Positive Largest Lyapunov Exponent Research Articles

Experimental study of the nonlinear dynamics of a harmonically forced double layer

A plasma double layer is created by an anode grid in the target chamber of the double plasma device. When the separation grid of the double plasma device is floating, very coherent plasma oscillations, triggered by the unstable double layer, are observed. The oscillations have a frequency close to 1 kHz and very high amplitude. The amplitude and frequency of these oscillations depend on the anode bias and on the discharge voltage of the plasma source. The anode characteristics exhibit hysteresis. When plasma oscillations are coupled to the external oscillator, by modulating the floating potential of the separation grid, synchronization and periodic pulling are observed. When external oscillations are tuned to the second, third or fourth harmonics, period doubling bifurcations are observed, when the external amplitude is increased. At synchronization with the third and fourth harmonics, oscillations become chaotic after the first period doubling bifurcation. The correlation dimension of the underlying attractor is found to be fractal between 2.3 and 2.8, with a positive largest Lyapunov exponent.

Analysis of seismic dynamical systems

A procedure is presented for the analysis of complex stationary time series for which the Fourier power spectra reveals broadband noise or broadened pulses. We first determine the Hurst exponent from which we may know whether the time series under study is mainly random or if the data points present correlations. If the data are correlated, a chaotic analysis will reveal whether they may be interpreted as a low dimensional nonlinear system (defined by a low correlation dimension and a finite and positive Kolmogorov entropy and largest positive Lyapunov exponent) or as a stochastic process. We have studied three kind of temporal series: inter-event time series of infrasonic pulses recorded at Stromboli volcano, and, S-coda waves and microseisms, that have been recorded at the eastern Pyrenees. Results show that microseisms and Coda waves can be modeled as a low dimensional deterministic system, Correlation dimensions 2.3, 3.2, respectively. At the contrary infrasonic has resulted stochastic. This chaotic character can be attributed to the medium properties. Coda waves with scattering through a fractal distribution of scatters or to multiple reflection inside resonators (for example sedimentary basins) and microseisms as a propagation of wave guide of variable cross section which have the same temporal characteristics as a nonlinear forced oscillator.

Comment on “Strange nonchaotic attractors in autonomous and periodically driven systems”

The problem of the existence of strange nonchaotic attractors (SNA's) in autonomous systems is discussed. It is demonstrated that the recently reported example of a SNA in an autonomous system [V. S. Anishchenko et al., Phys. Rev. E 54, 3231 (1996)] is in fact a chaotic attractor with positive largest Lyapunov exponent.

Some possible evidence for a chaotic geomagnetic field from observational data

The behaviour of the geomagnetic field as observed almost continuously at three European locations over the last 130 years is investigated by means of a non-linear forecasting approach. The analysis of the data in terms of first-differences (secular variation) of the horizontal magnetic components made in phase space with the simplex technique seems to exclude the pre-eminence of any stochastic or periodic behaviour. The dimensionality of the underlying non-linear process and the corresponding largest positive Lyapunov exponent are estimated. The results give some evidence that the geomagnetic field evolves as a non-linear chaotic system with unpredictable behaviour after times greater than a few years, confirming the common practice of updating global models of the geomagnetic field every 5 years.

Linear and non-linear properties of heart rate in postnatal maturation

An investigation was made of how postnatal maturation of cardiac control can be described by linear and non-linear methods of time series analysis. Sixteen healthy and term newborns were studied during their first 6 months of life. Power spectrum analysis including total power (TP), low frequency power (LF), high frequency power (HF) and LF/HF ratio were performed on the instantaneous heart rate (IHR) time series and mean heart rate (HR) was derived. The largest Lyapunov exponent (LLE) was calculated using a modified Wolf algorithm and checked by means of the surrogate-data test. There is an age dependency for the parameters LLE, HR, TP, HF, LF, LF/HF for active and LLE, HR, TP, HF, LF/HF for quiet sleep. HR is characterised by a steep increase between the 5th and 7th day. HF demonstrates a distinct development with high values around the first week and 90-180th day and low values around the 10-60th day. TP, LF and LF/HF show significantly higher values in active sleep in comparison to quiet sleep. For all ages and sleep stages, positive LLE were found, indicating sensitivity to initial conditions, a hall-mark of chaos. For the period between the 7 and 90th days of life, the LLE for active sleep took on larger values compared with the LLE of quiet sleep. This study shows that by linear as well as non-linear analysis one can reveal the complexity of the IHR development in humans and may gain insight into the system controlling the heart during the period considered. The positive LLE indicate that there is a non-linear component in the heart rate control. There is no "straight line" development for the parameters analysed within the first 6 months. This may result from manifold influences on the autonomic system, due to structural and functional maturation in this period of life.

