We present an approach that allows performing computations related to the Baker-Campbell-Haussdorff (BCH) formula and its generalizations in an arbitrary Hall basis, using labeled rooted trees. In particular, we provide explicit formulas (given in terms of the structure of certain labeled rooted trees) of the continuous BCH formula. We develop a rewriting algorithm (based on labeled rooted tree...

We consider perturbations of integrable, area preserving non-twist maps of the annulus (those are maps in which the twist condition changes sign). These maps appear in a variety of applications, notably transport in atmospheric Rossby waves. We show in suitable 2-parameter families the persistence of critical circles (invariant circles whose rotation number is the maximum of all the rotation nu...

Dynamic Topological Logic provides a context for studying the confluence of the topological semantics for S4, based on topological spaces rather than Kripke frames; topological dynamics; and temporal logic. In the topological semantics for S4, 2 is interpreted as topological interior: thus S4 can be understood as the logic of topological spaces. Topological dynamics studies the asymptotic prope...

Let ƒ: N —• M be a C°° map of oriented compact manifolds, and let L be an oriented closed submanifold of codimension q > 1 in M. If w is a closed form Poincaré dual to L, we show that f~L, with multiplicities counted, is Poincaré dual to ƒ *w in N and is even meaningful on a "secondary" level. This leads to generalized versions of the Hopf invariant, the Hopf index theorem and the Bezout theore...

In this paper we will find a continuous of periodic orbits passing near infinity for a class of polynomial vector fields in R3. We consider polynomial vector fields that are invariant under a symmetry with respect to a plane Σ and that possess a “generalized heteroclinic loop” formed by two singular points e+ and e at infinity and their invariant manifolds Γ and Λ. Γ is an invariant manifold of...

The angular momentum of light can be described by positions on a higher-order Poincaré sphere, where superpositions of spin and orbital angular momentum states give rise to laser beams that have many applications, from microscopy to materials processing. Many techniques exist to create such beams but none so far allow their creation at the source. Here we report on a new class of laser that is ...

We show that for planar dispersing billiards the return times distribution is in the limit Poisson for metric balls almost everywhere w.r.t. the SRB measure. Since the Poincaré return map is piecewise smooth but becomes singular at the boundaries of the partition elements, recent results on the limiting distribution of return times cannot be applied as they require the maps to have bounded seco...

In this study, a chaotic circuit suitable for an integrated circuit is proposed. The circuit consists of two CMOS ring oscillators and a pair of diodes. By using a simplified model of the circuit, the mechanism of generating chaos is explained and the exact solutions are derived. The exact expressions of the Poincaré map and its Jacobian matrix make those possible to confirm the generation of c...

We prove that the Frobenius–Perron operator of the cusp map F : [−1, 1] → [−1, 1], F (x) = 1 − 2 √ |x| (which is an approximation of the Poincaré section of the Lorenz attractor) has no analytic eigenfunctions corresponding to eigenvalues different from 0 and 1. We also prove that the spectrum of the Frobenius–Perron operator in the space of real-analytic functions on [0, 1] as well as in the s...