Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve $μ^2=ν^n-1, n\in{\Bbb Z}$: ergodicity, isochrony, periodicity and fractals

Details Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve $μ^2=ν^n-1, n\in{\Bbb Z}$: ergodicity, isochrony, periodicity and fractals Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve $μ^2=ν^n-1, n\in{\Bbb Z}$: ergodicity, isochrony, periodicity and fractals

2006-07-14

arxiv.org/abs/nlin/0607031v1

Nonlinear Sciences

Chaotic Dynamics

LaTex, 28 pages, 10 figures

Scientific paper

10.1016/j.physd.2007.05.002

We study the complexification of the one-dimensional Newtonian particle in a monomial potential. We discuss two classes of motions on the associated Riemann surface: the rectilinear and the cyclic motions, corresponding to two different classes of real and autonomous Newtonian dynamics in the plane. The rectilinear motion has been studied in a number of papers, while the cyclic motion is much less understood. For small data, the cyclic time trajectories lead to isochronous dynamics. For bigger data the situation is quite complicated; computer experiments show that, for sufficiently small degree of the monomial, the motion is generically periodic with integer period, which depends in a quite sensitive way on the initial data. If the degree of the monomial is sufficiently high, computer experiments show essentially chaotic behaviour. We suggest a possible theoretical explanation of these different behaviours. We also introduce a one-parameter family of 2-dimensional mappings, describing the motion of the center of the circle, as a convenient representation of the cyclic dynamics; we call such mapping the center map. Computer experiments for the center map show a typical multi-fractal behaviour with periodicity islands. Therefore the above complexification procedure generates dynamics amenable to analytic treatment and possessing a high degree of complexity.

Affiliated with

Also associated with

No associations

Rating

Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve $μ^2=ν^n-1, n\in{\Bbb Z}$: ergodicity, isochrony, periodicity and fractals does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve $μ^2=ν^n-1, n\in{\Bbb Z}$: ergodicity, isochrony, periodicity and fractals, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve $μ^2=ν^n-1, n\in{\Bbb Z}$: ergodicity, isochrony, periodicity and fractals will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-96561