Predictability experiments on a simplified thermal convection model: The role of spatial scales

The amplification of small initial errors in connection with mesoscale atmospheric variability is studied on a simplified thermal convection model. It is found that when the system is perturbed uniformly over the physical space, the initial regime of error dynamics follows a pathway far from the traditional exponential function whose rate is the largest positive Lyapunov exponent. It comprises two stages: (1) a transient decrease of the initial error as a result of the displacement of the system out of its attractor and (2) a superexponential increase whose origin is in the multifractal character of the attractor. Numerical experiments in which the initial error is introduced selectively in a particular scale reveal that perturbations at larger scales grow faster than those at the intermediate and small scales. A qualitative explanation of this unexpected behavior based on intrinsic properties of the dynamics such as the local Lyapunov vectors is advanced.

Chaos in counter-rotating Couette flow

A quasi-periodic transition to chaos has been observed experimentally in counter-rotating Taylor-Couette flow. This scenario occurs when a circular path is followed around a multicritical point in parameter space. (The multicritical point results from the nonlinear interaction between azimuthal wavenumbers m = 2 and m = 3.) To complement the experiments, the reduced amplitude equations that describe this mode interaction have been solved numerically. The numerical solutions follow a transition to chaos similar to the one observed experimentally. Both experimental and numerical chaotic flows are characterized by broadband spectra, low fractal dimension and a positive largest Lyapunov exponent. This result indicates that the chaos observed experimentally might be due to the mode interaction, and that the mathematical model might be appropriate to describe chaotic behavior in real flows.

A chaotic limit cycle paradox

Duffing's equation with sinusoidal forcing produces chaos for certain combinations of the forcing amplitude and frequency. To determine the most chaotic response achieveable for given bounds on the input force, an optimal control problem was investigated to maximize the largest Lyapunov exponent, which in this case also corresponds to maximizing the Kaplan-Yorke Lyapunov fractal dimension. The resulting bang-bang optimal feedback controller yielded a bounded attractor with a positive largest Lyapunov exponent and a fractional Lyapunov dimension, indicating a chaotic strange attractor. Indeed, the largest Lyapunov exponent was approximately twice as large as that achieved with sinusoidal forcing at the same amplitude. However, the resulting attractor is just a stable limit cycle and is not chaotic or fractal at all! this contradicts the basic idea that a bounded attractor with at least one positive Lyapunov exponent must be chaotic and fractal.

Investigations of attractors arising from the interaction of drift waves and potential relaxation instabilities

The nonlinear interaction between ion-drift waves and electrostatic potential relaxation instabilities (PRI) is studied in a magnetized double plasma with externally induced axial electron drift. The PRI is identified as unstable weak double layer which is determined by additional electron beam injection. The axial drift current Id is used as control parameter for the dynamical state of the plasma and shows a critical transition to anomalous resistivity. For various drift current values, time series of probe signal fluctuations have been recorded. The wave number and bispectral analysis is used to establish the interaction as nonresonant modulation of drift waves by PRI’s. The complexity of strongly nonlinear and fully turbulent regimes is investigated by statistical analysis of the reconstructed phase space. The nonlinear interaction is characterized by low correlation dimension values between 3≤D2≤5 and positive largest Lyapunov exponents. The results confirm earlier findings, which were obtained in a different device.

Quasiperiodic and chaotic oscillations of magnon systems driven by modulated pumping field

Parallel-pumping experiments in YIG have been performed with modulated pumping fields. Above the spin-wave instability threshold, frequency locking states and a quasiperiodic route to chaos are observed. In a quasiperiodic state, an obtained attractor is a torus. Above a transition to a chaotic state, a breakup of the torus occurs. And a positive largest Lyapunov exponent, which is one of quantitative measures of chaos, is obtained in the chaotic regime.