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- John A.G. Roberts, Franco Vivaldi
- Physical review letters
- 2003

We develop a method to detect the presence of integrals of the motion in symplectic rational maps, by representing these maps over finite fields and examining their orbit structure. We find markedly different orbit statistics depending upon whether the map is integrable or not.

- Danesh Jogia, John A.G. Roberts, Franco Vivaldi
- 2006

We show that the dynamics of a birational map on an elliptic curve over a field is, typically, conjugate to addition by a point (under the associated group law). When the field is taken to be the function field of rational complex functions of one variable, this amounts to an algebraic geometric version of the Arnold– Liouville integrability theorem for… (More)

- John A.G. Roberts, Danesh JOGIA, Franco Vivaldi
- 2003

We reduce planar measure-preserving rational maps over finite fields, and study their discrete dynamics. We show that application to the orbit analysis over these fields of the Hasse-Weil bound for the number of points on an algebraic curve gives a strong indication of the existence of an integral for the map. Moreover, the method is ideally suited to… (More)

- Franco Vivaldi
- Experimental Mathematics
- 1994

AMS Subject Classi cation: 58F20, 60J60, 65G05 We investigate the effects of round-off errors on quasi-periodic motions in a linear symplectic planar map. By discretizing coordinates uniformly we transform this map into a permutation of Z2 , and study motions near infinity, which correspond to a fine discretization. We provide numerical evidence that all… (More)

We investigate the reduction to finite fields of polynomial automorphisms of the plane, which lead to invertible dynamics (permutations) of a finite space. We provide evidence that, if the map of the plane is non-integrable, then the presence or absence of a type of time-reversal symmetry called R-reversibility produces a clear signature in the cycle… (More)

- Franco Vivaldi, Igor Vladimirov
- I. J. Bifurcation and Chaos
- 2003

We study a 4-fold symmetric kicked-oscillator map with sawtooth kick function. For the values of the kick amplitude λ = 2 cos(2πp/q) with rational p/q, the dynamics is known to be pseudochaotic, with no stochastic web of non-zero Lebesgue measure. We show that this system can be represented as a piecewise affine map of the unit square —the so-called local… (More)

We study propagation of pulses along one-way coupled map lattices, which originate from the transition between two superstable states of the local map. The velocity of the pulses exhibits a staircase-like behaviour as the coupling parameter is varied. For a piece-wise linear local map, we prove that the velocity of the wave has a Devil’s staircase… (More)

We consider issues of computational complexity that arise in the study of quasi-periodic motions ~Siegel discs! over the p-adic integers, where p is a prime number. These systems generate regular invertible dynamics over the integers modulo p, for all k , and the main questions concern the computation of periods and orbit structure. For a specific family of… (More)

We study time-reversal symmetry in dynamical systems with finite phase space, with applications to birational maps reduced over finite fields. For a polynomial automorphism with a single family of reversing symmetries, a universal (i.e. map-independent) distribution function R(x) = 1 − e−x(1 + x) has been conjectured to exist, for the normalized cycle… (More)