Vertically vibrating a liquid bath can give rise to a self-propelled wave–particle entity on its free surface. The horizontal walking dynamics of this wave–particle entity can be described adequately by an integro-differential trajectory equation. By transforming this integro-differential equation of motion for a one-dimensional wave–particle entity into a system of ordinary differential equations (ODEs), we show the emergence of Lorenz-like dynamical systems for various spatial wave forms of the entity. Specifically, we present and give examples of Lorenz-like dynamical systems that emerge when the wave form gradient is (i) a solution of a linear homogeneous constant coefficient ODE, (ii) a polynomial, and (iii) a periodic function. Understanding the dynamics of the wave–particle entity in terms of Lorenz-like systems may prove to be useful in rationalizing emergent statistical behavior from underlying chaotic dynamics in hydrodynamic quantum analogs of walking droplets. Moreover, the results presented here provide an alternative physical interpretation of various Lorenz-like dynamical systems in terms of the walking dynamics of a wave–particle entity.

1.
Y.
Couder
,
E.
Fort
,
C.-H.
Gautier
, and
A.
Boudaoud
, “
From bouncing to floating: Noncoalescence of drops on a fluid bath
,”
Phys. Rev. Lett.
94
,
177801
(
2005
).
2.
Y.
Couder
,
S.
Protière
,
E.
Fort
, and
A.
Boudaoud
, “
Dynamical phenomena: Walking and orbiting droplets
,”
Nature
437
,
208
(
2005
).
3.
R. N.
Valani
,
A. C.
Slim
, and
T.
Simula
, “
Superwalking droplets
,”
Phys. Rev. Lett.
123
,
024503
(
2019
).
4.
R. N.
Valani
,
J.
Dring
,
T. P.
Simula
, and
A. C.
Slim
, “
Emergence of superwalking droplets
,”
J. Fluid Mech.
906
,
A3
(
2021
).
5.
R. N.
Valani
,
A. C.
Slim
, and
T. P.
Simula
, “
Stop-and-go locomotion of superwalking droplets
,”
Phys. Rev. E
103
,
043102
(
2021
).
6.
E.
Fort
,
A.
Eddi
,
A.
Boudaoud
,
J.
Moukhtar
, and
Y.
Couder
, “
Path-memory induced quantization of classical orbits
,”
Proc. Natl. Acad. Sci.
107
,
17515
17520
(
2010
).
7.
D. M.
Harris
and
J. W. M.
Bush
, “
Droplets walking in a rotating frame: From quantized orbits to multimodal statistics
,”
J. Fluid Mech.
739
,
444
464
(
2014
).
8.
A. U.
Oza
,
D. M.
Harris
,
R. R.
Rosales
, and
J. W. M.
Bush
, “
Pilot-wave dynamics in a rotating frame: On the emergence of orbital quantization
,”
J. Fluid Mech.
744
,
404
429
(
2014
).
9.
S.
Perrard
,
M.
Labousse
,
E.
Fort
, and
Y.
Couder
, “
Chaos driven by interfering memory
,”
Phys. Rev. Lett.
113
,
104101
(
2014
).
10.
S.
Perrard
,
M.
Labousse
,
M.
Miskin
,
E.
Fort
, and
Y.
Couder
, “
Self-organization into quantized eigenstates of a classical wave-driven particle
,”
Nat. Commun.
5
,
3219
(
2014
).
11.
M.
Labousse
,
S.
Perrard
,
Y.
Couder
, and
E.
Fort
, “
Self-attraction into spinning eigenstates of a mobile wave source by its emission back-reaction
,”
Phys. Rev. E
94
,
042224
(
2016
).
12.
J.
Montes
,
F.
Revuelta
, and
F.
Borondo
, “
Bohr-Sommerfeld-like quantization in the theory of walking droplets
,”
Phys. Rev. E
103
,
053110
(
2021
).
13.
A.
Eddi
,
J.
Moukhtar
,
S.
Perrard
,
E.
Fort
, and
Y.
Couder
, “
Level splitting at macroscopic scale
,”
Phys. Rev. Lett.
108
,
264503
(
2012
).
14.
A. U.
Oza
,
R. R.
Rosales
, and
J. W. M.
Bush
, “
Hydrodynamic spin states
,”
Chaos
28
,
096106
(
2018
).
