Dynamics and chaos control of q-deformed Gaussian map via superior approach.

This research delved into the effects of topological modification on the nonlinear bifurcation properties of herringbone gear transmission system (HGTS). A model utilizing the potential energy method was initially developed to analyze the time-varying meshing stiffness (TVMS) of a herringbone gear following topological modification. Additionally, a dynamic model of a bending-torsion-axis-pendulum (BTAP) coupling of HGTS with 24 degrees of freedom (DOF) was established, which took into account various factors such as topological modification, meshing errors, TVMS, damping, backlash, and bearing clearance. The model employed the Runge-Kutta method to assess the pre-topological and post-topological modification nonlinear vibration responses and bifurcation characteristic, with a specific emphasis on backlash, and speed variations. The bifurcation diagram and maximum lyapunov exponent (MLE) graph were employed to elucidate these changes. Supplementary studies utilizing time domain diagram, frequency domain diagram, phase plane diagram, poincaré map analyses provided a comprehensive understanding of the HGTS’s nonlinear dynamics. The findings indicate that while the fundamental vibration and bifurcation trends persist post-topological modification, local bifurcation and vibration responses exhibit improvements. Specifically, an increase in backlash leads to a transition from single-periodic to multi-periodic, and then to chaotic regimes, the topological modification improves in reverting some chaotic behaviors to periodic, thereby reducing the extent of chaos. Increased rotational speeds result in periodic-chaotic-periodic sequences, but topological modification shifts bifurcation points, reducing chaos. Comparisons of experimental and theoretical data corroborate the model’s accuracy, demonstrating consistent trends.

The nonlinear characteristics and the influence factors of the floating wind turbine are studied under the condition of strong coupling between the environmental load, the load and the response of system and the multi degree of freedom. The aerodynamic load of the floating wind turbine is calculated by using the blade element and momentum theory, The five wave theory and Morision equation are used to calculate the wave load, so the nonlinear dynamic model of the floating wind turbine is established. The Poincare cross section is calculated, and the dynamic response of the three degrees of freedom of system under the action of the environment load is analyzed. The influence of the wave excitation frequency on the nonlinear characteristics of the system is analyzed by using the Lyapunov exponent and the bifurcation diagram. The study shows that the amplitude of the longitudinal swing is large, the amplitude of the longitudinal wave is small, and the wave excitation frequency has a significant effect on the motion of the system. When the wave excitation frequency is near 1.4, the system is in a chaotic state, the wave excitation frequency is 0.8, and the system is in the state of the class period.

As a controllable nonlinear system, autonomous Boolean network which is easy to generate chaotic signals has become a hot research topic. We realize an 18-node autonomous Boolean network circuit which can generate high entropy chaos by using the nonlinear characteristics of two-input XOR gate, and establish an improved mathematical model for the circuit by introducing the filter coefficient. Through numerical simulation and circuit comparison experiment, we study the influence of filter coefficient and delay parameters on network dynamic characteristics of this model. The numerical simulation results are basically consistent with those of the experimental phenomena. When the transmission delay between nodes is not equal, the filter coefficient has a modulation effect on the generation of chaos, which well indicates that the introduction of the filter coefficient makes the improved mathematical model more objective to describe the physical phenomena in the circuit. In addition, using bifurcation diagram, Lyapunov exponent and permutation entropy, we further analyze the nonlinear characteristics and randomness of the autonomous Boolean network. The output chaotic sequence of the network has the characteristics of high bandwidth, high entropy and strong robustness, which is used to generate high speed physical random sequences and has passed the NIST SP 800-22 randomness tests of National Institute of Standards and Technology of America.

A computational tool for nonlinear dynamical and bifurcation analysis of chemical engineering problems

The memristor is a nonlinear element and intrinsically possesses memory function. When it works as nonlinear part of a chaotic system, the complexity and the randomness of signal will be enhanced. In this paper memristor is introduced into a three-dimensional chaotic system based on the augmented L system. The interesting and promising behaviors of complex single, double and four-scroll chaotic attractors generated only by varying a parameter have not been reported in memristive chaotic system and thus they deserve to be further investigated. It is also obvious that such a simple change of one parameter could be used to generate a variety of quite complex attractors. Therefore, as a nonlinear device the memristor plays an important role in this system. Firstly, some basic dynamical properties of the memristive chaotic system, including symmetry and in-variance, the existence of attractor, equilibrium, and stability are investigated in detail. By numerically simulating the power spectrum, Lyapunov exponent, Poincare map and bifurcation diagram, in this paper we verify that the proposed system has abundant dynamical behaviors. The sensitivities of system parameters to the chaotic behaviors are further explored by calculating, in detail, its Lyapunov exponent spectrum and bifurcation diagrams. The results of simulation and experiment are in good agreement, thereby proving the veracity of analysis. The memristive chaotic circuit is designed using the memristor, operational amplifier, analog multiplier and other conventional components. The circuit implementation of the memristive system is simulated using SPICE (simulation program with integrated circuit emphasis). The SPICE simulation results and the theoretical analysis are found to be in good agreement, and thus verifying that the system can produce chaos. Pulse synchronization has the following characteristics: low energy consumption, fast synchronization and easy-to-implement single-channel transmission. Therefore, it is more practical in chaotic secure communication. Subsequently the pulse chaos synchronization is realized from the perspective of the maximum Lyapunov exponent, and numerical simulations show the existence of new memristive chaotic system and the feasibility of pulse synchronization control, and also provide an experimental basis for further studying the applications of the memristive chaotic system in voice secure communication and information processing.

