Existence Of Chaotic Attractors Research Articles

Chaotic solutions of the feedback driven Bloch equations

Recent acute manifestations of nonlinear spin behaviour in high field NMR spectroscopy of liquids have been observed. These were caused by collective effects, such as demagnetizing field effects or back action from the probe (radiation damping or radiation damping-based electronic feedback). These findings motivated the present study, in which the dynamics of a magnetization undergoing the effect of a radiation-damping based feedback field from the probe was numerically investigated. The existence of chaotic attractors is demonstrated and the type of transition to chaos investigated for various sets of parameters.

Using Lyapunov exponents to predict the onset of chaos in nonlinear oscillators.

An analytic technique for predicting the emergence of chaotic instability in nonlinear nonautonomous dissipative oscillators is proposed. The method is based on the Lyapunov-type stability analysis of an arbitrary phase trajectory and the standard procedure of calculating the Lyapunov characteristic exponents. The concept of temporally local Lyapunov exponents is then utilized for specifying the area in the phase space where any trajectory is asymptotically stable, and, therefore, the existence of chaotic attractors is impossible. The procedure of linear coordinate transform optimizing the linear part of the vector field is developed for the purpose of maximizing the stability area in the vicinity of a stable fixed point. By considering the inverse conditions of asymptotic stability, this approach allows formulating a necessary condition for chaotic motion in a broad class of nonlinear oscillatory systems, including many cases of practical interest. The examples of externally excited one- and two-well Duffing oscillators and a planar pendulum demonstrate efficiency of the proposed method, as well as a good agreement of the theoretical predictions with the results of numerical experiments. The comparison of the proposed method with Melnikov's criterion shows a potential advantage of using the former one at high values of dissipation parameter and/or multifrequency type of excitation in dynamical systems.

Resurrection of crushed magnetization and chaotic dynamics in solution NMR spectroscopy.

We show experimentally and theoretically that two readily observed effects in solution nuclear magnetic resonance (NMR)-radiation damping and the dipolar field-combine to generate bizarre spin dynamics (including chaotic evolution) even with extraordinarily simple sequences. For example, seemingly insignificant residual magnetization after a crusher gradient triggers exponential regrowth of the magnetization, followed by aperiodic turbulent spin motion. The estimated Lyapunov exponent suggests the onset of spatial-temporal chaos and the existence of chaotic attractors. This effect leads to highly irreproducible experimental decays that amplify minor nonuniformities such as temperature gradients. Imaging applications and consequences for other NMR studies are discussed.

On the existence of a low-dimensional chaotic attractor in the short-term solar UV time series

A mixture of periodic and chaotic components makes the detection of chaos difficult. The periodic components are sought in the solar UV time series by the Maximum Entropy Method and are removed by a digital notch filter. The filtered output is subjected to investigation for chaotic behavior by three different techniques. (1) Fixed-size method for attractor dimension determination; (2) sensitive initial dependence via prediction error; (3) trajectory direction estimation. All the investigation points to the existence of a chaotic attractor of fractional dimension.

Low-dimensional chaotic attractors in the rat brain.

The existence of chaotic attractors for discrete time series, derived from the occurrences of spikes during electrophysiological recordings, was investigated. The time series included between 800 and 5200 points per analyzed record. The spike trains were recorded in the substantia nigra pars reticulata (n = 13) and in the auditory thalamus (n = 14). The experiments were performed on anesthetized rats during spontaneous activity and during auditory stimulation. According to standard methods of dynamical systems theory, an embedding space was constructed using delay coordinates. The embedding and correlation dimensions were computed by means of the correlation integrals. For 7 of 27 samples, a deterministic structure with a low embedding dimension (ranging between 2 and 6) and a correlation dimension between 0.14 and 3.3 could be determined. Evidence was found that the sensory stimulation may affect the chaotic behavior. Single units recorded simultaneously from the same electrode tip may display different chaotic dynamics, even with a similar time-locked response to the stimulus onset.