15.
D. M.
Harris
,
J.
Moukhtar
,
E.
Fort
,
Y.
Couder
, and
J. W. M.
Bush
, “
Wavelike statistics from pilot-wave dynamics in a circular corral
,”
Phys. Rev. E
88
,
011001
(
2013
).
16.
T.
Gilet
, “
Quantumlike statistics of deterministic wave–particle interactions in a circular cavity
,”
Phys. Rev. E
93
,
042202
(
2016
).
17.
P. J.
Sáenz
,
T.
Cristea-Platon
, and
J. W. M.
Bush
, “
Statistical projection effects in a hydrodynamic pilot-wave system
,”
Nat. Phys.
14
,
315
319
(
2018
).
18.
T.
Cristea-Platon
,
P. J.
Sáenz
, and
J. W. M.
Bush
, “
Walking droplets in a circular corral: Quantisation and chaos
,”
Chaos
28
,
096116
(
2018
).
19.
M.
Durey
,
P. A.
Milewski
, and
Z.
Wang
, “
Faraday pilot-wave dynamics in a circular corral
,”
J. Fluid Mech.
891
,
A3
(
2020
).
20.
P. J.
Sáenz
,
T.
Cristea-Platon
, and
J. W. M.
Bush
, “
A hydrodynamic analog of Friedel oscillations
,”
Sci. Adv.
6
,
eaay9234
(
2020
).
21.
A.
Eddi
,
E.
Fort
,
F.
Moisy
, and
Y.
Couder
, “
Unpredictable tunneling of a classical wave–particle association
,”
Phys. Rev. Lett.
102
,
240401
(
2009
).
22.
A.
Nachbin
,
P. A.
Milewski
, and
J. W. M.
Bush
, “
Tunneling with a hydrodynamic pilot-wave model
,”
Phys. Rev. Fluids
2
,
034801
(
2017
).
23.
L.
Tadrist
,
T.
Gilet
,
P.
Schlagheck
, and
J. W. M.
Bush
, “
Predictability in a hydrodynamic pilot-wave system: Resolution of walker tunneling
,”
Phys. Rev. E
102
,
013104
(
2020
).
24.
P. J.
Sáenz
,
G.
Pucci
,
S. E.
Turton
,
A.
Goujon
,
R. R.
Rosales
,
J.
Dunkel
, and
J. W. M.
Bush
, “
Emergent order in hydrodynamic spin lattices
,”
Nature
596
,
58
62
(
2021
).
25.
R. N.
Valani
,
A. C.
Slim
, and
T.
Simula
, “
Hong–Ou–Mandel-like two-droplet correlations
,”
Chaos
28
,
096104
(
2018
).
26.
A.
Nachbin
, “
Walking droplets correlated at a distance
,”
Chaos
28
,
096110
(
2018
).
27.
Y.
Dagan
and
J. W. M.
Bush
, “
Hydrodynamic quantum field theory: The free particle
,”
C. R. Mec.
348
,
555
571
(
2020
).
28.
M.
Durey
and
J. W. M.
Bush
, “
Hydrodynamic quantum field theory: The onset of particle motion and the form of the pilot wave
,”
Front. Phys.
8
,
300
(
2020
).
29.
A.
Drezet
,
P.
Jamet
,
D.
Bertschy
,
A.
Ralko
, and
C.
Poulain
, “
Mechanical analog of quantum bradyons and tachyons
,”
Phys. Rev. E
102
,
052206
(
2020
).
30.
J. W. M.
Bush
, “
Pilot-wave hydrodynamics
,”
Annu. Rev. Fluid Mech.
47
,
269
292
(
2015
).
31.
J. W. M.
Bush
and
A. U.
Oza
, “
Hydrodynamic quantum analogs
,”
Rep. Prog. Phys.
84
,
017001
(
2020
).
32.
J. W. M.
Bush
,
Y.
Couder
,
T.
Gilet
,
P. A.
Milewski
, and
A.
Nachbin
, “
Introduction to focus issue on hydrodynamic quantum analogs
,”
Chaos
28
,
096001
(
2018
).
33.
