The mechanism of the exponential transient statistics of Poincaré recurrences in the presence of chaos border with its critical structure is studied using two simple models: separatrix map and the kicked rotator ('microtron'). For the exponential transient to exist the two conditions have been shown to be crucial: fast (ballistic) relaxation, and a small measure of the critical structure. The l...

In this paper, a mathematical analysis of a possible way to chaos (in the sense of Li and Yorke) for bounded piecewise smooth systems of dimension three submitted to nonsmooth transitions is proposed. This study is based on period doubling method applied to the relied Poincaré map defined on a Poincaré section chosen in the neighborhood of the bifurcation point and transverse to the switching m...

any (continuous) map N → M between stably parallelizable compact n-manifolds, n 6= 1, 2, 3, 7, is realizable in R, i.e. the composition of f with an embedding M ⊂ R is C-approximable by embeddings. It has been long believed that any degree 2 map S → S, obtained by capping off at infinity a time-symmetric (e.g. Shapiro’s) sphere eversion S×I → R, was non-realizable in R. We show that there exist...

Characteristics of the Poincaré return times are considered in a one-dimensional cubic map with a chaotic nonhyperbolic attractor. Two approaches, local one (Kac’s theorem) and global one related with the AP-dimension estimation of return times, are used. The return times characteristics are studied in the presence of external noise. The characteristics of Poincaré recurrences are compared with...

We give a detailed and mainly geometric proof of a theorem by N.N. Nekhoroshev for hamiltonian systems in n degrees of freedom with k constants of motion in involution, where 1 ≤ k ≤ n. This states persistence of k-dimensional invariant tori, and local existence of partial action-angle coordinates, under suitable nondegeneracy conditions. Thus it admits as special cases the Poincaré-Lyapounov t...

We study integral operators related to a regularized version of the classical Poincaré path integral and the adjoint class generalizing Bogovskiı̆’s integral operator, acting on differential forms in R. We prove that these operators are pseudodifferential operators of order −1. The Poincaré-type operators map polynomials to polynomials and can have applications in finite element analysis. For a ...

An ergodic study of Painlevé VI is developed. The chaotic nature of its Poincaré return map is established for almost all loops. The exponential growth of the numbers of periodic solutions is also shown. Principal ingredients of the arguments are a moduli-theoretical formulation of Painlevé VI, a Riemann-Hilbert correspondence, the dynamical system of a birational map on a cubic surface, and th...

It is known that unstable periodic orbits of a given map give information about the natural measure of a chaotic attractor. In this work we show how these orbits can be used to calculate the density function of the first Poincaré returns. The close relation between periodic orbits and the Poincaré returns allows for analytical and semi-analytical estimations of relevant quantities in dynamical ...

We study the incompressible Navier–Stokes equations with potential body forces on the three-dimensional torus. We show that the normalization introduced in the paper Ann. Inst. H. Poincaré Anal. Non Linéaire, 4(1):1–47, 1987, produces a Poincaré–Dulac normal form which is obtained by an explicit change of variable. This change is the formal power series expansion of the inverse of the normaliza...

For a Poincaré duality space X and a map X → B, consider the homotopy fiber product X × X . If X is orientable with respect to a multiplicative cohomology theory E, then, after suitably regrading, it is shown that the E-homology of X × X has the structure of a graded associative algebra. When X → B is the diagonal map of a manifold X , one recovers a result of Chas and Sullivan about the homolo...