Abstract

An essential starting point in understanding the dynamics of the magnetotail is the nature of the particle trajectories. It is this motion that ultimately determines the electric currents and subsequent magnetic fields. For years, this motion was calculated using approximate analytical techniques even though it was known that often times the approximations failed. Just over a decade ago, a more complete understanding of the "nonlinear dynamical" nature of particle motion was initiated by numerical experiments. Among the more significant results were numerical existence proof’s of chaotic behavior and the discovery that the particle phase space is partitioned into three dynamically distinct regions: transient, stochastic, and regular. Although many investigators have suggested applications of this newly recognized behavior, the underlying "cause" of the chaos remains hotly debated.

Using a computer simulation of charged particle dynamics in the modified Harris magnetic field,

(a standard model to the magnetotail magnetic field), we have begun an investigation into the nature and underlying causes of the chaos. In particular, we calculate the Lyapunov exponent, a Benettin and Strelcyn, in which the divergence of two numerical algorithms. First, we use the method of dimensional phase space. We then calculate the Lyapunov exponet by using the equations of deviation of the system:

where is the deviation vector in the phase space and is Jacobian Matrix of the equations of motion. Both calculations of the Lyapuvov exponent give nearly identical results and behaviors. One should be careful in the interpretation of the results, since the Lyapunov exponent is defined as a time asymptotic quantity and we are dealing with a chaotic scattering system where the particles have a finite residence time. It is important to note, however, that we are able to see distinctly different characteristics of the Lyapunov exponent for each of our orbit types (transient, stochastic, and regular.)

Introduction
Earth's Magnetosphere


Equations of Motion



reversal same as reversal

Poincare Surface of Section

The Three Disjoint Classes of Orbits

SOS as a Function of Energy

Fig.4.1. A sketch of trajectories in a three-dimensional state space. Notice how two nearby trajectories, starting near the origin, can continue to behave quite differently from each other yet remain boundec by weaving in and out and over and under each other.

Hilborn: Chaos and Nolinear Dynamics

The parameter is called the Lyapunov exponent.

Benettin And Strelcyn

Numerical experiments on the free motion of a point mass moving in a plane convex region: Stochastic transition and entropy

Method of calculating the Lyapunov exponent utilizing the divergence of two near by orbits.

     

Orbit

Lyapunov Exponent vs Time	bn = 0.1
Stochastic Orbits bn = 0.1
Transient Orbits	bn = 0.05
bn = 0.3	Stochastic Orbits

Conclusions

• Tthe Lyapunov exponent is reasonable measure of "chaos" of the system.
• Different classes of orbits exhibit different Lyapunov exponent behaviors.
• Resonnace surfaces have higher average stochastic Lyapunov exponents, as is expected because stochastic particles are trapped in the system longer and so have more interations with the current sheet.
• As the decreases the average Lyapunov exponent also decreases, as is expected because when the system becomes completely integrable (i.e. no chaos)

Further Work:

• Use double precision on the Bulirsch-Stoer integrator.
• Use a differnt integrator.
• Use other measures of chaos (i.e. Kolomagorov Entropy, Topological pressure).
• Attempt to understand the origin of the chaos (i.e. stretching, folding. separatrix crossings).