Lyapunov exponents in unstable systems

We investigate the dynamical behavior of unstable systems in the vicinity of the critical point associated with a liquid-gas phase transition. By considering a mean-field treatment, we first perform a linear analysis and discuss the instability growth times. Then, coming to complete Vlasov simulations, we investigate the role of nonlinear effects and calculate the Lyapunov exponents. As a main result, we find that near the critical point, the Lyapunov exponents exhibit a power-law behavior, with a critical exponent beta=0.5. This suggests that in thermodynamical systems the Lyapunov exponent behaves as an order parameter to signal the transition from the liquid to the gas phase.     

Lyapunov exponents of stochastic dynamical systems

It is shown that stochastic equations can have stable solutions. In particular, there exists stochastic dynamics for which the motion is both ergodic and stable, so that all trajectories merge with time. We discuss this in the context of Monte Carlo-type dynamics, and study the convergence of nearby trajectories as the number of degrees of freedom goes to infinity and as a critical point is approached. A connection with critical slowdown is suggested.     

Negative Lyapunov exponents for dissipative systems

By analyzing time-reversed trajectories from irreversible dissipative systems, we effectively reverse the order and signs of all the Lyapunov exponents. This reversal makes it possible to obtain the most negative Lyapunov exponents relatively easily. We illustrate the validity of this idea by studying the Lorenz model of Rayleigh-Bénard instability.     

Mixing and Lyapunov exponents of chaotic systems

Mixing of phase trajectories in chaotic systems is considered. The mixing rate is analytically estimated, and an algorithm for its computation in numerical experiment is descried. A relation between the local mixing rate and local Lyapunov exponents is considered. The applicability of the algorithm is demonstrated with a number of chaotic systems.     

On finite-size Lyapunov exponents in multiscale systems

We study the effect of regime switches on finite size Lyapunov exponents (FSLEs) in determining the error growth rates and predictability of multiscale systems. We consider a dynamical system involving slow and fast regimes and switches between them. The surprising result is that due to the presence of regimes, the error growth rate can be a non-monotonic function of initial error amplitude. In particular, troughs in the large scales of FSLE spectra are shown to be a signature of slow regimes, whereas fast regimes are shown to cause large peaks in the spectra where error growth rates far exceed those estimated from the maximal Lyapunov exponent. We present analytical results explaining these signatures and corroborate them with numerical simulations. We show further that these peaks disappear in stochastic parametrizations of the fast chaotic processes, and the associated FSLE spectra reveal that large scale predictability properties of the full deterministic model are well approximated, whereas small scale features are not properly resolved.     

Scale-invariant Lyapunov exponents for classical hamiltonian systems

The Lyapunov exponent for classical hamiltonian systems is made dimensionless by introducing a characteristic time τ_c. This modification yields an energy-independent exponent for systems with scale invariance.     

On the bound of the Lyapunov exponents for continuous systems

In this paper, both upper bounds and lower bounds for all the Lyapunov exponents of continuous differential systems are determined. Several examples are given to show the application of the estimates derived here.     

Lyapunov exponents,dual Lyapunov exponents,and multifractal analysis

It is shown that the multifractal property is shared by both Lyapunov exponents and dual Lyapunov exponents related to scaling functions of one-dimensional expanding folding maps. This reveals in a quantitative way the complexity of the dynamics determined by such maps. (c) 1999 American Institute of Physics.     

On estimates of Lyapunov exponents of synchronized coupled systems

Lyapunov exponents of a synchronized coupled system consist of those of the underlying individual systems and the transverse systems, based on a mode decomposition along the synchronization manifold. Estimates of bounds on the Lyapunov exponents (including transverse Lyapunov exponents) are derived. Several examples are used to validate the theoretical estimates.     

Scaling and interleaving of subsystem Lyapunov exponents for spatio-temporal systems

The computation of the entire Lyapunov spectrum for extended dynamical systems is a very time consuming task. If the system is in a chaotic spatio-temporal regime it is possible to approximately reconstruct the Lyapunov spectrum from the spectrum of a subsystem by a suitable rescaling in a very cost effective way. We compute the Lyapunov spectrum for the subsystem by truncating the original Jacobian without modifying the original dynamics and thus taking into account only a portion of the information of the entire system. In doing so we notice that the Lyapunov spectra for consecutive subsystem sizes are interleaved and we discuss the possible ways in which this may arise. We also present a new rescaling method, which gives a significantly better fit to the original Lyapunov spectrum. We evaluate the performance of our rescaling method by comparing it to the conventional rescaling (dividing by the relative subsystem volume) for one- and two-dimensional lattices in spatio-temporal chaotic regimes. Finally, we use the new rescaling to approximate quantities derived from the Lyapunov spectrum (largest Lyapunov exponent, Lyapunov dimension, and Kolmogorov-Sinai entropy), finding better convergence as the subsystem size is increased than with conventional rescaling. (c) 1999 American Institute of Physics.     

