  Replicating chaotic characteristics of non-linear dynamics by machine
learning (ML) has recently drawn wide attentions. In this work, we propose that
a ML model, trained to predict the state one-step-ahead from several latest
historic states, can accurately replicate the bifurcation diagram and the
Lyapunov exponents of discrete dynamic systems. The characteristics for
different values of the hyper-parameters are captured universally by a single
ML model, while the previous works considered training the ML model
independently by fixing the hyper-parameters to be specific values. Our
benchmarks on the one- and two-dimensional Logistic maps show that variational
quantum circuit can reproduce the long-term characteristics with higher
accuracy than the long short-term memory (a well-recognized classical ML
model). Our work reveals an essential difference between the ML for the chaotic
characteristics and that for standard tasks, from the perspective of the
relation between performance and model complexity. Our results suggest that
quantum circuit model exhibits potential advantages on mitigating over-fitting,
achieving higher accuracy and stability.
