Bifurcation Diagrams Research Articles

A generalized Bogdanov-Takens system with arbitrary degree

The aim of this paper is to give the bifurcation diagram and all global phase portraits in the Poincaré disc of a generalized Bogdanov-Takens system with arbitrary degree. Moreover, the system has abundant nonlinear phenomena, such as double limit cycle bifurcation, Hopf bifurcation, homoclinic bifurcation, heteroclinic bifurcation, and so on. Notice that the nonlinear phenomena of the system are more abundant and more complex than the classic Bogdanov-Takens system.

Design of a discrete memristive chaotic map: fractional-order memory, dynamics and application

In this paper, a discrete fracmemristor (DFM) model is derived based on the Caputo difference, and a new fractional-order chaotic map is designed. Dynamics of the proposed map is investigated in detail by means of Lyapunov exponent spectra, bifurcation diagrams, PE complexity and multistability analyses. Compared with the coupled discrete integer-order memristor (DIM), the map coupled with the DFM products richer dynamics, including larger attractor distribution, fewer numerically periodic windows, and higher complexity. Besides, the order becomes additional bifurcation parameter. Finally, the proposed map is implemented on Field-Programmable Gate Array (FPGA) platform, and applied in a pseudorandom number generator (PRNG), which further demonstrates its application value.

Dynamic analysis of a class of fractional‐order dry friction oscillators

This article investigates a class of Duffing nonlinear dynamic systems with fractional‐order dry friction and conducts in‐depth research on the stability, chaotic characteristics, and erosion of the safety basin of this system; the results are verified through numerical simulation. First, the average method is used to approximate the amplitude–frequency relationship of the system, and the accuracy of the analytical results is verified through numerical experiments. Second, the Melnikov method is used to obtain the conditions for the system to enter chaos in the Smale horseshoe sense, and the Melnikov curve is drawn for further verification. Then, bifurcation diagrams are drawn for the changes in various parameters in the system, with a focus on analyzing the influence of friction factors on chaotic bifurcation. By applying the definition and calculation principle of the maximum Lyapunov exponent, and drawing and utilizing the maximum Lyapunov exponent graph, the chaotic state that the system enters under different parameters is more clearly defined. Finally, the evolution law of the safety basin under various parameter changes, especially dry friction changes, is analyzed, and the erosion and bifurcation mechanism of the safety basin is studied. Comparing with the bifurcation diagram, it reveals that chaos primarily contributes to the erosion of the safety basin.

On the operational similarities of bladed rotor vibrations with casing contacts

AbstractRotor-to-stator rubbing in rotating machinery, resulting from tight clearances, introduces complex dynamics that can potentially lead to high vibrations and machine failure. Historically, the rubbing models were addressed using cylinder-to-cylinder contacts; however, recent attention has shifted towards examining blade-tip contact in turbines, which affects the systems dynamics and efficiency. This study investigates the impact of the variations in blade number on bladed rotor systems, emphasizing on the types of motion that occur as function of the operational speed in the sub-critical range. A simplified bladed rotor model has been developed, using a Jeffcott rotor with blades represented as damped elastic pendulums. The equations of motion are derived and numerical simulations are performed to explore the system’s behaviour with varying blade numbers (3, 5, 7, and 10) in order to analyse displacements, contact forces and bifurcation diagrams as function of the rotating speed. Results reveal distinct regions: periodic motion (I and III) and chaotic motion (II and IV) appear alternatively in the bifurcation diagram, with the chaotic regions occurring at specific fractions of the natural frequency and the number of blades. The study concludes that chaotic motions are associated with larger displacements and higher contact forces, and the vibrational behaviour becomes less hazardous as the number of blades increases. In addition, the appearance of periodic and chaotic motions occur in the same regions by scaling the rotating speed with the number of blades and natural frequency of the system. From an operational perspective, this dynamic investigation offers valuable insights into the severity of blade rubbing in industrial systems. It can guide the implementation of mitigation solutions to prevent worst-case failure scenarios and help to perform adjustments to either operational or design parameters.

Global Mittag-Leffler Attractive Sets, Boundedness, and Finite-Time Stabilization in Novel Chaotic 4D Supply Chain Models with Fractional Order Form

This research explores the complex dynamics of a Novel Four-Dimensional Fractional Supply Chain System (NFDFSCS) that integrates a quadratic interaction term involving the actual demand of customers and the inventory level of distributors. The introduction of the quadratic term results in significantly larger maximal Lyapunov exponents (MLE) compared to the original model, indicating increased system complexity. The existence, uniqueness, and Ulam–Hyers stability of the proposed system are verified. Additionally, we establish the global Mittag-Leffler attractive set (MLAS) and Mittag-Leffler positive invariant set (MLPIS) for the system. Numerical simulations and MATLAB phase portraits demonstrate the chaotic nature of the proposed system. Furthermore, a dynamical analysis achieves verification via the Lyapunov exponents, a bifurcation diagram, a 0–1 test, and a complexity analysis. A new numerical approximation method is proposed to solve non-linear fractional differential equations, utilizing fractional differentiation with a non-singular and non-local kernel. These numerical simulations illustrate the primary findings, showing that both external and internal factors can accelerate the process. Furthermore, a robust control scheme is designed to stabilize the system in finite time, effectively suppressing chaotic behaviors. The theoretical findings are supported by the numerical results, highlighting the effectiveness of the control strategy and its potential application in real-world supply chain management (SCM).

