Floris Takens

Singularities of vector fields

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Occurrence of strange AxiomA attractors near quasi periodic flows onTm,m≧3

It is shown that by a smallC2 (resp.C∞) perturbation of a quasiperiodic flow on the 3-torus (resp. them-torus,m>3), one can produce strange AxiomA attractors. Ancillary results and physical interpretation are also discussed.

BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS

We consider one parameter families or arcs of diffeomorphisms. For families starting with Morse-Smale diffeomorphisms we characterize various types of (structural) stability at or near the first bifurcation point. We also give a complete description of the stable arcs of diffeomorphisms whose limit sets consist of finitely many orbits. Universal models for the local unfoldings of the bifurcating periodic orbits (especially saddle-nodes) are established, as well as several results on the global dynamical structure of the bifurcating diffeomorphisms. Moduli of stability related to saddle-connections are introduced.

Reversibility as a criterion for discriminating time series

Abstract We propose a test for the hypothesis that a time series is reversible. If reversibility can be rejected all static transformations of linear Gaussian random processes can be excluded as a model for the time series.

Partially hyperbolic fixed points

WE COSSIDER 3 C”-diffeomorphism 10 : W” 432” with r?(O) = 0. The differential dq ( T,(W) induces a splitting T,(P) = T’ @ T’ @ T”, where T’. T’ and T” are invariant under dq and the eigenvalues of dq, restricted to T’, resp. T”, resp. T”, are, in absolute value,= 1, resp. < 1, resp. > 1. The fixed point 0 of IJJ is called hyperbolic if dim(T’) = 0. We shall consider the parfiall,v h,vperbolic case where dim(T’) # 0 and dim(T’ @ T”) # 0. Such partially hyperbolic fixed points arise for example as fixed points of the time t integral of a vectorfield Lvith a generic closed orbit with period f.

Dynamical systems and chaos

Examples and definitions of dynamical phenomena.- Qualitative properties and predictability of evolutions.- Persistence of dynamical properties.- Global structure of dynamical systems.- On KAM Theory.- Reconstruction and time series analysis.

Hyperbolicity and the creation of homoclinic orbits

We consider one-parameter families {sp,; t E R} of diffeomorphisms on surfaces which display a homoclinic tangency for yt = 0 and are hyperbolic for yt < 0 (i.e., (pf has a hyperbolic nonwandering set); the tangency unfolds into transversal homoclinic orbits for yt positive. For many of these families, we prove that (p is also hyperbolic for most small positive values of yt (which implies much regularity of the dynamical structure). A main assumption concerns the limit capacities of the basic set corresponding to the homoclinic tangency.

DETECTING NONLINEARITIES IN STATIONARY TIME SERIES

In this review we survey methods for detecting nonlinearities in stationary time series. These methods are based on the estimation of so-called correlation integrals. These correlation integrals provide a way of analyzing time series and reveal aspects which are often complementary to the information one obtains from power spectra and autocorrelations. So we also focus our attention on the meaning and the estimation of the correlation integrals.

Learning Chaotic Attractors by Neural Networks

An algorithm is introduced that trains a neural network to identify chaotic dynamics from a single measured time series. During training, the algorithm learns to short-term predict the time series. At the same time a criterion, developed by Diks, van Zwet, Takens, and de Goede (1996) is monitored that tests the hypothesis that the reconstructed attractors of model-generated and measured data are the same. Training is stopped when the prediction error is low and the model passes this test. Two other features of the algorithm are (1) the way the state of the system, consisting of delays from the time series, has its dimension reduced by weighted principal component analysis data reduction, and (2) the user-adjustable prediction horizon obtained by error propagationpartially propagating prediction errors to the next time step. The algorithm is first applied to data from an experimental-driven chaotic pendulum, of which two of the three state variables are known. This is a comprehensive example that shows how well the Diks test can distinguish between slightly different attractors. Second, the algorithm is applied to the same problem, but now one of the two known state variables is ignored. Finally, we present a model for the laser data from the Santa Fe time-series competition (set A). It is the first model for these data that is not only useful for short-term predictions but also generates time series with similar chaotic characteristics as the measured data.

Formally Symmetric Normal Forms and Genericity

We consider several classes of dynamical systems on manifolds, like Hamiltonian systems, volume preserving systems, etc. On each of these classes there is a more or less natural topology. The word generic is used for properties of dynamical systems which hold for almost all elements, in the topological sense, of a given class. Formal definitions will be given below. Generic properties are known which imply a certain local simplicity of the system, e.g. that fixed points or equilibria are isolated. Still for such a system the global dynamics can be very complicated.