S. E.
Turton
,
M. M. P.
Couchman
, and
J. W. M.
Bush
, “
A review of the theoretical modeling of walking droplets: Toward a generalized pilot-wave framework
,”
Chaos
28
,
096111
(
2018
).
34.
A.
Rahman
and
D.
Blackmore
, “
Walking droplets through the lens of dynamical systems
,”
Mod. Phys. Lett. B
34
,
2030009
(
2020
).
35.
A. U.
Oza
,
R. R.
Rosales
, and
J. W. M.
Bush
, “
A trajectory equation for walking droplets: Hydrodynamic pilot-wave theory
,”
J. Fluid Mech.
737
,
552
570
(
2013
).
36.
M.
Labousse
,
A. U.
Oza
,
S.
Perrard
, and
J. W. M.
Bush
, “
Pilot-wave dynamics in a harmonic potential: Quantization and stability of circular orbits
,”
Phys. Rev. E
93
,
033122
(
2016
).
37.
A. U.
Oza
,
R. R.
Rosales
, and
J. W. M.
Bush
, “
Hydrodynamic spin states
,”
Chaos
28
,
096106
(
2018
).
38.
K. M.
Kurianski
,
A. U.
Oza
, and
J. W. M.
Bush
, “
Simulations of pilot-wave dynamics in a simple harmonic potential
,”
Phys. Rev. Fluids
2
,
113602
(
2017
).
39.
L. D.
Tambasco
and
J. W. M.
Bush
, “
Exploring orbital dynamics and trapping with a generalized pilot-wave framework
,”
Chaos
28
,
096115
(
2018
).
40.
R. N.
Valani
,
A. C.
Slim
,
D. M.
Paganin
,
T. P.
Simula
, and
T.
Vo
, “
Unsteady dynamics of a classical particle-wave entity
,”
Phys. Rev. E
104
,
015106
(
2021
).
41.
M.
Durey
, “
Bifurcations and chaos in a Lorenz-like pilot-wave system
,”
Chaos
30
,
103115
(
2020
).
42.
R. N.
Valani
and
A. C.
Slim
, “
Pilot-wave dynamics of two identical, in-phase bouncing droplets
,”
Chaos
28
,
096114
(
2018
).
43.
A. U.
Oza
,
Ø.
Wind-Willassen
,
D. M.
Harris
,
R. R.
Rosales
, and
J. W. M.
Bush
, “
Pilot-wave hydrodynamics in a rotating frame: Exotic orbits
,”
Phys. Fluids
26
,
082101
(
2014
).
44.
L. D.
Tambasco
,
D. M.
Harris
,
A. U.
Oza
,
R. R.
Rosales
, and
J. W. M.
Bush
, “
The onset of chaos in orbital pilot-wave dynamics
,”
Chaos
26
,
103107
(
2016
).
45.
J.
Arbelaiz
,
A. U.
Oza
, and
J. W. M.
Bush
, “
Promenading pairs of walking droplets: Dynamics and stability
,”
Phys. Rev. Fluids
3
,
013604
(
2018
).
46.
A. U.
Oza
,
E.
Siéfert
,
D. M.
Harris
,
J.
Moláček
, and
J. W. M.
Bush
, “
Orbiting pairs of walking droplets: Dynamics and stability
,”
Phys. Rev. Fluids
2
,
053601
(
2017
).
47.
E. N.
Lorenz
, “
Deterministic nonperiodic flow
,”
J. Atmos Sci.
20
,
130
141
(
1963
).
48.
C.
Zou
,
Q.
Zhang
,
X.
Wei
, and
C.
Liu
, “
Image encryption based on improved Lorenz system
,”
IEEE Access
8
,
75728
75740
(
2020
).
49.
X.-Y.
Wang
,
P.
Li
,
Y.-Q.
Zhang
,
L.-Y.
Liu
,
H.
Zhang
, and
X.
Wang
, “
A novel color image encryption scheme using DNA permutation based on the Lorenz system
,”
Multimed. Tools Appl.
77
,
6243
6265
(
2018
).
50.
M.
Kaur
and
V.
Kumar
, “
Efficient image encryption method based on improved Lorenz chaotic system
,”
Electron. Lett.
54
,
562
564
(
2018
).
51.
M.