Discrete model and non‐linear characteristics analysis of magnetic coupled resonant wireless power transfer system with constant power load

AbstractMagnetically coupled resonant wireless power transfer (MCR‐WPT) system with constant power load (CPL), finds extensive applications in military, industrial, and medical treatment. However, this system can easily exhibit complex non‐linear behaviors under certain parameter conditions due to the influences of constant power loads, switching devices, and feedback controllers. These behaviors limit the effectiveness of controllers and reduce the efficiency and stability of the system. It is necessary to study the non‐linear characteristics of the MCR‐WPT system with CPL. The stroboscopic discrete mapping of the MCR‐WPT system with CPL is established. Then the system's non‐linear dynamics are analyzed theoretically using a bifurcation diagram, the maximal Floquet multipliers, and the maximal Lyapunov exponent spectrum obtained from the proposed discrete mapping. The results show that the MCR‐WPT system with CPL will exhibit rich non‐linear dynamics with the variation of the power of CPL, such as cyclic fold bifurcation, Neimark–Scaker bifurcation, border collision bifurcations, chaos, etc. The excellent alignment of experimental and theoretical outcomes in corresponding states confirms the accuracy of the proposed discrete mapping and the nonlinear analysis of the system. The results of this study can provide a reference for selecting parameters for an actual MCR‐WPT system with CPL.

Simulink Modeling and Analysis of a Three-Dimensional Discrete Memristor Map

The memristor, a novel device, has been widely utilized due to its small size, low power consumption, and memory characteristics. In this paper, we propose a new three-dimensional discrete memristor map based on coupling a one-dimensional chaotic map amplifier with a memristor. Firstly, we analyzed the memristor model to understand its characteristics. Then, a Simulink model for this three-dimensional discrete memristor map was developed. Lastly, the complex dynamical characteristics of the system were analyzed via equilibrium points, bifurcation diagrams, Lyapunov exponent spectra, complexity, and multistability. This study revealed the phenomena of coexisting attractors and hyperchaotic attractors. Simulink modeling confirmed that the discrete memristors effectively enhanced the chaos complexity in the three-dimensional discrete memristor map. This approach addresses the shortcomings of randomness, the lack of ergodicity, and the small key space in a one-dimensional chaotic map, thereby enriching the theoretical analysis and circuit implementation of chaos.

Dynamics of memristor coupled discrete fractional symmetric neural network model

Neural networks, also known as artificial neural networks (ANNs), represent innovative computational models inspired by the functioning of biological neurons in the human body. Currently, neural network models are employed to address intricate real-world challenges. This article focuses on constructing and dynamically analyzing interconnected symmetric neural network models inspired by the Hopfield type. These models incorporate discrete fractional memristor elements featuring quadratic memductance. The primary objective is to comprehend system characteristics by introducing nonlinearity to the inter-neuron weights in a functional manner. Additionally, the study explores the influence of electromagnetic radiation on neurons. Bifurcation diagrams, considering fractional order and Lyapunov exponents, illustrate the chaotic nature of the interconnected neural network model. The investigation delves into coexisting system states, examining coexisting bifurcations and attractors. The article offers potential for developing bio-inspired, energy-efficient, and adaptive neural network architectures, thereby contributing to advancements in artificial intelligence and neuromorphic applications.

A secure communication scheme based on generalized modified projective synchronization of a new 4-D fractional-order hyperchaotic system

This paper focuses on the exploration of the chaotic behavior of a new 4-D fractional-order hyperchaotic system with five nonlinearities. The Adomian decomposition method is employed to solve the fractional-order hyperchaotic system. The stability of equilibrium points in this system is analyzed. Through bifurcation diagrams, Lyapunov exponent spectra, chaotic attractors, 0-1 test, C 0 complexity, and spectral entropy, the chaotic dynamics of the proposed system are investigated. Using these tools, we demonstrate the fractional-order system’s sensitivity to variation in both the derivative order and initial conditions. Moreover, a modified generalized projective synchronization is developed to implement chaos synchronization between two coupled fractional-order hyperchaotic systems. Furthermore, this work presents an application of synchronization scheme in secure communication. Numerical simulations were implemented to validate the effectiveness of the proposed secure communication scheme.

Dynamics of memristive Liu system and its DSP implementation

In this paper, a novel four-dimensional memristive system is investigated to generate abundant dynamical behaviors. By combining the Liu chaotic system with an ideal memristor, an enhanced chaotic system is proposed. Dynamical analysis indicates that the new system sustains stable chaotic states and exhibits complex behaviors, with the help of the Lyapunov exponents, bifurcation diagrams, Poincaré section, parametric offset boosting, and SE complexity. The coexistence of attractors is investigated by the variation of parameters. The chaotic performance is enhanced in the proposed system, broadens the range of parameters for chaotical oscillations, and transforms periodic and quasiperiodic states into chaos. The practical applicability and feasibility of the system are validated via NIST testing and DSP implementation. The system exhibits resilient dynamical characteristics that make it highly suitable for deployment in various domains, including secure communication and signal detection.

