Calculation of the Lyapunov exponents and their dependence on the initial condition

Question

On page 4 of this online book of Arnold Neumaier about classical and quantum mechanics via Lie algebras, it says that the Lyapunov exponent $\lambda$, which characterizes the evolution in time of the errors in the initial conditions a classical system is assumed to start with, depends on these initial conditions (positions and velocities).

Even though I roughly understand what Lyapunov exponents are, I am not sure how they are generally calculated for a system which is given for example by a Hamilton function $H$. I guess that they are obtained from doing some kind of stability analysis, but can somebody enlighten me in more detail about how the Lyapunov exponents are generally calculated?

And how exactly do the Lyapunov exponents depend on the initial condition in phase space?

Comments

I can answer this (as probably most questions related to my online book), but need some time to organize my knowledge on this.

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@ArnoldNeumaier ok, looking forward to it :-)

Answer 1

I found an excellent survey paper that answers your question is full detail: http://arxiv.org/abs/0811.0882 . Thus I'll be brief in the following overview.

Every orbit of an autonomous dynamical system has an associated sequence of Lyapunov exponents. The first (or maximal) Lyapunov exponent $\lambda_1$ tells how fast the distance of points in an infinitesimal neighborhood of a point of the orbit increases or decreases with time in the limit of very long times. (''Very long'' may be a second for a microscopic system, or many hundred millions of years for a planetary system.)

 The asymptotic law is exponential, with dominating term $ce^{\lambda_1 t}$. For example, a dissipative system that contracts to a point has $\lambda_1<0$, a distance-preserving system has $\lambda_1<0$, and a chaotic system has $\lambda_1>0$. The latter condition says that there is a very sensitive dependence on initial conditions, and is a bit more general than chaoticity. The other Lyapunov exponents express similar properties related to areas, 3-volumes, etc. in place of distances.

The Lyapunov exponents are independent of the initial condition on the orbit, but may be different for initial conditions defining different orbits. They are a property of the system only if the latter is topologically transitive (ergodic), so that all orbits behave essentially the same. (An example of the distribution of $\lambda_1$ for a large sample of initial conditions in an application in meteorology is given in Figure 9 of a paper by Shepherd et al..

For their computation of Lyapunov exponents in case of a conservative system with a given Hamiltonian one indeed needs to do a stability analysis, but in a globalized form. One linearizes the system in a neighborhood of the orbit under study, and represents the resulting linear, time-dependent system in a product form, writing (for systems with finitely many degrees of freedom) the evolution matrices $A(t)$ (that tell how an initial deviation vector $d(0)$ is mapped to $d(t)=A(t)d(0)$) in the form $A(t)=Q(t)R(t)$ with orthogonal $Q(t)$ and upper triangular $R(t)$. (This is a continuous version of the orthogonal factorization $A=QR$ of a matrix $A$, heavily used in numerical analysis.) The Lyapunov exponents can then be read off from the time-dependence of the diagonal elements of $R(t)$. To find $Q(t)$ and $R(t)$ one must solve a coupled system of ordinary differential equations for the components.