Chaotic and steady state behaviour of a nonlinear controlled gyro subjected to harmonic disturbances

This work studies a forced generalised Lienard oscillator with $$\phi ^8$$ potential with order 8 dissipation. The fixed points and their stability have been analysed for autonomous and non-dissipative Lienard oscillator. The system can exhibit three, five or seven fixed points and the corresponding stability diagram is checked and analysed. The effect of restoring parameters on the potential is also studied. Periodic, multiperiodic and chaotic monostable and bistable attractors and their coexistence have been checked through the bifurcation diagram, Lyapunov exponent, phase space and Poincare section using the fourth-order Runge–Kutta algorithm. The results obtained by the analytical methods are validated and complemented by the numerical simulations.

In this study, a 4D hyperchaotic system is constructed based on the foundation of a 3D Lü chaotic system. The newly devised hyperchaotic system possesses a sole equilibrium point, showcasing a simplified system structure that reduces complexity. This simplification offers a clearer opportunity for in-depth analysis of dynamic behaviors in the realm of scientific research. The proposed hyperchaotic system undergoes an in-depth examination of its dynamical characteristics, including chaotic attractors, equilibrium point stability, Lyapunov exponents’ spectrum, and bifurcation diagram. Numerical analysis results reveal that the attractor of this hyperchaotic system exhibits highly complex, non-periodic, and fractal structural dynamics. Its motion demonstrates extreme sensitivity and randomness, even within a wide range of variations in parameter d, affirming its hyperchaotic properties with two positive Lyapunov exponents. Hyperchaotic bifurcation diagrams typically exhibit highly intricate structures, such as fractals, branches, and period doubling characteristics, signifying that even minor parameter adjustments can lead to significant changes in system behavior, presenting diversity and unpredictability. Subsequently, to further investigate the practical utility of this hyperchaotic system, a linear feedback control strategy is implemented. Through linear feedback control, the hyperchaotic system is stabilized at its unique equilibrium point. Experimental validation is conducted using both computer software simulation Matlab, electronic circuit simulation Multisim, and embedded hardware STM32. The results of these experiments consistently align, providing theoretical support for the application of this hyperchaotic system in practical domains. Finally, leveraging the hyperchaotic keys generated by this hyperchaotic system, audio encryption is achieved using a cross-XOR algorithm, which is then realized on the embedded hardware platform STM32. The results show that the audio encryption scheme based on the hyperchaotic system is feasible, and the method is simple to implement, has nonlinear characteristics and certain algorithm complexity, which can be applied to audio encryption, image encryption, video encryption, and more.

In this paper, bifurcation points of two chaotic maps are studied: symmetric sine map and Gaussian map. Investigating the properties of these maps shows that they have a variety of dynamical solutions by changing the bifurcation parameter. Sine map has symmetry with respect to the origin, which causes multistability in its dynamics. The systems’ bifurcation diagrams show various dynamics and bifurcation points. Predicting bifurcation points of dynamical systems is vital. Any bifurcation can cause a huge wanted/unwanted change in the states of a system. Thus, their predictions are essential in order to be prepared for the changes. Here, the systems’ bifurcations are studied using three indicators of critical slowing down: modified autocorrelation method, modified variance method, and Lyapunov exponent. The results present the efficiency of these indicators in predicting bifurcation points.

Based on the study of Chua’s circuit, a novel chaotic system is reported. Basic dynamical properties of the new system are further investigated via theoretical analysis and numerical simulation, including Lyapunov exponent, Lyapunov dimension, portrait diagrams, Lyapunov exponent spectrum, bifurcation diagrams, Poincaré mapping and power spectrum. Finally, an electronic circuit is designed by the Orcad-PSpice softeware to implement the new system. The investigation results show that the new chaotic system has broad parameter regions, an maximum Lyapunov exponent approaching one, and is not topologically equivalent to Chua’s circuit. It also shows a good agreement between numerical simulation and circuit experimental simulation, which proves the existence and physical realizability of the new chaotic system.

