dimensional-printed multistable mechanical metamaterials with a deterministic deformation sequence
,”
J. Appl. Mech.
84
(
1
),
011004
(
2017
).
52.
L.
Jin
,
R.
Khajehtourian
,
J.
Mueller
,
A.
Rafsanjani
,
V.
Tournat
,
K.
Bertoldi
, and
D. M.
Kochmann
, “
Guided transition waves in multistable mechanical metamaterials
,”
Proc. Natl. Acad. Sci. U. S. A.
117
(
5
),
2319
2325
(
2020
).
53.
H.
Zhang
,
J.
Wu
,
D.
Fang
, and
Y.
Zhang
, “
Hierarchical mechanical metamaterials built with scalable tristable elements for ternary logic operation and amplitude modulation
,”
Sci. Adv.
7
(
9
),
eabf1966
(
2021
).
54.
H.
Morowitz
and
E.
Smith
, “
Energy flow and the organization of life
,”
Complexity
13
(
1
),
51
59
(
2007
).
55.
J.
Shim
,
S.
Shan
,
A.
Košmrlj
,
S. H.
Kang
,
E. R.
Chen
,
J. C.
Weaver
, and
K.
Bertoldi
, “
Harnessing instabilities for design of soft reconfigurable auxetic/chiral materials
,”
Soft Matter
9
(
34
),
8198
8202
(
2013
).
56.
E.
Medina
,
P. E.
Farrell
,
K.
Bertoldi
, and
C. H.
Rycroft
, “
Navigating the landscape of nonlinear mechanical metamaterials for advanced programmability
,”
Phys. Rev. B
101
(
6
),
064101
(
2020
).
57.
S.
Fang
,
S.
Zhou
,
D.
Yurchenko
,
T.
Yang
, and
W.-H.
Liao
, “
Multistability phenomenon in signal processing, energy harvesting, composite structures, and metamaterials: A review
,”
Mech. Syst. Signal Process.
166
,
108419
(
2022
).
58.
A. N.
Pisarchik
and
U.
Feudel
, “
Control of multistability
,”
Phys. Rep.
540
(
4
),
167
218
(
2014
).
59.
A. N.
Pisarchik
and
B. K.
Goswami
, “
Annihilation of one of the coexisting attractors in a bistable system
,”
Phys. Rev. Lett.
84
(
7
),
1423
1426
(
2000
).
60.
A. N.
Pisarchik
, “
Controlling the multistability of nonlinear systems with coexisting attractors
,”
Phys. Rev. E
64
(
4
),
046203
(
2001
).
61.
B. K.
Goswami
,
S.
Euzzor
,
K.
Al Naimee
,
A.
Geltrude
,
R.
Meucci
, and
F. T.
Arecchi
, “
Control of stochastic multistable systems: Experimental demonstration
,”
Phys. Rev. E
80
(
1
),
016211
(
2009
).
62.
B. K.
Goswami
, “
Controlled destruction of chaos in the multistable regime
,”
Phys. Rev. E
76
(
1
),
016219
(
2007
).
63.
B. P.
Bernard
,
M. J.
Mazzoleni
,
N.
Garraud
,
D. P.
Arnold
, and
B. P.
Mann
, “
Experimental investigation of bifurcation induced bandgap reconfiguration
,”
J. Appl. Phys.
116
(
8
),
084904
(
2014
).
64.
M. J.
Mazzoleni
,
B. P.
Bernard
,
N.
Garraud
,
D. P.
Arnold
, and
B. P.
Mann
, “
Theoretical and experimental analysis of bifurcation induced passive bandgap reconfiguration
,” in
Nonlinear Dynamics, Volume 1
,
Conference Proceedings of the Society for Experimental Mechanics Series
, edited by
G.
Kerschen
(
Springer International Publishing
,
Cham
,
2016
), pp.
155
162
.
65.
F. C.
Moon
, “
Experiments on chaotic motions of a forced nonlinear oscillator: Strange attractors
,”
J. Appl. Mech.
47
(
3
),
638
644
(
1980
).
66.
P. J.
Holmes
and
F. C.
Moon
, “
Strange attractors and chaos in nonlinear mechanics
,”
J. Appl. Mech.
50
(
4b
),
1021
1032
(
1983
).
67.
F. C.
Moon
, “
Fractal boundary for chaos in a two-state mechanical oscillator
,”
Phys. Rev. Lett.
