Ice Crystals Inside the Bell
Identification of discrete chaotic maps with singular points.
Identification of periodic and cyclic fractional stable motions
We consider an important subclass of self-similar, non-gaussian stable processes with stationary increments known as self-similar stable mixed moving averages. As previously shown by the authors, following the seminal approach of Jan Rosiński, these processes can be related to nonsingular flows through their minimal representations. Different types of flows give rise to different classes of self-similar mixed moving averages, and to corresponding general decompositions of these processes. Self-similar...
Identification of quasiperiodic processes in the vicinity of the resonance
In nonlinear dynamical systems, strong quasiperiodic beating effects appear due to combination of self-excited and forced vibration. The presence of symmetric or asymmetric beatings indicates an exchange of energy between individual degrees of freedom of the model or by multiple close dominant frequencies. This effect is illustrated by the case of the van der Pol equation in the vicinity of resonance. The approximate analysis of these nonlinear effects uses the harmonic balance method and the multiple...
Identification of stochastic loads applied to a nonlinear dynamical system using an uncertain computational model.
Identifying generalized Fitzhugh-Nagumo equation from a numerical solution of Hodgkin-Huxley model.
Identifying points of a pseudo-Anosov homeomorphism
We investigate the question, due to S. Smale, of whether a hyperbolic automorphism T of the n-dimensional torus can have a compact invariant subset homeomorphic to a compact manifold of positive dimension, other than a finite union of subtori. In the simplest case such a manifold would be a closed surface. A result of Fathi says that T can sometimes have an invariant subset which is a finite-to-one image of a closed surface under a continuous map which is locally injective except possibly at a finite...
If the [T,Id] automorphism is Bernoulli then the [T,Id] endomorphism is standard
For any 1-1 measure preserving map T of a probability space we can form the [T,Id] and automorphisms as well as the corresponding endomorphisms and decreasing sequence of σ-algebras. In this paper we show that if T has zero entropy and the [T,Id] automorphism is isomorphic to a Bernoulli shift then the decreasing sequence of σ-algebras generated by the [T,Id] endomorphism is standard. We also show that if T has zero entropy and the [T²,Id] automorphism is isomorphic to a Bernoulli shift then the...
Imbalances in Arnoux-Rauzy sequences
In a 1982 paper Rauzy showed that the subshift generated by the morphism , and is a natural coding of a rotation on the two-dimensional torus , i.e., is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in each domain being translated by the same vector modulo a lattice. It was believed more generally that each sequence of block complexity satisfying a combinatorial criterion known as the condition of Arnoux and Rauzy codes the orbit of a point...
Immediate and Virtual Basins of Newton’s Method for Entire Functions
We investigate the well known Newton method to find roots of entire holomorphic functions. Our main result is that the immediate basin of attraction for every root is simply connected and unbounded. We also introduce “virtual immediate basins” in which the dynamics converges to infinity; we prove that these are simply connected as well.
Immersed spheres in symplectic 4-manifolds
We discuss conditions under which a symplectic 4-manifold has a compatible Kähler structure. The theory of -holomorphic embedded spheres is extended to the immersed case. As a consequence, it is shown that a symplectic 4-manifold which has two different minimal reductions must be the blow-up of a rational or ruled surface.
Implicit Differential Equations From the Singularity Theory Viewpoint
Impulsivity in binary choices and the emergence of periodicity.
Incompressibilité des feuilles de germes de feuilletages holomorphes singuliers
Nous considérons un germe de feuilletage holomorphe singulier non-dicritique défini sur une boule fermée , satisfaisant des hypothèses génériques, de courbe de séparatrice . Nous démontrons l’existence d’un voisinage ouvert de dans tel que, pour toute feuille de , l’inclusion naturelle induit un monomorphisme au niveau du groupe fondamental. Pour cela, nous introduisons la notion géométrique de « connexité feuilletée » avec laquelle nous réinterprétons la notion d’incompressibilité....
Index estimates and critical points of functionals not satisfying Palais-Smale
Index filtrations and Morse decompositions for discrete dynamical systems
On a Morse decomposition of an isolated invariant set of a homeomorphism (discrete dynamical system) there are partial orderings defined by the homeomorphism. These are called admissible orderings of the Morse decomposition. We prove the existence of index filtrations for admissible total orderings of a Morse decomposition and introduce the connection matrix in this case.
Index pairs and the fixed point index for semidynamical systems with discrete time
Induced expansion for quadratic polynomials
Induced subsystems associated to a Cantor minimal system
Let (X,T) be a Cantor minimal system and let (R,) be the associated étale equivalence relation (the orbit equivalence relation). We show that for an arbitrary Cantor minimal system (Y,S) there exists a closed subset Z of X such that (Y,S) is conjugate to the subsystem (Z,T̃), where T̃ is the induced map on Z from T. We explore when we may choose Z to be a T-regular and/or a T-thin set, and we relate T-regularity of a set to R-étaleness. The latter concept plays an important role in the study of...
Inductorless Chua's circuit: experimental time series analysis.