Trajectory exponential sensitivity

Local Trajectory Instability That is, the local (in phase space) exponential sensitivity of trajectories to changes in initial conditions. This property is the primary characteristic of a C system. Specifically, consider, in an N degree of freedom system, two trajectories with initial coordinates and momenta q0, p0 and qo, Po- For convenience we denote the column vector [q(f), p(t)] associated with p0, q0 as x(t) and the trajectory [q ( ), p (t)] associated with Po, q 0 as x (t). The phase space separation d(t) between trajectories is given by... 

As already mentioned, the motion of a chaotic flow is sensitive to initial conditions [H] points which initially he close together on the attractor follow paths that separate exponentially fast. This behaviour is shown in figure C3.6.3 for the WR chaotic attractor at /c 2=0.072. The instantaneous rate of separation depends on the position on the attractor. However, a chaotic orbit visits any region of the attractor in a recurrent way so that an infinite time average of this exponential separation taken along any trajectory in the attractor is an invariant quantity that characterizes the attractor. If y(t) is a trajectory for the rate law fc3.6.2] then we can linearize the motion in the neighbourhood of y to get... 

The most important quantitative measure for the degree of chaotic-ity is provided by the Lyapunov exponents (LE) (Eckmann and Ru-elle, 1985 Wolf et. al., 1985). The LE defines the rate of exponential divergence of initially nearby trajectories, i.e. the sensitivity of the system to small changes in initial conditions. A practical way for calculating the LE is given by Meyer (Meyer, 1986). This method is based on the Taylor-expansion method for solving differential equations. This method is applicable for systems whose equations of motion are very simple and higher-order derivatives can be determined analytically (Schweizer et.al., 1988). 

The Lyapunov exponents provide a computable measure of the sensitivity to initial conditions, i.e. characterize the mean exponential rate of divergence of two nearby trajectories if there is at least one positive Lyapunov exponent, or convergence when all Lyapunov exponents are negative. The Lyapunov exponents are defined for autonomous dynamical systems and can be described by ... 

Results demonstrate that when agitators are switched the slope of the pathline becomes discontinuous. We will see later in this chapter how this mechanism may produce an essentially stochastic response in the Lagrangian sense. Aref termed this chaotic advection, which he suggested to be a new intermediate regime between turbulent and laminar advection. The chaos has a kinematic origin, it is temporal—that is, along trajectories associated with the motion of individual fluid particles. Chaos is used in the sense of sensitivity of the motion to the initial position of the particle, and exponential divergence of adjacent trajectories. 

The spectrum of Lyapunov exponents provides fundamental and quantitative characterization of a dynamical system. Lyapunov exponents of a reference trajectory measure the exponential rates of principal divergences of the initially neighboring trajectories [1], Motion with at least one positive Lyapunov exponent has strong sensitivity to small perturbations of the initial conditions, and is said to be chaotic. In contrast, the principal divergences in regular motion, such as quasi-periodic motion, are at most linear in time, and then all the Lyapunov exponents are vanishing. The Lyapunov exponents have been studied both theoretically and experimentally in a wide range of systems [2-5], to elucidate the connections to the physical phenomena of importance, such as transports in phase spaces and nonequilibrium relaxation [6,7]. 

Sensitive dependence on initial conditions means that nearby trajectories separate exponentially fast, i.e., the system has a positive Liapunov exponent. 

In the phase space, the trajectory followed by the system never passes again through the same point, but remains confined to a finite portion of this space (fig. 4.10) the system evolves towards a strange attractor (Ruelle, 1989). The unpredictability of the time evolution in the chaotic regime is associated with the sensitivity to initial conditions two points, initially close to each other on the strange attractor, will diverge exponentially in the course of time. 

Due to the chaotic nature of molecular dynamics, which implies a sensitivity to perturbations of the initial condition or the differential equations themselves, it is to be expected that the global error due to using a numerical method will always grow rapidly (exponentially) in time. As we shall see in later chapters, this does not necessarily mean that a long trajectory is entirely without value. In molecular dynamics it turns out that the real importance of the trajectory is that it provides a mechanism for calculating averages that maintain physical parameters. The simplest example of such a parameter is the energy. 

Chaos is characterized by sensitivity to initial conditions and consequent exponential diversions of initially adjacent phase space trajectories. A(x)o, which is called the Lyapunov exponent (LE), is a measure of such divergence which is defined by... 

The previous analysis is confirmed when one measures the quasiclassical probability Pn(t) for remaining in the initial state (n, 0) for an ensemble of two hundred trajectories defined with initial conditions J = n + 1/2, = random J = 1/2, = random. The decay is close -fo exponential when the central fixed point is unstable and on the contrary it is distinctively non exponential with prominent oscillations ("beats") when the fixed point is stable. Thus, the sensitivity of short time relaxation to potential energy coupling derives from the drastic effect of a change from instability to stability on the stretch-bend energy flow in quasiperiodic trajectories. In this way, we have shown that the short time overtone decay dynamics of the two mode model exhibit the same sensitivity to potential energy coupling as does the trajectory calculations for the full planar benzene Hamiltonian of Hase and coworkers(10). 

A strange (or chaotic) attractor is by definition an attractor for which the largest Lyapunov exponent is positive. Then trajectories starting from nearby points will separate exponentially fast as time evolves. Therefore, all information about the initial conditions is rapidly lost, since any uncertainty, no matter how small, will be magnified until it becomes as large as the attractor thus there is sensitive dependence on initial conditions (RUELLE [38]). Long term predictions about the state of the system are impossible. 

We now suppose that the phenomenological equations of evolution for the macrovariables of the system give rise to a chaotic attractor possessing at least one positive Lyapunov exponent. It is well known that under these conditions the system manifests sensitivity to initial conditions [32], reflected by the divergence of initially nearby trajectories which in the double limit of initial deviations going to zero (to be taken first) and of time going to infinity (to be taken next) follows on the average an exponential law. 

The equations of motion of a molecular system formally represent a coupled set of nonlinear differential equations. (The nonlinearity comes from the complicated distance-dependence of the pair-potentials.) It is a property of nonlinear differential equations that they are extremely sensitive to small differences in their initial conditions. In nature, these small differences are most generally created by the perturbations of the surroundings while in the computer simulations they are produced by the finite accuracy of the numerical computation. The sensitivity is manifested in the fast increase of these initial differences nearby trajectories separate exponentially until the system boundaries force them to turn back. This mechanism quickly mixes the trajectories and after a short initial period the behavior of the system forgets its past. This obviously happens for equilibrium systems when their macroscopic properties relax to fixed average values. It also occurs for NESS systems because after short transients their distribution function also becomes stationary. ... 

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