53
(
10
),
962
964
(
1984
).
68.
F. C.
Moon
and
G. X.
Li
, “
Fractal basin boundaries and homoclinic orbits for periodic motion in a two-well potential
,”
Phys. Rev. Lett.
55
(
14
),
1439
1442
(
1985
).
69.
N.
Grønbech-Jensen
and
O.
Farago
, “
A simple and effective Verlet-type algorithm for simulating Langevin dynamics
,”
Mol. Phys.
111
(
8
),
983
991
(
2013
).
70.
See https://github.com/hrideshkedia/bistable-spring-net.git for the GitHub repository containing the code for reproducing the results of this paper.
71.
W.
Lohmiller
and
J.-J. E.
Slotine
, “
On contraction analysis for non-linear systems
,”
Automatica
34
(
6
),
683
696
(
1998
).
72.
W.
Szemplińska-Stupnicka
, “
The analytical predictive criteria for chaos and escape in nonlinear oscillators: A survey
,”
Nonlinear Dyn.
7
(
2
),
129
147
(
1995
).
73.
M.
Baggioli
,
R.
Milkus
, and
A.
Zaccone
, “
Vibrational density of states and specific heat in glasses from random matrix theory
,”
Phys. Rev. E
100
(
6
),
062131
(
2019
).
74.
D.
Kondepudi
,
B.
Kay
, and
J.
Dixon
, “
End-directed evolution and the emergence of energy-seeking behavior in a complex system
,”
Phys. Rev. E
91
(
5
),
050902(R)
(
2015
).
75.
T.
Kachman
,
J. A.
Owen
, and
J. L.
England
, “
Self-organized resonance during search of a diverse chemical space
,”
Phys. Rev. Lett.
119
(
3
),
038001
(
2017
).
76.
J. L.
England
, “
Dissipative adaptation in driven self-assembly
,”
Nat. Nanotechnol.
10
(
11
),
919
923
(
2015
).
77.
N.
Perunov
,
R. A.
Marsland
, and
J. L.
England
, “
Statistical physics of adaptation
,”
Phys. Rev. X
6
(
2
),
021036
(
2016
).
78.
P.
Chvykov
and
J.
England
, “
Least-rattling feedback from strong time-scale separation
,”
Phys. Rev. E
97
(
3
),
032115
(
2018
).
79.
L.
Tociu
,
É.
Fodor
,
T.
Nemoto
, and
S.
Vaikuntanathan
, “
How dissipation constrains fluctuations in nonequilibrium liquids: Diffusion, structure, and biased interactions
,”
Phys. Rev. X
9
(
4
),
041026
(
2019
).
80.
É.
Fodor
,
T.
Nemoto
, and
S.
Vaikuntanathan
, “
Dissipation controls transport and phase transitions in active fluids: Mobility, diffusion and biased ensembles
,”
New J. Phys.
22
(
1
),
013052
(
2020
).
81.
T.
Nemoto
,
É.
Fodor
,
M. E.
Cates
,
R. L.
Jack
, and
J.
Tailleur
, “
Optimizing active work: Dynamical phase transitions, collective motion, and jamming
,”
Phys. Rev. E
99
(
2
),
022605
(
2019
).
82.
P.
Chvykov
,
T. A.
Berrueta
,
A.
Vardhan
,
W.
Savoie
,
A.
Samland
,
T. D.
Murphey
,
K.
Wiesenfeld
,
D. I.
Goldman
, and
J. L.
England
, “
Low rattling: A predictive principle for self-organization in active collectives
,”
Science
371
(
6524
),
90
95
(
2021
).
83.
Z.
Zhu
,
J.
Wu
,
B.
Yu
,
L.
Wu
, and
J.
Ma
, “
The anisotropic noise in stochastic gradient descent: Its behavior of escaping from sharp minima and regularization effects
,” in Proceedings of the 36th International Conference on Machine Learning, Long Beach, California, 9–15 June 2019 (PMLR,
2019
), Vol. 97, pp.
7654
7663
.
84.
P.
Foret
,
A.
Kleiner
,
H.
Mobahi
, and
B.
Neyshabur
, “
Sharpness-aware minimization for efficiently improving generalization
,” in 9th International Conference on Learning Representatives, 3–7 May 2021 (ICLR,
2021
); available at https://iclr.cc/virtual/2021/spotlight/3497.

Supplementary Material

You do not currently have access to this content.