Feki
, “
An adaptive chaos synchronization scheme applied to secure communication
,”
Chaos, Solitons Fractals
18
,
141
148
(
2003
).
52.
J.
Pan
,
Q.
Ding
, and
B.
Du
, “
A new improved scheme of chaotic masking secure communication based on Lorenz system
,”
Int. J. Bifurcat. Chaos
22
,
1250125
(
2012
).
53.
X.-F.
Li
,
Y.-D.
Chu
,
J.-G.
Zhang
, and
Y.-X.
Chang
, “
Nonlinear dynamics and circuit implementation for a new Lorenz-like attractor
,”
Chaos, Solitons Fractals
41
,
2360
2370
(
2009
).
54.
S.
Yu
,
W. K. S.
Tang
,
J.
, and
G.
Chen
, “
Generating 2n-wing attractors from Lorenz-like systems
,”
Int. J. Circuit Theory Appl.
38
,
243
258
(
2010
).
55.
J. N.
Blakely
,
M. B.
Eskridge
, and
N. J.
Corron
, “
A simple Lorenz circuit and its radio frequency implementation
,”
Chaos
17
,
023112
(
2007
).
56.
X.
Zang
,
S.
Iqbal
,
Y.
Zhu
,
X.
Liu
, and
J.
Zhao
, “
Applications of chaotic dynamics in robotics
,”
Int. J. Adv. Robot. Syst.
13
,
60
(
2016
).
57.
D.
Poland
, “
Cooperative catalysis and chemical chaos: A chemical model for the Lorenz equations
,”
Physica D
65
,
86
99
(
1993
).
58.
J.
Moláček
and
J. W. M.
Bush
, “
Drops walking on a vibrating bath: Towards a hydrodynamic pilot-wave theory
,”
J. Fluid Mech.
727
,
612
647
(
2013
).
59.
A. P.
Damiano
,
P.-T.
Brun
,
D. M.
Harris
,
C. A.
Galeano-Rios
, and
J. W. M.
Bush
, “
Surface topography measurements of the bouncing droplet experiment
,”
Exp. Fluids
57
,
163
(
2016
).
60.
O.
Rossler
, “
An equation for hyperchaos
,”
Phys. Lett. A
71
,
155
157
(
1979
).
61.
G.
Zhang
,
F.
Zhang
,
X.
Liao
,
D.
Lin
, and
P.
Zhou
, “
On the dynamics of new 4D Lorenz-type chaos systems
,”
Adv. Differ. Equ.
2017
,
217
.
62.
Q.
Yang
,
K.
Zhang
, and
G.
Chen
, “
Hyperchaotic attractors from a linearly controlled Lorenz system
,”
Nonlinear Anal. Real World Appl.
10
,
1601
1617
(
2009
).
63.
Q.
Jia
, “
Hyperchaos generated from the Lorenz chaotic system and its control
,”
Phys. Lett. A
366
,
217
222
(
2007
).
64.
Y.
Chen
, “
The existence of homoclinic orbits in a 4D Lorenz-type hyperchaotic system
,”
Nonlinear Dyn.
87
,
1445
1452
(
2017
).
65.
C.
Xu
,
J.
Sun
, and
C.
Wang
, “
An image encryption algorithm based on random walk and hyperchaotic systems
,”
Int. J. Bifurcat. Chaos
30
,
2050060
(
2020
).
66.
X.
Wang
and
M.
Wang
, “
A hyperchaos generated from Lorenz system
,”
Phys. A: Stat. Mech. Appl.
387
,
3751
3758
(
2008
).
67.
M.
Durey
,
P. A.
Milewski
, and
J. W. M.
Bush
, “
Dynamics, emergent statistics, and the mean-pilot-wave potential of walking droplets
,”
Chaos
28
,
096108
(
2018
).
68.
Y.
Aizawa
, “
Global aspects of the dissipative dynamical systems. I: Statistical identification and fractal properties of the Lorenz chaos
,”
Prog. Theor. Phys.
68
,
64
84
(
1982
).
69.
F.
Olver
,
D.
Lozier
,
R.
Boisvert
, and
C.
Clark
,
The NIST Handbook of Mathematical Functions
(
Cambridge University Press
,
New York
,
2010
).
You do not currently have access to this content.