I Variational Equations, Chaos Indicators

[dreens]

I work with an electromagnetic molecule trap, and I'd like to determine which orbits are chaotic. To this end, I intend to study the evolution of a perturbation on a trajectory with time.

I'd like to compute something called the fast lyapunov indicator for various trajectories y(t), where I have a force law y''(t)=f(y).

I'm told I need to consider a variation dy, and follow its evolution d(dy)/dt = df(y)/dy * dy.

My questions are:
1. how do I deal with the fact that I have a second order equation not first.
2. Could I just evolve two trajectories separated by dy initially in order to evolve dy, or is this much less precise than evolving dy according to its specific higher order evolution equation?

 

[dreens]

By the way, I've heard that chaotic orbits only exist in a conservative system like mine when they can escape eventually. Can anyone confirm this or point me to a reference? I think I read it in this article, they say something about Arnold's webs and KAM tori but I couldn't follow it.

 

[wrobel]

Dynamical chaos is a very large set of very different effects that express a complicated behavior of the trajectories. Chaos in Hamiltonian systems is not the same as chaos in general systems; chaos in infinite dimensional systems is not the same as in finite dimensional ones. In each concrete system or concrete class of systems one should specify what one means when he says "chaos". One must formulate a mathematical definition of chaos basing on physical phenomena which one sees. It does not make sense to use the term "chaos" before this

 

[dreens]

I mean variation of initial conditions, and my definition of chaos is that dy increases exponentially with time. For conservative hamiltonian systems people often adopt less strict definitions, such as linear increase of dy with time, but I'd rather leave that out for now.

 

[wrobel]

ok. trajectories of a system $\dot x=x$ diverge exponentially. Thus it is a chaotic system in accordance to your definition.
Studying of the dynamical chaos requires for professional mathematical background

 

[dreens]

I'm interested in systems that support stable orbits, for example traps, unlike the anti trap force example you've brought up. I don't think I need to be a professional mathematician in order to use chaos indicators to learn about orbits in my trap. I can follow the work of other professionals, who have established lyapunov based indicators as a useful tool for quantifying the chaos present in a system.

My question pertains to the details of computation of these indicators. In particular whether it is sufficient to calculate two different trajectories and subtract them to get the evolution of the variational vector, or whether evolving the variational vector on its own with a higher order equation is essential.

 

[dreens]

I'm answering my own questions just in case someone else comes across this.

1. Always treat the dynamical system as a first order one when taking derivatives of the flow function f (where y' = f(t,y) ).

2. It may not be absolutely necessary to use the higher order variational equations compared with say a trajectory subtraction technique, but the former provides much higher numerical precision for the same tolerances, and worked well for my application.