P. We-acknowledge-discussions-with, O. Ern, and . Marquet, Auguste for providing the DNS data used for comparison, This work was supported by the Agence Nationale de la Recherche under Grant No. ANR-09- BLAN-0132 OBLIC

H. K. Moffatt, Three coins in a fountain, Journal of Fluid Mechanics, vol.720, issue.1, 2013.
DOI : 10.1017/jfm.2012.602

P. Ern, F. Risso, D. Fabre, and J. Magnaudet, Wake-Induced Oscillatory Paths of Bodies Freely Rising or Falling in Fluids, Annual Review of Fluid Mechanics, vol.44, issue.1, p.97, 2012.
DOI : 10.1146/annurev-fluid-120710-101250

U. Lacis, N. Brosse, N. Ingremeau, A. Mazzino, F. Lundell et al., Passive appendages generate drift through symmetry breaking, Nature Communications, vol.30, p.5310, 2014.
DOI : 10.1103/PhysRevLett.93.144501

URL : https://hal.archives-ouvertes.fr/hal-01091659

Y. Tanabe and K. Kaneko, Behavior of a Falling Paper, Physical Review Letters, vol.73, issue.10, p.1372, 1994.
DOI : 10.1103/PhysRevLett.73.1372

L. Mahadevan, Tumbling of a falling card, C. R. Acad. Sci. Paris, vol.323, p.729, 1996.

A. Belmonte, H. Eisenberg, and E. Moses, From Flutter to Tumble: Inertial Drag and Froude Similarity in Falling Paper, Physical Review Letters, vol.81, issue.2, p.345, 1998.
DOI : 10.1103/PhysRevLett.81.345

U. Pesavento and Z. J. Wang, Falling Paper: Navier-Stokes Solutions, Model of Fluid Forces, and Center of Mass Elevation, Physical Review Letters, vol.93, issue.14, p.144501, 2004.
DOI : 10.1103/PhysRevLett.93.144501

A. Andersen, U. Pesavento, and Z. J. Wang, Unsteady aerodynamics of fluttering and tumbling plates, Journal of Fluid Mechanics, vol.541, issue.-1, p.65, 2005.
DOI : 10.1017/S002211200500594X

P. Ern, F. Risso, P. C. Fernandes, and J. Magnaudet, Dynamical Model for the Buoyancy-Driven Zigzag Motion of Oblate Bodies, Physical Review Letters, vol.102, issue.13, p.134505, 2009.
DOI : 10.1103/PhysRevLett.102.134505

W. L. Shew and J. Pinton, Dynamical Model of Bubble Path Instability, Physical Review Letters, vol.97, issue.14, p.144508, 2006.
DOI : 10.1103/PhysRevLett.97.144508

G. Mougin and J. Magnaudet, Path Instability of a Rising Bubble, Physical Review Letters, vol.88, issue.1, p.14502, 2001.
DOI : 10.1103/PhysRevLett.88.014502

J. Tchoufag, J. Magnaudet, and D. Fabre, Linear instability of the path of a freely rising spheroidal bubble, Journal of Fluid Mechanics, vol.88, p.4, 2014.
DOI : 10.1146/annurev.fluid.32.1.659

URL : https://hal.archives-ouvertes.fr/hal-01015463

J. Tchoufag, D. Fabre, and J. Magnaudet, Global linear stability analysis of the wake and path of buoyancy-driven disks and thin cylinders, Journal of Fluid Mechanics, vol.48, p.278, 2014.
DOI : 10.1017/S0022112093002150

URL : https://hal.archives-ouvertes.fr/hal-00958683

D. Sipp and A. Lebedev, Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows, Journal of Fluid Mechanics, vol.6, p.333, 2007.
DOI : 10.1007/BF00127673

P. Meliga, J. Chomaz, and D. Sipp, Global mode interaction and pattern selection in the wake of a disk: a weakly nonlinear expansion, Journal of Fluid Mechanics, vol.102, p.159, 2009.
DOI : 10.1017/S0022112093002150

URL : https://hal.archives-ouvertes.fr/hal-01002592

J. Tchoufag, J. Magnaudet, and D. Fabre, Linear stability and sensitivity of the flow past a fixed oblate spheroidal bubble, Physics of Fluids, vol.25, issue.5, p.54108, 2013.
DOI : 10.1063/1.4804552

URL : https://hal.archives-ouvertes.fr/hal-00874779

F. Hecht, New development in freefem++, Journal of Numerical Mathematics, vol.20, issue.3-4, p.251, 2012.
DOI : 10.1515/jnum-2012-0013

D. Fabre, J. Tchoufag, and J. Magnaudet, The steady oblique path of buoyancy-driven disks and spheres, Journal of Fluid Mechanics, vol.707, p.24, 2012.
DOI : 10.1017/S0022112093002150

URL : https://hal.archives-ouvertes.fr/hal-00877022

P. Luchini and A. Bottaro, Adjoint Equations in Stability Analysis, Annual Review of Fluid Mechanics, vol.46, issue.1, p.493, 2014.
DOI : 10.1146/annurev-fluid-010313-141253

M. Golubitsky, I. Stewart, and D. Schaeffer, Singularities and Groups in Bifurcation Theory, 1988.

S. B. Field, M. Klaus, M. G. Moore, and F. Nori, Chaotic dynamics of falling disks, Nature, vol.388, issue.6639, p.252, 1997.
DOI : 10.1038/40817

P. C. Fernandes, F. Risso, P. Ern, and J. Magnaudet, Oscillatory motion and wake instability of freely rising axisymmetric bodies, Journal of Fluid Mechanics, vol.573, p.479, 2007.
DOI : 10.1017/S0022112006003685

F. Auguste, J. Magnaudet, and D. Fabre, Falling styles of disks, Journal of Fluid Mechanics, vol.312, p.388, 2013.
DOI : 10.1016/S0301-9322(02)00078-2

URL : https://hal.archives-ouvertes.fr/hal-00908124