Lyapunov landscape elucidates disorder-taming-chaos in arrays of coupled pendula

Disordering coupled arrays of nonlinear elements can convert chaotic dynamics to periodic dynamics. Disorder-taming-chaos (DTC) was originally demonstrated by disordering the lengths of torqued and damped coupled pendula. Guided by a possibility space of the lengths’ mean and standard deviation, we investigate periodicity and chaos in such arrays by estimating their maximal Lyapunov exponents via numerical integration of the corresponding variational equations, after identifying a plateau in the variational algorithm’s parameter space where these exponents are well defined. In mapping the Lyapunov landscape as a function of disorder and coupling, we classify the dynamics as periodic or chaotic—exhaustively for binary and trinary arrays and typically for longer arrays, including the continuum limit—and connect with the arrays’ spatiotemporal patterns. We thereby locate DTC at bifurcations of the single-pendulum dynamics and find that both mean-changing and mean-preserving length-disorders can tame chaos, but the latter may require large disorder at large coupling.