Analytic Dynamical Systems Research Articles

Stable discrete observation programs in analytic dynamical systems

For analytic dynamical systems, we consider the problem of finding discrete observation programs providing unique reconstruction of the motion and stable under small perturbations. A relationship is established between these programs and finite bases of observable functions. The results are based on a description of ideals in the ring of germs of analytic functions at a point and the theory of complex analytic sets.

Analyticity, scaling and renormalization for some complex analytic dynamical systems

We review some results about the analytic structure of Lindstedt series for some complex analytic dynamical systems: in particular, we consider Hamiltonian maps (like the standard map and its generalizations), the semi-standard map and Siegel's problem of the linearization of germs of holomorphic diffeomorphisms of ( C ,0). The analytic structure of those series can be studied numerically using Padé approximants, and one can show the existence of natural boundaries for real, diophantine values of the rotation number; by complexifying the rotation number, we show how these natural boundaries arise from the accumulation of singularities due to resonances, providing a new intuitive insight into the mechanism of the break-down of invariant KAM curves. Moreover, we study the Lindstedt series at resonances, i.e. for rational values of the rotation number, by suitably rescaling to 0 the value of the perturbative parameter, and a simple analytic structure emerges. Finally, we present some proofs for the simplest models and relate these results to renormalization ideas.

Generalized Counterexamples to the Seifert Conjecture

Using the theory of plugs and the self-insertion construction due to the second author, we prove that a foliation of any codimension of any manifold can be modified in a real analytic or piecewise-linear fashion so that all minimal sets have codimension 1. In particular, the 3-sphere S^3 has a real analytic dynamical system such that all limit sets are 2-dimensional. We also prove that a 1-dimensional foliation of a manifold of dimension at least 3 can be modified in a piecewise-linear fashion so that there are no closed leaves but all minimal sets are 1-dimensional. These theorems provide new counterexamples to the Seifert conjecture, which asserts that every dynamical system on S^3 with no singular points has a periodic trajectory.

The differential field associated to a general analytic nonlinear dynamical system

A rigorous and detailed construction of the differential field K/sub Sigma /(u,y) associated, in a differential-algebraic context, to a nonlinear system Sigma described in state-space form with analytic coefficients is presented. The field K/sub Sigma /(u,y) represents an input-output description of Sigma , and its study is of key importance in problems concerning the input-output behavior of Sigma . Technically, the basic step of the construction of K/sub Sigma /(u,y) consists in proving that the differential ideal associated to the state-evolution equation of Sigma is prime. >

Dynamical systems and stability: The improvements given by the near-resonance theorem

The near-resonance theorem represents a major advance in the analysis of analytic dynamical systems. It leads to great simplifications in the study of the vicinity and the stability of periodic solutions, it leads also to the following. 1. (i) The asymptotic solutions lying in the stable and unstable subsets are integrable and simple. 2. (ii) Almost all periodic, “quasi-periodic” and “almost-periodic” solutions of conservative systems (such as the quasi-periodic solutions of the “Arnold tori”) are without neighbouring asymptotic solutions and have the first-order stability: all their Liapunov characteristic exponents are zero. Conversely almost all bounded solutions with one or several positive Liapunov characteristic exponents are chaotic. 3. (iii) In analytic Hamiltonian problems the near-resonance theorem leads to the notion of quasi-integral (i.e. state functions without secular variations to any order) and gives a sign to the eigenfrequencies. In most critical cases the quasi-integrals lead to the “all-order stability”, i.e. either the true stability or the Arnold diffusion. However there exist destabilizing resonances, the “positive resonances” related to the signs of the eigenfrequencies.

The continuous, desingularized Newton method for meromorphic functions

For any (nonconstant) meromorphic function, we present a real analytic dynamical system, which may be interpreted as an infinitesimal version of Newton's method for finding its zeros. A fairly complete description of the local and global features of the phase portrait of such a system is obtained (especially, if the function behaves not too bizarre at infinity). Moreover, in the case of rational functions, structural stability aspects are studied. For a generic class of rational functions, we give a complete graph-theoretical characterization, resp. classification, of these systems. Finally, we present some results on the asymptotic behaviour of meromorphic functions.

On the quasiconformal surgery of rational functions

The method of quasiconformal surgery for rational functions, considered as complex analytic dynamical systems, is developed. This applied to give the sharpest estimates for the numbers of cycles of stable regions corresponding to D. Sullivan's classification. As another application, rational functions having Herman rings are constructed.

Unstable manifolds of analytic dynamical systems

Unstable manifolds of analytic dynamical systems

STABLE AND OSCILLATING MOTIONS IN NONAUTONOMOUS DYNAMICAL SYSTEMS. A GENERALIZATION OF C. L. SIEGEL'S THEOREM TO THE NONAUTONOMOUS CASE

In this paper we generalize to the nonautonomous case a theorem of C. L. Siegel on the reducibility of an analytic dynamical system to normal form in a neighborhood of an equilibrium point. In fact, under certain concrete assumptions with respect to the behavior of the system as t→∞, we show that in a neighborhood of an equilibrium we can reduce the system to a linear system by means of a change of coordinates that depends on the time t and is analytic in the remaining variables. The results obtained are applicable to the problem of the stability of an equilibrium point. Bibliography: 16 items.

Motion of two weakly coupled nonlinear oscillators

Abstract : The purpose of this investigation was to establish the existence of almost periodic solutions for analytic dynamical systems consisting of two weakly coupled oscillators and to describe some qualitative features of these solutions. Hamiltonian formalism was used in the mathematics of this dynamical problem to realize simplicity of presentation and to agree with much of the literature concerning the problem. The method consisted primarily in the application of an invariant curve theorem due to Moser. The basic problem was given in terms of a Hamiltonian with the usual displacement and momentum variables. By changing to action-angle variables, by selecting a particular value of the energy and by eliminating time from the equations of motion, the study of the free oscillations of the two weakly coupled oscillators was reduced to the consideration of a one dimensional oscillator which is weakly forced with periodic forcing.