Related papers: Renormalization in Lorenz maps -- completely invar…

The general term of the Poincare normalizing series is explicitly constructed for non-resonant systems of ODE's in a large class of equations. In the resonant case, a non-local transformation is found, which exactly linearizes the ODE's and…

The reconnection process in the dynamics of cubic nontwist maps, introduced in [3], is studied. The present paper extends the work presented in [8]. As in that work, in order to describe the route to reconnection of the involved…

We face the problem of characterizing the periodic cases in parametric families of (real or complex) rational diffeomorphisms having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the…

We study the dynamics of the renormalization operator for multimodal maps. In particular, we prove the exponential convergence of this operator for infinitely renormalizable maps with same bounded combinatorial type.

We introduce a renormalization model which explains how the behavior of a discrete-time continuous dynamical system changes as the dimension of the system varies. The model applies to some two-dimensional systems, including H\'enon and Lozi…

The critical behavior for intermittency is studied in two coupled one-dimensional (1D) maps. We find two fixed maps of an approximate renormalization operator in the space of coupled maps. Each fixed map has a common relavant eigenvaule…

We consider the Lorenz equations, a system of three dimensional ordinary differential equations modeling atmospheric convection. These equations are chaotic and hard to study even numerically, and so a simpler "geometric model" has been…

We present a general analysis of the bifurcation sequences of periodic orbits in general position of a family of reversible 1:1 resonant Hamiltonian normal forms invariant under $\Z_2\times\Z_2$ symmetry. The rich structure of these…

We generalize the two dimensional Lozi map in order to systematically obtain piece-wise continuous maps in three and higher dimensions. Similar to higher-dimensional generalizations of the related Henon map, these higher-dimensional Lozi…

When the Poincar\'{e} map associated with a periodic orbit of a hybrid dynamical system has constant-rank iterates, we demonstrate the existence of a constant-dimensional invariant subsystem near the orbit which attracts all nearby…

We introduce a class of infinitely renormalizable, unicritical diffeomorphisms of the disk (with a non-degenerate "critical point"). In this class of dynamical systems, we show that under renormalization, maps eventually become…

We present a geometric proof of the Poincar\'e-Dulac Normalization Theorem for analytic vector fields with singularities of Poincar\'e type. Our approach allows us to relate the size of the convergence domain of the linearizing…

Renormalizations can be considered as building blocks of complex dynamical systems. This phenomenon has been widely studied for iterations of polynomials of one complex variable. Concerning non-polynomial hyperbolic rational maps, a recent…

This paper will study topological, geometrical and measure-theoretical properties of the real Fibonacci map. Our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely…

We study dynamical systems that switch between two different vector fields depending on a discrete variable and with a delay. When the delay reaches a problem-dependent critical value so-called event collisions occur. This paper classifies…

We classify the possible behaviour of Poincar\'e-Dulac normal forms for dynamical systems in $R^n$ with nonvanishing linear part and which are equivariant under (the fundamental representation of) all the simple compact Lie algebras and…

The internal symmetry of composite relativistic systems is discussed. It is demonstrated that Lorentz-Poincar\'e symmetry implies the existence of internal moments associated with the Lorentz boost, which are Laplace-Runge-Lenz (LRL)…

We formulate the problem of renormalization of Feynman integrals and its relation to periods of motives in configuration space instead of momentum space. The algebro-geometric setting is provided by the wonderful compactifications of…

We study the dynamics of the Forced Logistic Map in the cylinder. We compute a bifurcation diagram in terms of the dynamics of the attracting set. Different properties of the attracting set are considered, as the Lyapunov exponent and, in…

Lorenz attractors play an important role in the modern theory of dynamical systems. The reason is that they are robust, i.e. preserve their chaotic properties under various kinds of perturbations. This means that such attractors can exist…