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Cross section of the basins of attraction (a) corresponding to the four asymmetric attractors of Fig. 5: The blue and green colors are related to the one-band chaotic attractors, while the magenta and yellow colors correspond to the three-band chaotic attractors. An enlargement of this diagram is provided in (b). The perfect symmetry of the basins of attraction is visible (Color figure online)
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Nonlinear dynamical systems with symmetry have been studied extensively yielding rich and striking bifurcation patterns such as period-doubling sequence, merging crisis, crisis-induced intermittency, spontaneous symmetry breaking, and coexisting pairs of mutually symmetric attractors as well. However, very little is known unfortunately about the be...
Citations
... Some complex dynamic behaviors, such as periodic bursting and chaotic bursting, can also be generated in the generalized ternary memristive circuit [23]. The influence of a specific symmetry break on the dynamics of a fourth-order autonomous memristive chaotic circuit was evaluated [24]. A novel conservative memristive system was discovered by introducing the three-terminal memristor into a newly constructed 4D Euler equation [25]. ...
This paper proposes a novel five-dimensional (5D) memristor-based chaotic system by introducing a flux-controlled memristor into a 3D chaotic system with two stable equilibrium points, and increases the dimensionality utilizing the state feedback control method. The newly proposed memristor-based chaotic system has line equilibrium points, so the corresponding attractor belongs to a hidden attractor. By using typical nonlinear analysis tools, the complicated dynamical behaviors of the new system are explored, which reveals many interesting phenomena, including extreme homogeneous and heterogeneous multistabilities, hidden transient state and state transition behavior, and offset-boosting control. Meanwhile, the unstable periodic orbits embedded in the hidden chaotic attractor were calculated by the variational method, and the corresponding pruning rules were summarized. Furthermore, the analog and DSP circuit implementation illustrates the flexibility of the proposed memristic system. Finally, the active synchronization of the memristor-based chaotic system was investigated, demonstrating the important engineering application values of the new system.
Memristors have several chaotic dynamic models and have been used successfully in various fields, including secure communication systems, information storage, and artificial neural networks. The memristor-based four-dimensional chaotic (FDMC) systems generate unpredictable and intricate time domain signals. Parameter fluctuations in the FDMC system may give birth to chaos, making it difficult to suppress. Stabilizing chaos in the FDMC system improves the circuit’s performance. This paper synthesizes a novel time-efficient chatter-free nonlinear robust adaptive control (NLRAC) technique that stabilizes chaos in the FDMC system affected by time-varying unknown bounded exogenous disturbances and model uncertainties. The proposed NLRAC strategy decimates the time-varying unknown bounded exogenous disturbances and model uncertainties effects; it establishes a faster, smoother state-variable trajectories convergence to the zero vicinity. The theoretical analysis and mathematical proofs are based on the Lyapunov stability technique. Computer simulation results show that the proposed NLRAC technique effectively brings the FDMC system's state-variable trajectories to zero with reduced fluctuations for control input signals and state-variable trajectories. This feedback controller’s attribute enhances closed-loop stability performance, improves precision, and reduces risk overshoot. The paper includes comparative computer simulation results to endorse the proposed controller performance.
