Ms. Jolly et al., CONVERGENCE OF A CHAOTIC ATTRACTOR WITH INCREASED SPATIAL-RESOLUTION OF THE GINZBURG-LANDAU EQUATION, Chaos, solitons and fractals, 5(10), 1995, pp. 1833-1845
The convergence of chaotic attractors is demonstrated for increasingly
finer spatial discretizations of a dissipative partial differential e
quation using both the traditional and nonlinear Galerkin methods. Thi
s is done for the complex Ginzburg-Landau equation which describes amp
litude evolution of instability waves in fluid flow. Density functions
of instantaneous Lyapunov exponents are used to establish the converg
ence. These exponents measure the local variations in the contraction/
expansion rates along an orbit. The results indicate that the converge
nce of the density functions requires no more iterations in time than
is needed for convergence of the classical Lyapunov exponent. Thus the
density function, which gives a more detailed description of the orbi
t, can serve as a viable means of comparing different methods of spati
al discretization as well as the effect of finer resolution. The densi
ty function converged faster, i.e. with fewer modes, in the case of th
e nonlinear Galerkin method than in the traditional Galerkin method.