Differentiable dynamical systems

Abstract

1.1. Introduction to conjugacy problems for diffeomorphisms. This is a survey article on the area of global analysis defined by differentiable dynamical systems or equivalently the action (differentiable) of a Lie group G on a manifold M. An action is a homomorphism G->Diff(M) such that the induced map GXM->M is differentiable. Here Diff(M) is the group of all diffeomorphisms of Mand a diffeomorphism is a differentiable map with a differentiable inverse. Everything will be discussed here from the C 00 or C r point of view. All manifolds maps, etc. will be differentiable (C r , l^r^co) unless stated otherwise.In the beginning we will be restricted to the discrete case, G = Z. Here Z denotes the integers, Z + the positive integers. By taking a generator ƒ G Diff(ikT), this amounts to studying diffeomorphisms on a manifold from the point of view of orbit structure. The orbit of %(EM, relative to/, is the subset {f m (x) \m^Z} of M or else the map Z->M which sends m into f m (x). The finite orbits are called periodic orbits and their points, periodic points. Thus xÇ£M is a periodic point if f m (x)=x for some w£Z + , Here m is called a period of x and if m = l, x is Si fixed point. Our problem is to study the global orbit structure, i.e,, all of the orbits on M.The main motivation for this problem comes from ordinary differential equations, which essentially corresponds to G~R> R the reals acting on M. There are two reasons for this leading to the diffeomorphism problem. One is that certain differential equations have cross-sections (see, e.g., [114]) and in this case the qualitative study of the differential equation reduces to the study of an associated diffeomorphism of the cross-section. This is the reason why Poincaré [90] and Birkhoff [19] studied diffeomorphisms of surfaces, I believe there is a second and more important reason for studying the diffeomorphism problem (besides its great natural beauty). That is, the same phenomena and problems of the qualitative theory of ordinary differential equations are present in their simplest form in the diffeomorphism problem. Having first found theorems in the dif-

Hyperbolic chaotic attractor in amplitude dynamics of coupled self-oscillators with periodic parameter modulation

“…(8). This confirms the applicability of the slow-amplitude method in the operation modes we deal with.…”

“…Attractors of this type occur in systems of the so-called axiom A class and are considered in the hyperbolic theory [8][9][10][11][12][13][14][15][16][17]. The chaotic nature of dynamics on these attractors is proved rigorously.…”

“…In textbooks and reviews, examples of the uniformly hyperbolic attractors are traditionally represented by artificial mathematical constructions [8][9][10][11][12][13][14][15]20]. One is the SmaleWilliams solenoid, which can occur in the phase space of maps of dimension 3 or more.…”

Hyperbolic Chaos of Turing Patterns

“…PACS numbers: 89.75.Kd, 05.45.Jn In nonlinear dynamics the notion of structural stability, or robustness, is one of the key tools allowing one to specify systems and effects that are really significant for theoretical and numerical researches, and especially for practical applications [1,2]. Among chaotic attractors, structural stability is intrinsic to those possessing the uniform hyperbolicity ("the systems with axiom A"), mathematical examples of which were advanced already since 60's -70's [3][4][5][6]. That time, such attractors were expected to be relevant for various physical situations (such as hydrodynamic turbulence), but later it became clear that the chaotic attractors, which normally occur in applications, do not relate to the class of structurally stable ones.…”

Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom

“…Here we merely note that the Smale-Birkhoff homoclinic theorem asserts the existence, near any transverse homoclinic point, of a zero dimensional invariant Cantor set A on which some power of the map, Pf , is homeomorphic to a shift on [Moser (1973)] that (3.5) possesses no analytic second integral. Also see Smale (1967).…”