Extreme multistability in a fractional-order thin magnetostrictive actuator (TMA)

Hysteretic dynamics inducing coexistence of attractors in a thin magnetostrictive actuator system with quintic nonlinearity

Aero-engine rotor system contains a variety of connectors such as bolts and bearings. However, the dynamic behaviors of the connection are complex in actual engineering. In this paper, the dynamic behavior of the rotor-bearing system with bolted joints and the influence of the parameters of the joints on it are clarified. The parameters include the bearing clearance and the deflection caused by the uneven bolt preload. A rotor-bearing model with bolted joints is established by using Lagrange equations. The bifurcation diagram is solved to explore the tendency of vibration of the system at different speeds by considering the Hertz contact force. Furthermore, the phase diagram, Poincaré map, time-domain steady-state response curve, and spectrum diagram are used to discuss the dynamic behaviors of the system specifically. The influence of the bearing clearance change and uneven bolt preload on the dynamic behaviors is investigated by using the bifurcation diagram. The result shows that the motion of the system is extremely complex, which includes chaotic motion. The Lyapunov exponent is then calculated to verify whether the system enters chaos. The correctness of the model is verified by comparing the result in this work with those in the literature. The model can clarify the dynamic behaviors of the system well and has high accuracy, which can provide the theoretical guidance for the design of rotor-bearing systems with bolted joints.

A non-autonomous mega-extreme multistable chaotic system

Memristor, as a nonlinear element, provides many advantages thanks to its superior properties to design different chaotic circuits. Thus, a novel four-dimensional double-scroll chaotic system with line equilibria as well as two unstable equilibria based on the flux-memristor model is proposed in this paper. The effects of initial values and parameters on the dynamic characteristics of the system are studied in detail by means of phase diagrams, Lyapunov exponent spectrums, bifurcation diagrams, two-parameter bifurcation diagrams and basins of attraction. Besides, a series of complex phenomena in the system, such as sustained chaos, bistability, transient chaos and coexisting attractors are observed, proving that the chaotic system has rich dynamic characteristics. Also, spectral entropy (SE) complexity algorithm and [Formula: see text] complexity algorithm based on structure complexity are adopted to analyze the complexity of the system. Additionally, PSPICE circuit simulation and Micro-Controller Unit (MCU) hardware experiment are carried out to verify the correctness of theoretical analysis and numerical simulation. Finally, the pulse chaos synchronization is achieved from the perspective of maximum Lyapunov exponent, and numerical simulations demonstrate the occurrence of the proposed system and practicability of the pulse synchronization control.

Understanding and characterizing multistabilities, whether homogeneous or heterogeneous, is crucial in various fields as it helps to unveil complex system behaviors and provides insights into the resilience and adaptability of these systems when faced with perturbations or changes. Homogeneous and heterogeneous multistabilities refer, respectively, to situation in which various multiple stable states within a system are qualitatively similar or distinct. Generating such complex phenomena with multi-scrolls from inherent circuits is less reported. This paper aims to investigate extreme multistability dynamics with homogeneous and heterogeneous multi-scrolls in two coupled resonant oscillators through a shunted Josephson junction. Analysis of equilibrium points revealed that the system supports both hidden and self-excited attractors. Various dynamical tools, including bifurcation diagrams, spectrum of Lyapunov exponents, and phase portraits, are exploited to establish the connection between the system parameters and various complicated dynamical features of the system. By tuning both system parameters and initial conditions, some striking phenomena, such as homogeneous and heterogeneous extreme multistability, along with the emergence of multi-scrolls, are illustrated. Furthermore, it is observed that one can readily control the number of scrolls purely by varying the initial conditions of the investigated system. A multi-metastable phenomenon is also captured in the system and confirmed using the finite-time Lyapunov exponents. Finally, the microcontroller implementation of the system demonstrates strong alignment with the numerical investigations.

