  A thermostatted dynamical model with five degrees of freedom is used to test
both the Evans-Searles and the Gallavotti-Cohen fluctuation relations (ESFR and
GCFR respectively). In the absence of an external driving field, the model
generates a time independent ergodic equilibrium state with two conjugate pairs
of Lyapunov exponents. Each conjugate pair sums to zero. The fluctuation
relations are tested numerically both near and far from equilibrium. As
expected from previous work, near equilibrium the ESFR is verified by the
simulation data while the GCFR is not confirmed by the data. Far from
equilibrium where a positive exponent in one of these conjugate pairs becomes
negative, we test a conjecture regarding the GCFR made by Gallavotti and
co-workers. They conjectured that where the number of nontrivial Lyapunov
exponents that are positive becomes less than the number of such negative
exponents, then the form of the GCFR needs to be corrected. We show that there
is no evidence for this conjecture in the empirical data. In fact as the field
increases, the uncorrected form of the GCFR appears to become more accurate.
The real reason for this observation is likely to be that as the field
increases, the argument of the GCFR more and more accurately approximates the
argument of the ESFR. Since the ESFR works for arbitrary field strengths, the
uncorrected GCFR appears to become ever more accurate as the field increases.
The final point of evidence against the conjecture is that when the smallest
positive exponent changes sign, the conjecture predicts a discontinuous change
in the "correction factor" for GCFR. We see no evidence for a discontinuity at
this field strength; only a gradual improvement of degree of agreement as the
field increases.
