Normal forms, nonlocal chaotic behaviour in sportt and NMR systems

Abstract

Poincaré's theory of normal forms is applied to a number of simple chaotic Sprott flows that have resonant eigenvalues. It is shown that the normal form expansion can give significant information not limited to the local properties of chaotic attractors, but also, on nonlocal properties such as positive and zero Liapunov exponents for systems that have the Hopf bifurcation property. Existence of a zero Liapunov exponent is indicated if the system has hyperbolic fixed points. The method is not directly applicable where an eigenvalue of the linearized part vanishes, because of the complexity of the normal form. Rational transformations that change the eigenvalue spectrum of the linearized parts are employed on the Sprott C and E systems to obtain simpler systems. A proposel on the possible use of fractal analysis methods on functional MRI data and preliminary results on possible source of chaotic behavior inherent in nuclear spin systems are presented.