Narrow Results

This paper presents a Ying–Yang theory for nonlinear discrete dynamical systems considering both positive and negative iterations of discrete iterative maps. In the existing analysis, the solutions relative to "Yang" in nonlinear dynamical systems are extensively investigated. However, the solutions pertaining to "Ying" in nonlinear dynamical systems are investigated. A set of concepts on "Ying" and "Yang" in discrete dynamical systems are introduced to help one understand the hidden dynamics in nonlinear discrete dynamical systems. Based on the Ying–Yang theory, the periodic and chaotic solutions in nonlinear discrete dynamical system are discussed, and all possible, stable and unstable periodic solutions can be analytically predicted. A discrete dynamical system with the Henon map is investigated, as an example.

This paper studies a new class of three-dimensional maps in a Jerk-like structure with a special concern of their hidden chaotic dynamics. Our investigation focuses on the hidden chaotic attractors in three typical scenarios of fixed points, namely no fixed point, single fixed point, and two fixed points. A systematic computer search is performed to explore possible hidden chaotic attractors, and a number of examples of the proposed maps are used for demonstration. Numerical results show that the routes to hidden chaotic attractors are complex, and the basins of attraction for the hidden chaotic attractors could be tiny, so that using the standard computational procedure for localization is impossible.

A general formalism describing a type of energy-conservative system is established. Some possible dynamic behaviors of such energy-conservative systems are analyzed from the perspective of geometric invariance. A specific 4D chaotic energy-conservative system with a line of equilibria is constructed and analyzed. Typically, an energy-conservative system is also conservative in preserving its phase volume. The constructed system however is conservative only in energy but is dissipative in phase volume. It produces energy-conservative attractors specifically exhibiting chaotic 2-torus and quasiperiodic behaviors including regular 2-torus and 3-torus. From the basin of attraction containing a line of equilibria, the hidden nature of chaotic attractors generated from the system is further discussed. The energy hypersurface on which the attractors lie is determined by the initial value, which generates complex dynamics and multistability, verified by energy-related bifurcation diagrams and Poincaré sections. A new type of coexistence of attractors on the equal-energy hypersurface is discovered by turning the system parameter values to their opposite. The basins of attraction under three sets of parameter values demonstrate that the Hamiltonian is the leading factor predominating the dynamic behaviors of the system with a closed energy hypersurface. Finally, an analog circuit is designed and implemented to demonstrate the consistent theoretical and simulation results.

Reliable neuron models play an essential role in identifying the electrical activities, global bifurcation patterns, and dynamic mechanisms of neurons in electromagnetic environments. Considering that memristive autapse can characterize the self-induced effect of neurons, a five-dimensional Hindmarsh–Rose (HR) neuron model involving electric and magnetic fields is established. The detailed existence and stability analyses for equilibrium points are performed, and the complex time-varying stability, saddle-node bifurcation, and Hopf bifurcation behaviors are demonstrated. Interestingly, the bistable structures consisting of quiescent state and periodic bursting modes near the subcritical Hopf bifurcation and counterintuitive dynamic phenomena can be induced via appropriately adjusting the memristive current. Accordingly, the mechanism of positive feedback autaptic current decreases its firing frequency, while negative feedback autaptic current promotes its excitability and is revealed by the fast–slow dynamic analysis. Generally, the system possesses period-adding bifurcation patterns and comb-shaped chaotic structures as demonstrated by the numerical results. Importantly, it can be confirmed that the electrical activities and multistability of the system can be accurately predicted by analyzing the global dynamic behaviors of the Hamilton energy. Furthermore, it is verified that the unidirectional coupling controller involving energy is far more efficient and consumes lower energy than electrical synaptic coupling in achieving complete synchronization with mismatched parameters. These results provide potential guidance and help for further research in computational neuroscience and the design and control of intelligent sensors.

Considering the existence of magnetic induction effect with different intensities in the process of subthreshold and suprathreshold oscillations of bioelectrical activities, a non-smooth feedback strategy for memristive current with time delay is proposed, and then a four-dimensional Filippov Hindmarsh–Rose (HR) neuron model is established. The local stability and bifurcation patterns of delayed subsystems are qualitatively analyzed. Accordingly, the discriminant formula for the direction and stability of periodic solutions generated by Hopf bifurcation is obtained on the center manifold. Importantly, the stability of subsystems has switching behavior, which is accompanied by abundant hidden electrical activities under the effect of time delay. The theoretical analysis clarifies that the proposed feedback strategy leads to complex sliding mode dynamics, including sliding segments, various equilibrium points and sliding bifurcations. Meanwhile, the analytical conditions for motions of grazing, sliding, and crossing are developed and verified based on the flow switching theory. Moreover, the mechanism and evolutive rule of the self-excited and hidden sliding electrical activities are revealed by the fast-slow variable dissection method. Finally, it is verified that the time delay can not only induce bistable structures composed of the quiescent state and periodic bursting, but also eliminate the hidden sliding dynamics.