The onset of the back-firing instability is studied in a one-dimensional spatially extended and dissipative system, where propagating localized solutions become unstable. It corresponds to the emission in the tail of a solitary wave of a new wave propagating in the opposite direction. The transition is illustrated, in geometrical terms, using a model normal form equation.

1.
I. S.
Aranson
and
L.
Kramer
,
Rev. Mod. Phys.
74
,
99
(
2002
).
2.
L.
Brusch
,
M.
Zimmermann
,
M.
van Hecke
,
M.
Bär
, and
A.
Torcini
,
Phys. Rev. Lett.
85
,
86
(
2000
).
3.
L.
Brusch
,
A.
Torcini
,
M.
van Hecke
,
M. G.
Zimmermann
, and
M.
Bär
,
Physica D
160
,
127
(
2001
).
4.
See, e.g., A. A. Nepomnyashchy, M. G. Velarde, and P. Colinet, Interfacial Phenomena and Convection (CRC-Chapman and Hall, Boca Raton, 2002), Chap. 7.
5.
H.
Chaté
,
A. S.
Pikovsky
, and
O.
Rudzick
,
Physica D
131
,
17
(
1999
).
6.
M.
Bär
,
M.
Hildebrand
,
M.
Eiswirth
,
M.
Falcke
,
H.
Engel
, and
M.
Neufeld
,
Chaos
4
,
499
(
1994
).
7.
Y.
Hayase
and
T.
Ohta
,
Phys. Rev. Lett.
81
,
1726
(
1998
).
8.
H.
Chaté
and
P.
Manneville
,
Phys. Rev. Lett.
58
,
112
(
1987
).
9.
M.
van Hecke
,
Phys. Rev. Lett.
80
,
1896
(
1998
).
10.
M. G.
Zimmermann
,
S. O.
Firle
,
M. A.
Natiello
,
M.
Hildebrand
,
M.
Eiswirth
,
M.
Bär
,
A. K.
Bangia
, and
I. G.
Kevrekidis
,
Physica D
110
,
92
(
1997
).
11.
V.
Petrov
,
S. K.
Scott
, and
K.
Showalter
,
Philos. Trans. R. Soc. London, Ser. A
347
,
631
(
1994
).
12.
L. A.
Lugiato
,
L. M.
Narducci
,
D. K.
Bandy
, and
C. A.
Pennise
,
Opt. Commun.
46
,
64
(
1983
);
see also
H. G.
Solari
and
G. L.
Oppo
,
Opt. Commun.
111
,
173
(
1994
).
13.
H. Meinhardt, The Algorithmic Beauty of Seashells (Springer, Berlin, 1995).
14.
J. Keener and J. Sneyd, Mathematical Physiology (Springer, Berlin, 1998).
15.
C.
Morris
and
H.
Lecar
,
Biophys. J.
35
,
193
(
1981
).
16.
D. P.
Vallette
,
G.
Jacobs
, and
J. P.
Gollub
,
Phys. Rev. E
55
,
4274
(
1997
).
17.
J. M.
Gambaudo
,
J. Diff. Eqns.
57
,
172
(
1985
).
18.
P.
Coullet
and
K.
Emilsson
,
Physica D
61
,
119
(
1992
);
P.
Coullet
and
K.
Emilsson
, see also
Physica A
188
,
190
(
1992
).
19.
S. Rasband, Chaotic Dynamics of Nonlinear Systems (Wiley, New York, 1990), pp. 130 and 131.
20.
O. Rudzick, Ph.D. thesis, Universität Potsdam, 1998.
21.
M. C.
Cross
and
P. C.
Hohenberg
,
Rev. Mod. Phys.
65
,
851
(
1993
).
22.
E. J.
Doedel
,
H. B.
Keller
, and
J. P.
Kernévez
,
Int. J. Bifurcation Chaos Appl. Sci. Eng.
1
,
493
(
1991
);
E. J.
Doedel
,
H. B.
Keller
, and
J. P.
Kernévez
,
Int. J. Bifurcation Chaos Appl. Sci. Eng.
1
,
745
(
1991
).
23.
Y.
Pomeau
,
Physica D
23
,
1
(
1986
).
24.
M.
Argentina
,
P.
Coullet
, and
L.
Mahadevan
,
Phys. Rev. Lett.
79
,
2803
(
1997
).
25.
M.
Argentina
,
P.
Coullet
, and
V.
Krinsky
,
J. Theor. Biol.
205
,
47
(
2000
).
26.
Y.
Nishiura
,
T.
Teramoto
, and
K.
Ueda
,
Chaos
13
,
962
(
2003
).
27.
M.
van Hecke
and
M.
Howard
,
Phys. Rev. Lett.
86
,
2018
(
2001
).
28.
Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer, Berlin, 1984).
29.
A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Theory of Bifurcation of Dynamical Systems on the Plane (Wiley, New York, 1971).
30.
M.
Or-Guil
,
J.
Krishnan
,
I. G.
Kevrekidis
, and
M.
Bär
,
Phys. Rev. E
64
,
046212
(
2001
).
This content is only available via PDF.
You do not currently have access to this content.