  The Sachdev-Ye-Kitaev (SYK) model describes Majorana fermions with random
interaction, which displays many interesting properties such as non-Fermi
liquid behavior, quantum chaos, emergent conformal symmetry and holographic
duality. Here we consider a SYK model or a chain of SYK models with $N$
Majorana fermion modes coupled to another SYK model with $N^2$ Majorana fermion
modes, in which the latter has many more degrees of freedom and plays the role
as a thermal bath. For a single SYK model coupled to the thermal bath, we show
that although the Lyapunov exponent is still proportional to temperature, it
monotonically decreases from $2\pi/\beta$ ($\beta=1/(k_BT)$, $T$ is
temperature) to zero as the coupling strength to the thermal bath increases.
For a chain of SYK models, when they are uniformly coupled to the thermal bath,
we show that the butterfly velocity displays a crossover from a
$\sqrt{T}$-dependence at relatively high temperature to a linear $T$-dependence
at low temperature, with the crossover temperature also controlled by the
coupling strength to the thermal bath. If only the end of the SYK chain is
coupled to the thermal bath, the model can introduce a spatial dependence of
both the Lyapunov exponent and the butterfly velocity. Our models provide
canonical examples for the study of thermalization within chaotic models.
