A walker is the association of a sub-millimetric bouncing drop moving along with a co-evolving Faraday wave. When confined in a harmonic potential, its stable trajectories are periodic and quantised both in extension and mean angular momentum. In this article, we present the rest of the story, specifically the chaotic paths. They are chaotic and show intermittent behaviors between an unstable quantised set of attractors. First, we present the two possible situations we find experimentally. Then, we emphasise theoretically two mechanisms that lead to unstable situations. It corresponds either to noise-driven chaos or low-dimensional deterministic chaos. Finally, we characterise experimentally each of these distinct situations. This article aims at presenting a comprehensive investigation of the unstable paths in order to complete the picture of walkers in a two dimensional harmonic potential.

1.
Y.
Couder
,
S.
Protiere
,
E.
Fort
, and
A.
Boudaoud
,
Nature
437
,
208
(
2005
).
2.
E.
Fort
,
A.
Eddi
,
J.
Moukhtar
,
A.
Boudaoud
, and
Y.
Couder
,
Proc. Natl. Acad. Sci.
107
,
17515
(
2010
).
3.
S.
Perrard
,
M.
Labousse
,
M.
Miskin
,
E.
Fort
, and
Y.
Couder
,
Nat. Commun.
5
,
3219
(
2014
).
4.
J. W. M.
Bush
,
Annu. Rev. Fluid Mech.
47
,
269
292
(
2015
).
5.
D.
Harris
,
J.
Moukhtar
,
E.
Fort
,
Y.
Couder
, and
J. W. M.
Bush
,
Phys. Rev. E
88
,
011001(R)
(
2013
).
6.
D.
Harris
and
J. W. M.
Bush
,
J. Fluid Mech.
739
,
444
(
2014
).
7.
S.
Perrard
,
M.
Labousse
,
E.
Fort
, and
Y.
Couder
,
Phys. Rev. Lett.
113
,
104101
(
2014
).
8.
L. D.
Tambasco
,
D. M.
Harris
,
A. U.
Oza
,
R. R.
Rosales
, and
J. W. M.
Bush
,
Chaos
26
,
103107
(
2016
).
9.
A. U.
Oza
,
O.
Wind-Willassen
,
D. M.
Harris
,
R. R.
Rosales
, and
J. W. M.
Bush
,
Phys. Fluids
26
,
082101
(
2014
).
10.
M.
Faraday
,
Philos. Trans. R. Soc. Lond.
52
,
299
(
1831
).
11.
12.
K.
Kumar
and
L.
Tuckerman
,
J. Fluid Mech.
279
,
49
(
1994
).
13.
S.
Protiere
,
A.
Boudaoud
, and
Y.
Couder
,
J. Fluid Mech.
554
,
85
(
2006
).
14.
J.
Browaeys
, “
Les ferrofluides: ondes de surface, résistance de vague et simulation de la convection dans le manteau terrestre
,” Ph.D. thesis (
University of Paris-Diderot
,
2000
).
15.
A. U.
Oza
,
D. M.
Harris
,
R. R.
Rosales
, and
J. W. M.
Bush
,
J. Fluid Mech.
744
,
404
(
2014
).
16.
J. W. M.
Bush
,
A. U.
Oza
, and
J.
Moláček
,
J. Fluid Mech.
755
,
R7
(
2014
).
17.
M.
Labousse
,
S.
Perrard
,
Y.
Couder
, and
E.
Fort
,
New J. Phys.
16
,
113027
(
2014
).
18.
M.
Durey
and
P.
Milewski
,
J. Fluid Mech.
821
,
296
(
2017
).
19.
K. M.
Jurianski
,
A. U.
Oza
, and
J. W. M.
Bush
,
Phys. Rev. Fluids
2
,
113602
(
2017
).
20.
M.
Labousse
,
A. U.
Oza
,
S.
Perrard
, and
J. W. M.
Bush
,
Phys. Rev. E
93
,
033122
(
2016
).
21.
S. H.
Strogatz
,
Nonlinear Dynamics and Chaos: With Applications to Physics, Biology and Chemistry
(
Perseus Books Group
,
1994
).
22.
P.
Bergé
,
Y.
Pomeau
, and
C.
Vidal
,
Order within Chaos: Towards a Deterministic Approach to Turbulence
(
Wiley and Sons
,
New York
,
1984
).
23.
P.
Manneville
,
Instabilités, Chaos et Turbulence
(
Les éditions de l’école Polytechnique
,
2006
).
24.
A.
Eddi
,
E.
Sultan
,
J.
Moukhtar
,
E.
Fort
,
M.
Rossi
, and
Y.
Couder
,
J. Fluid Mech.
674
,
433
(
2011
).
25.
P.
Milewski
,
C.
Galeano-Rios
,
A.
Nachbin
, and
J. W. M.
Bush
,
J. Fluid Mech.
778
,
361
(
2015
).
26.
A. U.
Oza
,
R. R.
Rosales
, and
J. W. M.
Bush
,
J. Fluid Mech.
737
,
552
(
2013
).
27.
J.
Molàček
and
J. W. M.
Bush
,
J. Fluid Mech.
727
,
582
(
2013
).
28.
J.
Molàček
and
J. W. M.
Bush
,
J. Fluid Mech.
727
,
612
(
2013
).
29.
M.
Labousse
, “
Investigation of a path-memory dynamics: A theoretical trial
,” Ph.D. thesis (
Université Pierre et Marie Curie
,
2014
).
31.
P.
Castiglione
,
M.
Falcioni
,
A.
Lesne
, and
A.
Vulpiani
,
Physique Statisque, Chaos et Approches Multiéchelle
(
Éditions Belin
,
2008
).
32.
G.
Floquet
,
Ann. École Norm. Sup.
12
,
47
88
(
1883
).
33.
H. D. I.
Abarbanel
,
Analysis of Observed Chaotic Data
(
Springer-Verlag
,
Heidelberg
,
1996
).
34.
J.
Eckmann
,
Rev. Mod. Phys.
53
,
643
(
1981
).
35.
E. P.
Lorenz
,
J. Atmos. Sci.
20
,
130
141
(
1963
).
36.
O. E.
Rössler
and
P. J.
Ortoleva
,
Theoretical Approaches to Complex Systems, Lecture Notes in Biomathematics Vol. 21 (Springer, 1978), p. 67
.
37.
M.
Berhanu
et al.,
Europhys. Lett.
77
,
59001
(
2007
).
39.
N.
Platt
,
E. A.
Spiegel
, and
C.
Tresser
,
Phys. Rev. E
70
,
279
(
1993
).
40.
J.
Heagy
,
N.
Platt
, and
S. M.
Hammel
,
Phys. Rev. E
49
,
1140
(
1994
).
41.
M.
Labousse
,
S.
Perrard
,
Y.
Couder
, and
E.
Fort
,
Phys. Rev. E
94
,
063017
(
2016
).
42.
F.
Pétrélis
,
S.
Fauve
,
E.
Dormy
, and
J. P.
Valet
,
Phys. Rev. Lett.
102
,
144503
(
2009
).
43.
D.
Ruelle
and
F.
Takens
,
Commun. Math. Phys.
20
,
167
(
1971
).
44.
S.
Newhouse
,
D.
Ruelle
, and
F.
Takens
,
Commun. Math. Phys.
64
,
35
(
1978
).
You do not currently have access to this content.