  We propose theoretically an experimentally realizable method to demonstrate
the Lyapunov instability and to extract the value of the largest Lyapunov
exponent for a chaotic many-particle interacting system. The proposal focuses
specifically on a lattice of coupled Bose-Einstein condensates in the classical
regime describable by the discrete Gross-Pitaevskii equation. We suggest to use
imperfect time-reversal of system's dynamics known as Loschmidt echo, which can
be realized experimentally by reversing the sign of the Hamiltonian of the
system. The routine involves tracking and then subtracting the noise of
virtually any observable quantity before and after the time-reversal. We
support the theoretical analysis by direct numerical simulations demonstrating
that the largest Lyapunov exponent can indeed be extracted from the Loschmidt
echo routine. We also discuss possible values of experimental parameters
required for implementing this proposal.
