PapersNo Access

ELLIPTIC AND AUTOMORPHIC DYNAMICAL SYSTEMS ON SURFACES

S. P. BANKS

Department of Automatic Control and Systems Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK

Search for more papers by this author

and

SONG YI

Department of Automatic Control and Systems Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK

Search for more papers by this author

https://doi.org/10.1142/S0218127406015209Cited by:4 (Source: Crossref)

Abstract

We derive explicit differential equations for dynamical systems defined on generic surfaces applying elliptic and automorphic function theory to make uniform the surfaces in the upper half of the complex plane with the hyperbolic metric. By modifying the definition of the standard theta series we will determine general meromorphic systems on a fundamental domain in the upper half plane the solution trajectories of which "roll up" onto an appropriate surface of any given genus.

Keywords:

References

R. Abraham and J. Marsden , Foundations of Mechanics ( Benjamin/Cummings , Menlo Park, CA , 1978 ) . Google Scholar
V. I. Arnold , Mathematical Methods of Classical Mechanics , 2nd edn. ( Springer-Verlag , NY , 1989 ) . Crossref, Google Scholar
S. P. Banks, Int. J. Bifurcation and Chaos 12, 1 (2002), DOI: 10.1142/S0218127402004152. Link, Web of Science, Google Scholar
S. P. Banks and D. Diaz, Int. J. Bifurcation and Chaos 14, 1267 (2004), DOI: 10.1142/S0218127404009776. Link, Web of Science, Google Scholar
A. T. Fomenko , Topological Classification of Integrable Systems , Advances in Soviet Mathematics 6 ( AMS , Providence, RI , 1991 ) . Crossref, Google Scholar
L. R. Ford , Automorphic Functions ( McGraw-Hill , NY , 1929 ) . Google Scholar
G. A. Jones and D. Singerman , Complex Functions: An Algebraic and Geometric Viewpoint ( CUP , Cambridge , 1987 ) . Crossref, Google Scholar
L. Perko , Differential Equations and Dymanical Systems ( Springer-Verlag , NY , 1991 ) . Crossref, Google Scholar
J. G. Ratcliffe , Foundations of Hyperbolic Manifolds , GTM 149 ( Springer-Verlag , NY , 1994 ) . Crossref, Google Scholar
G. Shimura , Introduction to the Arithmetic Theory of Automorphic Functions ( Princeton University Press , 1971 ) . Google Scholar