Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: From period to quasi-period to chaos: A continuous spectrum of orbits of
  charged particles trapped in a dipole magnetic field   Via evaluation of the Lyapunov exponent, we report the discovery of three
prominent sets of phase space regimes of quasi-periodic orbits of charged
particles trapped in a dipole magnetic field. Besides the low energy regime
that has been studied extensively and covers more than 10% in each dimension of
the phase space of trapped orbits, there are two sets of high energy regimes,
the largest of which covers more than 4% in each dimension of the phase space
of trapped orbits. Particles in these high energy orbits may be observed in
space and be realized in plasma experiments on the Earth. It is well-known that
there are quasi-periodic orbits around stable periodic orbits in Hamiltonian
systems with 2 degrees of freedom and these quasi-periodic orbits are stable as
well. Since periodic orbits appear to have a negligible measure in the phase
space, they are difficult to realize in nature. Quasi-periodic orbits, on the
other hand, may occupy a finite volume in the 4 dimensional (4D) phase space
and be readily detectable. A chaotic orbit has at least one positive Lyapunov
exponent. The Lyapunov exponents of quasi-periodic orbits, on the other hand,
should be zero. Via calculation of the Lyapunov exponent of orbits of trapped
charged particles in a dipole magnetic field, we scanned the corresponding
phase space and found several prominent regimes of quasi-periodic orbits
associated with stable periodic orbits in the equatorial plane. These regimes
appear to be connected to some small regimes of quasi-periodic orbits
associated with stable periodic orbits in the Meridian plane. Our numerical
results also show a continuous spectrum of these orbits from stable periodic,
to quasi-periodic with vanishing Lyapunov exponents, and eventually to chaotic
ones with at least one positive Lyapunov exponent and there are unstable
periodic orbits with a positive maximum Lyapunov exponent.

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The categories of this paper are: cond-mat
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The categories of this paper are: bayes-an
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The categories of this paper are: quant-ph
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The categories of this paper are: adap-org
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The categories of this paper are: math
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The categories of this paper are: patt-sol
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: physics
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: plasm-ph