A flexible rotor bearings model supported by nonlinear oil film forces is presented, where gyroscopic effect, static unbalance and couple unbalance are all taken into consideration. The dynamic equations are solved using the Runge–Kutta method. Phase portraits, Poincare maps, bifurcation diagram and the maximum Lyapunov exponents are employed to analyze the nonlinear dynamical behavior of the rotor system under different system parameters. The numerical results show that the system exhibits rich nonlinear dynamical behavior including periodic motions, period doubling motions and chaotic motions as the system parameters are varied and the main route that leads the system to chaos is period doubling bifurcation.

SummaryThis paper constructs a three‐dimensional (3D) memristor‐based system for creating multiwing chaotic attractors. A second‐degree polynomial memristance function and a sixth‐order exponent internal state memristor function with one parameter are employed, and the complexity of attractors is increased. A detailed analysis on dynamical behaviors of the proposed system are described, such as the bifurcation diagrams, finite‐time local Lyapunov exponents, time series, phase portraits, and Poincaré maps. By adjusting the design parameters, the system displays two‐to‐eight–wing chaotic attractors, especially the five‐wing and seven‐wing attractors, which have never been found in the known systems. Further, we provide the calculation formula of the number of wings in the system, discuss the distribution of the involving inner holes on the plane, and design an electronic circuit to realize the proposed system. The experimental results of the circuit implementation agree with the numerical simulations on Matlab well. It indicates the potential engineering applications for various chaos‐based information systems.

In this paper, a new memristor-based multi-scroll hyper-chaotic system is designed. The proposed memristor-based system possesses multiple complex dynamic behaviors compared with other chaotic systems. Various coexisting attractors and hidden coexisting attractors are observed in this system, which means extreme multistability arises. Besides, by adjusting parameters of the system, this chaotic system can perform single-scroll attractors, double-scroll attractors, and four-scroll attractors. Basic dynamic characteristics of the system are investigated, including equilibrium points and stability, bifurcation diagrams, Lyapunov exponents, and so on. In addition, the presented system is also realized by an analog circuit to confirm the correction of the numerical simulations.

Image privacy is getting more and more attention, so it is necessary to design a secure and efficient image encryption algorithm. This paper proposes a novel Logistics-Henon-Memristor Coupled Chaotic Map (LHM-CCM), which exhibits high sensitivity to system parameters as demonstrated by Lyapunov exponents, bifurcation diagrams, offset-boosting analysis and extreme multistability. Based on this system, a non-uniform pixel segmentation image encryption algorithm with multiple randomization elements is introduced, dynamically controlling pixel segmentation using a hash matrix. The Josephus scrambling and bit-plane reconstruction achieve disruption and diffusion, respectively. It shows excellent performance in terms of sequence randomness, resistance to attacks, key security, and robustness against interference. Moreover, encrypting a 256 × 256 grayscale image only takes 0.0696 s. The proposed algorithm demonstrates significant advantages in encryption and decryption efficiency while ensuring security.

Recently, chaotic systems with hidden attractors and multistability have been of great interest in the field of chaos and nonlinear dynamics. Two special categories of systems with multistability are systems with extreme multistability and systems with megastability. In this paper, the simplest (yet) megastable chaotic oscillator is designed and introduced. Dynamical properties of this new system are completely investigated through tools like bifurcation diagram, Lyapunov exponents, and basin of attraction. It is shown that between its countable infinite coexisting attractors, only one is self-excited and the rest are hidden.