精确配置离散动力系统的所有Lyapunov指数

针对给定的确定性离散时间动力学系统,提出一种可以精确配置受控系统所有Lyapunov指数的混沌控制与反控制新方法.本文不限制给定系统的具体形式,仅利用取模运算和预设的一组Lyapunov指数,设计出一种新的状态反馈控制器.在控制器的作用下,受控系统的Lyapunov指数与预设的Lyapunov指数一致.控制器的形式简单,可以用于解决混沌控制和混沌反控制两种问题.给出了数学证明和两个实例,仿真结果显示了算法的有效性.     

Velocity-dependent Lyapunov exponents as a measure of chaos for open-flow systems

Many flows in nature are “open flows” (e.g. pipe flow). We study two open-flow systems driven by low-level external noise: the time-dependent generalized Ginzburg-Landau equation and a system of coupled logistic maps. We find that a flow which gives every appearance of being chaotic may nonetheless have no positive Lyapunov exponents. By generalizing the notions of convective instability and Lyapunov exponents we define a measure of chaos for these flows.     

Direct measurement of Lyapunov exponents

Benettin, Calgani and Strelcyn studied the dynamical separation of neighboring phase-space trajectories, determining the corresponding Lyapunov exponents by discrete rescaling of the intertrajectory separation. We incorporate rescaling directly into the equations of motion, preventing Lyapunov instability by using an effective constraint force.     

Lyapunov exponents from quantum dynamics

We extract classical Lyapunov exponents from the time dependence of quantum mechanical expectation values. Classical chaos is revealed as a quantum transient with a liftetime ～? ln ?. Our strategy is shown to work for the example of a periodically kicked top.     

Symmetry-breaking in local Lyapunov exponents

Integrable dynamical systems, namely those having as many independent conserved quantities as freedoms, have all Lyapunov exponents equal to zero. Locally, the instantaneous or finite time Lyapunov exponents are nonzero, but owing to a symmetry, their global averages vanish. When the system becomes nonintegrable, this symmetry is broken. A parallel to this phenomenon occurs in mappings which derive from quasiperiodic Schr?dinger problems in 1-dimension. For values of the energy such that the eigenstate is extended, the Lyapunov exponent is zero, while if the eigenstate is localized, the Lyapunov exponent becomes negative. This occurs by a breaking of the quasiperiodic symmetry of local Lyapunov exponents, and corresponds to a breaking of a symmetry of the wavefunction in extended and critical states. Received 25 October 2001 / Received in final form 8 December 2001 Published online 2 October 2002 RID="a" ID="a"e-mail: r.ramaswamy@mail.jnu.ac.in     

Transport coefficients and Lyapunov exponents

After introducing the viscosity of a fluid macroscopically and microscopically as well as the Lyapunov exponents of the fluid, the SLLOD equations of motion with a Gaussian thermostat and Lees-Edwards boundary conditions for the motion of particles in a sheared fluid in a nonequilibrium stationary state are discussed. An explicit expression, due to Posch and Hoover, for the viscosity is then derived in terms of the sum of all Lyapunov exponents, illustrating the direct connection between irreversible entropy production (due to viscous heating) and phase space contraction. A symmetry of the Lyapunov spectrum allows this expression to be reduced to a simple relation between the viscosity and the two maximal Lyapunov exponents of the fluid in the stationary state. A numerical check of this relation for fluids consisting of 108 and 864 particles is presented. Finally, similar relations for other transport coefficients and the connection with other work are discussed.     

Entropy potential and Lyapunov exponents

According to a previous conjecture, spatial and temporal Lyapunov exponents of chaotic extended systems can be obtained from derivatives of a suitable function, the entropy potential. The validity and the consequences of this hypothesis are explored in detail. The numerical investigation of a continuous-time model provides a further confirmation to the existence of the entropy potential. Furthermore, it is shown that the knowledge of the entropy potential allows determining also Lyapunov spectra in general reference frames where the time-like and space-like axes point along generic directions in the space-time plane. Finally, the existence of an entropy potential implies that the integrated density of positive exponents (Kolmogorov-Sinai entropy) is independent of the chosen reference frame. (c) 1997 American Institute of Physics.