Premium
A global condition for quasi‐random behavior in a class of conservative systems
Author(s) -
Buffoni B.,
Séré E.
Publication year - 1996
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199603)49:3<285::aid-cpa3>3.0.co;2-9
Subject(s) - homoclinic orbit , transversality , mathematics , homoclinic bifurcation , hamiltonian system , heteroclinic orbit , chaotic , torus , saddle , manifold (fluid mechanics) , orbit (dynamics) , attractor , mathematical analysis , focus (optics) , pure mathematics , bifurcation , geometry , physics , computer science , nonlinear system , mechanical engineering , mathematical optimization , optics , quantum mechanics , artificial intelligence , engineering , aerospace engineering
Devaney has shown that an autonomous Hamiltonian system in dimension 4, with an orbit homoclinic to a saddle‐focus equilibrium, admits a chaotic behavior as soon as the homoclinic orbit is the transverse intersection of the stable and unstable manifolds. In this paper we deal with two classes of saddle‐focus systems: Lagrangian systems defined on a two‐manifold in the presence of a gyroscopic force, and fourth‐order systems arising in water‐wave theory. We first establish, by a standard variational method, the existence of a homoclinic orbit. Then, under a weak nondegeneracy condition, we show that it gives rise to an infinite family of multibump homoclinic solutions and that the dynamics are chaotic. Our condition is much easier to check than transversality. For example, it is automatically satisfied for gyroscopic systems on a two‐torus, for topological reasons. © 1996 John Wiley & Sons, Inc.