Determination Of Lyapunov Exponents Research Articles

Chaotic behavior in classical Yang-Mills dynamics

Understanding the underlying mechanisms causing rapid thermalization deduced for high-energy heavy ion collisions is still a challenge. To estimate the thermalization time, entropy growth for classical Yang-Mills theories is studied, based on the determination of Lyapunov exponents. Distinct regimes for short, medium and long sampling times are characterized by different properties of their spectrum of Lyapunov exponents. Clarifying the existence of these regimes and their implications for gauge-field dynamics is one of the results of this contribution. As a phenomenological application we conclude that for pure gauge theories with random initial conditions thermalization occurs within few fm/c, an estimate which can be reduced by the inclusion of fermions, specific initial conditions, etc.

Evaluating Noise Sensitivity on the Time Series Determination of Lyapunov Exponents Applied to the Nonlinear Pendulum

This contribution presents an investigation on noise sensitivity of some of the most disseminated techniques employed to estimate Lyapunov exponents from time series. Since noise contamination is unavoidable in cases of data acquisition, it is important to recognize techniques that could be employed for a correct identification of chaos. State space reconstruction and the determination of Lyapunov exponents are carried out to investigate the response of a nonlinear pendulum. Signals are generated by numerical integration of the mathematical model, selecting a single variable of the system as a time series. In order to simulate experimental data sets, a random noise is introduced in the signal. Basically, the analyses of periodic and chaotic motions are carried out. Results obtained from mathematical model are compared with the one obtained from time series analysis, evaluating noise sensitivity. This procedure allows the identification of the best techniques to be employed in the analysis of experimental data.

Chaotic motion of nonspherical particles settling in a cellular flow field

Small, spheroidal particles undergo a tumbling motion as they settle out under gravity through a fluid flow. This motion is in response to the fluid forces that act on the particle causing it to rotate due to the local fluid vorticity and rate of strain. In this paper the nonlinear system of differential equations governing the particle motion in a spatially periodic, cellular flow field is considered. The motion of spherical particles is completely regular, and in some regions of the flow they may be permanently suspended. Using recent results on volume-preserving maps, it is shown that this behavior can persist for nonspherical particles. More usually the tumbling motion is chaotic. This is characterized by results on Poincar\'e sections of the motion and determination of Lyapunov exponents. The response to flow vorticity and rate of strain is analyzed for both full three-dimensional motion and restricted planar motion to determine the main factors governing the chaotic motion.

REDUCTION ENTROPY

In this paper we study the numerical determination of the different reduction entropies (or α-entropies) of dynamical systems from time series. This method is advocated as a more suitable way to investigate the different aspects of sensitive dependence on initial positions than the determination of Lyapunov exponents, especially in the case of “noisy” time series.

Determination of Lyapunov exponents by weak solutions of Fokker-Planck equations

To determine the asymptotic stability of the zero position of a linear dynamic system one has to calculate the top Lyapunov exponent of the system. For this purpose, the state vector of the system is projected onto the unit hypersphere of Hasminskji's angle coordinates, which allows separation of the non-stationary solution part of the amplitude and representation of the top Lyapunov exponent in terms of the expectation values calculable from the density of the stationary angle processes. It turns out that these expected values are coefficients of orthogonal expansion series applied to the associated Fokker-Planck equation. Utilizing symbolic computation, the projection and the expansion can be performed automatically. The remaining numerical evaluation is restricted to the solution of weakly coupled linear equations.

Chaotic sedimentation of spheroidal particles in a cellular flow field

Chaotic mixing of Lagrangian fluid elements has been shown to be a feature of several unsteady two-dimensional flows or steady three-dimensional flows. There is also an interesting group of problems relating to the sedimentation of discrete particles, and for nonspherical particles which respond to the local flow conditions we have found the particle motion to be chaotic even in simple, steady two-dimensional flows. Specifically, we have studied the motion of rigid spheroidal particles settling under gravity in a spatially periodic, cellular flow field. The particles are small compared to the cell size and sufficiently small that their motion relative to the fluid satisfies conditions for Stokes flow. The position of the particle X(t) and the symmetry axis m are determined by (dX/dt)=u[X(t),t]+W1 (ĝ⋅m)m+W2 (ĝ−ĝ⋅mm), (dm/dt)=( (1)/(2) ω+Dm×E⋅m)×m. The flow field considered is u=(sin πx cos πy, −cos πx sin πy, 0) and ω and E are the corresponding local vorticity and rate of strain; W1 and W2 are the terminal fall speeds parallel and normal to the symmetry axis m, while ĝ is (0, 1, 0). The ratio W2/W1, and the parameter D are set by the aspect ratio of the particle, but for the purpose of analysis may be varied separately. Previous work1 has dealt with the restricted problem of planar motion, where m3=0, and given general numerical results on settling rates and particle suspension by the flow. Clear evidence was found from the trajectories of selected particles of chaotic motion in certain parts of the flow. More recently we have investigated in more detail the transition from regular to chaotic motion in this restricted problem. Based on perturbation analysis of periodic solutions, Poincaré sections, and determination of Lyapunov exponents we conclude that regions of regular motion where particles are suspended by the flow persist but the extent of these regions is greatly reduced for increasingly nonspherical shapes. Chaotic tumbling motion occurs outside these regions, as the particle turns in response to the local vorticity and rate of strain. Interestingly, chaotic motion is found even if the particle turns only in response to the local vorticity. Besides summarizing these results for the restricted problem, we present results for the unrestricted problem of three-dimensional motion which could be used for comparison with experiments. We find that the motion is qualitatively similar with both regular and chaotic regions.

Nonlinear and chaotic fluidelastic vibrations of a flexible pipe conveying fluid

The dynamics of a cantilevered pipe conveying fluid have been investigated, both theoretically and experimentally, focusing on the nonlinear effects introduced by motion-limiting constraints. The experiments have indicated that, with increasing flow velocity, beyond the Hopf bifurcation (flutter) there are regions of period-doubling and chaos. Fractal dimension calculations of the recorded vibration signals showed that finite-dimensionalmodelling of the system is possible. Accordingly, a two-degree-of-freedom, four-dimensional analytical model was implemented, with the aim of investigating the existence of chaotic vibrations in the parameter space of this autonomous system. Chaotic regions were indeed found to exist, with the aid of modern numerical techniques, involving the construction of bifurcation diagrams and the determination of Lyapunov exponents and power spectra. The analytical results are in qualitative agreement with experimental observations.