A novel memristive elements-based chaotic circuit only containing three elements of memristor, memcapacitor and meminductor is highlighted in this paper. The state equations of the memristive circuit are described, and the dynamical behaviors such as equilibrium set, Lyapunov exponents and bifurcation diagram are revealed by theoretical analyses and numerical simulations. Simulation results demonstrate that the sum of five Lyapunov exponents in this circuit is negative and the maximal Lyapunov exponent is equal to 0.1773, which verifies typical characteristics of limit cycle and 1-scroll chaotic attractor. The most striking feature is that this system to build a high-dimensional dynamic equation only using three circuit elements and exhibit rich phenomenon of period and chaotic behavior. Moreover, equivalent realization circuit are design to verify the theoretical analyses and numerical simulations.

The nonlinear dynamic characteristics of a peak current regulation fractional-order (FO) flyback converter, considering the fractional nature of inductance and capacitance, are investigated in detail. First, the discrete iterative model of the fractional-order (FO) flyback converter under 10 kHz operating conditions is accomplished using the application of the Generalized Euler Method (GEM). On this basis, bifurcation diagrams, phase diagrams, and simulated time domain diagrams are used to describe the nonlinear dynamic behavior of the converter. The nonlinear dynamics of the converter are investigated through bifurcation and phase diagram analyses. A comprehensive examination is conducted to evaluate the impact of key parameters, including input voltage, reference current, and the fractional orders of inductance and capacitance, on the system’s stability. Furthermore, a comparative analysis is performed with conventional integer-order (IO) flyback converters to highlight the distinctive characteristics. The findings demonstrate that the FO converter manifests distinct nonlinear characteristics, including period-doubling bifurcation and chaotic behavior. Moreover, for identical parameter sets, the FO flyback converter is found to possess a smaller stability domain but a larger parameter region for bifurcation and chaos compared to its IO counterpart. This behavior allows the FO converter to more accurately capture the nonlinear dynamic characteristics of the flyback converter. Simulation results further substantiate the theoretical predictions.

This paper presents a novel four-dimensional autonomous conservative model characterized by an infinite set of equilibrium points and an unusual algebraic structure in which all eigenvalues of the Jacobian matrix are zero. The linearization of the proposed model implies that classical stability analysis is inadequate, as only the center manifolds are obtained. Consequently, the stability of the system is investigated through both analytical and numerical methods using Lyapunov functions and numerical simulations. The proposed model exhibits rich dynamics, including hyperchaotic behavior, which is characterized using the Lyapunov exponents, bifurcation diagrams, sensitivity analysis, attractor projections, and Poincaré map. Moreover, in this paper, we explore the model with fractional-order derivatives, demonstrating that the fractional dynamics fundamentally change the geometrical structure of the attractors and significantly change the system stability. The Grünwald–Letnikov formulation is used for modeling, while numerical integration is performed using the Caputo operator to capture the memory effects inherent in fractional models. Finally, an analog electronic circuit realization is provided to experimentally validate the theoretical and numerical findings.

Nonlinear vehicle dynamics are omnipresent and significantly affect driving performance and safety. Jumping vehicles, in particular, exhibit strong nonlinearities that can lead to severe vibrations and steering instabilities. This study investigates the dynamics of nonlinear jumping vehicle models with and without time delay to clarify their fundamental characteristics. Quarter- and half-car models with jumping nonlinearity are considered, and delayed feedback is introduced as an active suspension control. Numerical simulations under periodic and random excitations reveal several key findings as follows. Both models exhibit sudden, discontinuous transitions from periodic to chaotic dynamics in their bifurcation diagrams. A general relationship between the time period of periodic motions and the forcing period is also identified and further validated using a forced Duffing oscillator with time delay, confirming its generic nature. With the introduction of delay, both in-phase and out-of-phase motions emerge between the front and rear, even under simultaneous excitation. From a practical standpoint, in-phase motion is undesirable, making the realization of out-of-phase behavior within a specific delay range important for improving ride performance. This study identifies the stability regions of nonlinear quarter- and half-car models using bifurcation diagrams, Lyapunov exponents, and frequency response curves. In addition, the inclusion of a time delay is shown to effectively stabilize chaotic motions into periodic responses and to induce both in-phase and out-of-phase oscillations. These findings demonstrate that time delay plays a significant role in enhancing the stability of vehicle models with jumping nonlinearities.

This article presents the bifurcation and chaos phenomenon of the three-dimensional generalized Hénon map. We establish the existence and stability conditions for the fixed points of the system. According to the center manifold theorem and bifurcation theory, we get the existence conditions for fold bifurcation, flip bifurcation, and Naimark–Sacker bifurcation of the system. Finally, the bifurcation diagrams, Lyapunov exponents, phase portraits are carried out to illustrate these theoretical results. Furthermore, as parameter varies, new interesting dynamics behaviors, including from stable fixed point to attracting invariant cycle and to chaos, from periodic-10 to chaos, etc., are observed from the numerical simulations. In particular, we find the double-cycle phenomenon from bifurcation diagrams and phase portraits.