<abstract><p>In this work, a five-parameter jerk system with one hyperbolic sine nonlinearity is proposed, in which $ \varepsilon $ is a small parameter, and $ a $, $ b $, $ c $, $ d $ are some other parameters. For $ \varepsilon = 0 $, the system is $ Z_{2} $ symmetric. For $ \varepsilon \neq {0} $, the system loses the symmetry. For the symmetrical case, the pitchfork bifurcation and Hopf bifurcation of the origin are studied analytically by Sotomayor's theorem and Hassard's formulas, respectively. These bifurcations can be either supercritical or subcritical depending on the governing parameters. In comparison, it is much more restrictive for the origin of the Lorenz system: Only a supercritical pitchfork bifurcation is available. Thus, the symmetrical system can exhibit very rich local bifurcation structures. The continuation of local bifurcations leads to the main contribution of this work, i.e., the discovery of two basic mechanisms of chaotic motions for the jerk systems. For four typical cases, Cases A–D, by varying the parameter $ a $, the mechanisms are identified by means of bifurcation diagrams. Cases A and B are $ Z_{2} $ symmetric, while Cases C and D are asymmetric (caused by constant terms). The forward period-doubling routes to chaos are observed for Cases A and C; meanwhile, the backward period-doubling routes to chaos are observed for Cases B and D. The dynamical behaviors of these cases are studied via phase portraits, two-sided Poincaré sections and Lyapunov exponents. Using Power Simulation (PSIM), a circuit simulation model for a chaotic jerk system is created. The circuit simulations match the results of numerical simulations, which further validate the dynamical behavior of the jerk system.</p></abstract>

A new 4-D memristive system is presented in this paper. The peculiarity of the model is that it displays a line of equilibrium points and it has hyperjerk dynamics in a particular range of the parameters space. The behavior of the proposed system is investigated through numerical simulations, by using phase portraits, Lyapunov exponents and bifurcation diagrams. Also, its circuital implementation confirms the memristive systems expected dynamics.

Since the concept of memristor was proposed, the memristor, as the fourth-generation electronic component, has attracted great attention from researchers. Memristors can be used not only for nonvolatile memory but also to mimic the behavior of nervous system. The inherent nonlinearity of memristor makes it valuable in nonlinear circuits. Since it is still difficult to realize the commercial application of memristors at present, designing suitable memristor models can play a guiding role in practical application. This work proposes a novel memristor model with a time-delay state variable. The proposed memristor cannot only generate pinched hysteresis loop under periodic signal excitation, but also can generate chaotic current or even hyperchaotic current under DC voltage. The nonlinear dynamics of the proposed time-delay memristor are studied by phase portraits, bifurcation diagram and Lyapunov exponents. Furthermore, the proposed memristor is used in the HR neuron model to study neuronal electrical activities with electromagnetic induction. Multiple firing patterns of the memristive neuron can be generated such as periodic, bursting and chaotic. Finally, the memristor emulator circuit and the HR neuron model circuit are designed and simulated by Pspice.

ABSTRACTThis paper investigates a three-dimensional autonomous chaotic flow without linear terms. Dynamical behaviour of the proposed system is investigated through eigenvalue structures, phase portraits, bifurcation diagram, Lyapunov exponents and basin of attraction. For a suitable choice of the parameters, the proposed system can exhibit anti-monotonicity, periodic oscillations and double-scroll chaotic attractor. Basin of attraction of the proposed system shows that the chaotic attractor is self-excited. Furthermore, feasibility of double-scroll chaotic attractor in the real word is investigated by using the OrCAD-PSpice software via an electronic implementation of the proposed system. A good qualitative agreement is illustrated between the numerical simulations and the OrCAD-PSpice results. Finally, a finite-time control method based on dynamic sliding surface for the synchronisation of master and slave chaotic systems in the presence of external disturbances is performed. Using the suggested control technique, the superior master–slave synchronisation is attained. Illustrative simulation results on the studied chaotic system are presented to indicate the effectiveness of the suggested scheme.

The design of systems without equilibrium or with line of equilibrium points is a subject which has started to attract the interest of the research community the last decade. In this direction, various chaotic systems with hidden attractors, which are based on memristors or memristive systems, have been proposed. In this chapter a new 4-D memristive system is presented. The peculiarity of the model is that it displays a line of equilibrium points for a range of the parameters as well as no-equilibrium for another range of the parameters. System in both occasions presents a chaotic behavior with hidden attractors. The behavior of the proposed system is investigated through numerical simulations, by using phase portraits, Lyapunov exponents and bifurcation diagrams. The adaptive control scheme of the system is presented in order to prove that the memristive system’s dynamical behavior can be controlled. Also, we have designed an electronic circuit to confirm the feasibility of the system in both cases.