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• The features of multiple time-scales in a memristor-based Shinriki's circuit are classified. • The generic structural properties of the MMO cascades in the control parameter space are revealed. • Two kinds of interesting self-organizations are discovered. The structure of complex oscillations with two time scales generated by a novel memristor-based Shinriki's circuit is studied in detail. Employing the stability phase diagrams, we identify the parameter regions of mixed-mode oscillations (MMOs) produced by each state variable, and classify the fast and slow properties of state variables according to the characteristics of these MMOs. Additionally, we exhibit the self-organizations of phases and describe how they are interconnected and how their complexity unfolds with control parameters varying. For several parameter combinations, in addition to recently reported shrimp-shaped cascades and "eye" of chaos, two kinds of interesting self-organizations, the shrimp cascade covered by four distinct numbers of peaks and "eye" of anti-chaos, are also discovered. A significant feature of these structures indicates that the fast-slow dynamics of the system depend not only on the state variables but also strongly on the parameter interval. The results of theoretical analyses obtained are verified by the Multisim simulation experiments. This paper will contribute to understanding the profound dynamical behaviors of the memristor-based circuit and open the possibility of studying hitherto unobserved phenomena. [ABSTRACT FROM AUTHOR]

• Based on GPU parallel computing technology, the topology of mode-locking in a hybrid discrete/continuous non-smooth system is investigated. • A new cloned dynamical approach is employed to calculate the Lyapunov exponent of the hybrid system. • The mode-locking topology is organized regularly according to the elusive Stern-Brocot sum tree. • The mode-locking order is unfolded in a way that never encountered before to emerge in perfect agreement with the reciprocal of the Stern-Brocot sum tree. We report the topology of stable periodic solutions of an SIR epidemic dynamics model with impulsive vaccination control, a hybrid discrete/continuous non-smooth system, in the parameter plane spanned by the pulse period T and the quantity of pulse vaccination b. This mode-locking topology is governed by an invariant torus (a pair of frequencies) initiated from Hopf bifurcations rather than the fast-slow time scales, and its periodicity is in perfect agreement with the so-called Stern-Brocot sum tree, a derived tree from "Farey algorithm". More surprisingly, the mode-locking order is unfolded in a way that never encountered before to emerge organized according to the reciprocal of the Stern-Brocot sum tree. Furthermore, the global organization of mode-locking is not isolated structure but an infinite Stern-Brocot sum tree cascade when tuning the control parameter T. The results obtained contribute a new framework to classify mode-locking oscillations observed in the hybrid system. [ABSTRACT FROM AUTHOR]