The definition of fractional calculus is introduced into the 5D chaotic system, and the 5D fractional-order chaotic system is obtained. The new 5D fractional-order chaotic system has no equilibrium, multi-scroll hidden attractor and multi-stability. By analyzing the time-domain waveform, phase diagram, bifurcation diagram and complexity, it is found that the system has no equilibrium but is very sensitive to parameters and initial values. With the variation of different parameters, the system can produce attractors of different scroll types accompanied by bursting oscillation. Secondly, the multi-stability of the hidden attractor is studied. Different initial values lead to the coexistence of attractors of different scroll number, which shows the advantages of the system. The correctness and realizability of the fractional-order chaotic system are proved by analog circuit and physical implement. Finally, because of the high security of multi-scroll attractor and hidden attractor, finite-time synchronization based on the fractional-order chaotic system is studied, which has a good application prospect in the field of secure communication.
By introducing a sine function, a non-autonomous multi-wing chaotic system is proposed. The system has an infinite number of equilibrium points and produces symmetrical attractors. The complex dynamical behaviors of the system are demonstrated by phase portraits, Lyapunov exponents spectrum and bifurcation diagram. The effect of driving amplitudes and initial conditions on the resulting system dynamics is then thoroughly investigated. The resulting attractors will enter different oscillatory states or have topological changes. The rotational coexisting attractors depend on the initial conditions and external amplitudes. Besides, a variety of interesting symmetrical transient behaviors and initial-offset boosting behaviors are also found. The driving amplitudes of the system affect the number of attractor wings, a weak signal detection circuit is accordingly designed to estimate the amplitude of periodic weak signals at diverse frequencies. The circuit operates near the switching threshold between the four-wing and two-wing chaotic attractor. Finally, an experimental investigation of the proposed design is performed that demonstrates the theoretical and simulation results.
A four-dimensional autonomous hyperchaotic system is constructed, and the basic characteristics of the system are studied by the power spectrum, Poincaré maps, 0–1 test and Lyapunov exponents. The system has rich dynamical behaviors, such as bursting oscillations, offset boosting, transient chaos, intermittent chaos and coexistence of attractors. In addition, by studying the coexisting phenomenon and spectral entropy (SE) complexity of different initial values, an initial value that is more suitable for chaotic secure communication is selected. The circuit simulation of the system using Multisim and then the actual hardware implementation of the system by Field Programmable Gate Array (FPGA), these prove the practical existence of the system. Finally, combining the methods of backstepping control, multi-switching synchronization and synchronization of different systems, a set of controllers are proposed which can realize the backstepping multi-switching synchronization of this system with a memristive chaotic system.
An autonomous memristive circuit is implemented by an active third-order generalized memristor. The mathematical model is established and the stability of the equilibrium point and divergence are analyzed. Lyapunov exponents and bifurcation analysis demonstrate the complex dynamical behaviors of the system. As an internal parameter of voltage controlled memristor is changed, the system changes from bursting chaos to general chaos, which includes chaotic bursting attractor and periodic bursting attractor. This system produces periodic bursting similar to the clusters discharge of biological neurons. Interestingly, the system differs from the single helical clusters discharge of neurons. The bifurcation mechanism of the periodic bursting behavior is explored by constructing equilibrium trajectories of the fast-scale subsystem to verify the Fold bifurcation and to establish the Hopf bifurcation sets. Finally, it is shown that a circuit experiment based on Multisim is consistent with the theoretical analysis and numerical simulations, which proves the feasibility of the real circuit.
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