  In this paper, we establish the multiplicative ergodic theorem for
McKean-Vlasov stochastic differential equations, in which the Lyapunov exponent
is defined using the upper limit. The reasonability of this definition is
illustrated through an example; i.e., even when the coefficients are regular
enough and their first-order derivatives are bounded, the upper limit cannot be
replaced by a limit, as the limit may not exist. Furthermore, the example
reveals how the dependence on distribution significantly influences the
dynamics of the system and evidently distinguishes McKean-Vlasov stochastic
differential equations from classical stochastic differential equations.
