Control and cryptographic application of a memristor-induced second-order discrete map

A second-order chaotic map with a completely new structure is constructed by introducing a discrete memristor model. The steady-state dynamic distributions and local bifurcation analysis of the model are performed and simulated on two-dimensional parameter planes. The extended isocline method is employed to identify codimension-1 and codimension-2 bifurcations on the parametric planes. It is also very effective on identifying and sketching resonant entrainments and cusps in combination with continuation method. The tangency and intersections of invariant manifolds expound the evolutions of basins of attraction. Global dynamics analyses reveal the inherent complex structure and multi-stability. A finite-time control scheme is proposed specifically for the nonlinear discrete systems with such a structure. The control scheme is disposed to be instructive and flexible. Both numerical simulations and physical implementations have verified the rationality and effectiveness of the controller. In the end, an improved image encryption algorithm with parallel structure is employed to demonstrate its potential applications. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2025.