Nonlinear Dynamics and Meshing State Transition Analysis of Spalling Defect Gears with Multi-Lubrication Conditions

The purpose of this paper is to analyze the dynamic characteristics and the meshing state transition characteristics of spalling defect gears under various lubrication conditions. The time-varying geometric parameters, motion parameters and load distribution of the gears in the state of positive meshing and reverse impact are analyzed. The time-varying meshing stiffness considering the time-varying friction coefficient under elastohydrodynamic lubrication (EHL), mixed lubrication and boundary lubrication are calculated, and the spalling defect is introduced in the stiffness calculation process. The equivalent lubrication model of the gear is established, the switching condition of the lubrication state is derived, and the distribution of the lubrication state in the torque-speed plane is discussed. The multimeshing state dynamic model of the spalling defect gear is established, and the evolution characteristics of the coexistence response and the transition of the meshing state under different lubrication conditions are simulated by using the attraction domain, phase trajectory, meshing force and bifurcation diagram. The results show that the phase trajectory of the gear with the spalling defect is deformed compared with that of the healthy gear. The change of lubrication state has little effect on the phase trajectory and attraction domain of the gear with spalling defect, but has a great influence on the meshing force. At the spalling boundary, the meshing force is changed abruptly, and the worse the lubrication state, the more obvious the mutation. Due to the influence of spalling on tooth stiffness, it will also affect the meshing force in the nonspalling area.

Complex Dynamics of a Discretized Predator–Prey System with Prey Refuge Using a Piecewise Constant Argument Method

The objective of this work is to investigate the complex dynamics of a discrete predator–prey system using the method of piecewise constant argument for discretization. An analysis is conducted to examine the presence and stability of fixed points. Furthermore, the system is shown to undergo period-doubling (PD) and Neimark–Sacker (NS) bifurcations by the use of center manifold and bifurcation theories. The feedback and hybrid control strategies are used to regulate the system’s bifurcating and chaotic behaviors. Both strategies seem to be effective in managing bifurcation and chaos inside the system. Finally, the main results are validated by numerical evidence. Parameters of the system are varied to produce time graphs, phase portraits, bifurcation diagrams, and maximum Lyapunov exponent (MLE) graphs. The discrete model displays rich dynamics, as seen in the numerical simulations and graphs, indicating a complex and chaotic system.

Chaotic Dynamics of the Fractional Order Predator-Prey Model Incorporating Gompertz Growth on Prey with Ivlev Functional Response

This paper examines dynamic behaviours of a two-species discrete fractional order predator-prey system with functional response form of Ivlev along with Gompertz growth of prey population. A discretization scheme is first applied to get Caputo fractional differential system for the prey-predator model. This study identifies certain conditions for the local asymptotic stability at the fixed points of the proposed prey-predator model. The existence and direction of the period-doubling bifurcation, Neimark-Sacker bifurcation, and Control Chaos are examined for the discrete-time domain. As the bifurcation parameter increases, the system displays chaotic behaviour. For various model parameters, bifurcation diagrams, phase portraits, and time graphs are obtained. Theoretical predictions and long-term chaotic behaviour are supported by numerical simulations across a wide variety of parameters. This article aims to offer an OGY and state feedback strategy that can stabilize chaotic orbits at a precarious equilibrium point.

Periodic response and stability analysis of a bistable viscoelastic von Mises truss

This paper examines the effect of viscoelasticity on the periodic response of a lumped parameter viscoelastic von Mises truss. The viscoelastic system is described by a second-order equation that governs the mechanical motion coupled to a first-order equation that governs the time evolution of the viscoelastic forces. The viscoelastic force evolves at a much slower rate than the elastic oscillations in the system. This adds additional time scales and degrees of freedom to the system compared to its viscous counterparts. The focus of this study is on the system’s behavior under harmonic loading, which is expected to show both regular and chaotic dynamics for certain combinations of forcing frequency and amplitude. While the presence of chaos in this system has already been demonstrated, we shall concentrate only on the periodic solutions. The presence of the intrawell and interwell periodic oscillations is revealed using the Harmonic Balance method. The study also looks at the influence of parameter changes on the system’s behavior through bifurcation diagrams, which enable us to identify optimal system parameters for maximum energy dissipation. Lastly, we formulate an equivalent viscous system using an energy-based approach. We observe that a naive viscous model fails to capture the behavior accurately depending on the system and excitation parameters, as well as the type of excitation. This underscores the necessity to study the full-scale viscoelastic system.

Design of chaotic circuit based on Knowm memristor

AbstractMost previous works in the field of nonlinear memristive chaotic systems mainly focus on non‐material memristor while seldom on material memristor. In this letter, the model and the corresponding parameter identification of the Knowm memristor is presented. The chaotic circuit based on the Knowm memristor is designed. Equilibrium points and stability of this Knowm‐memristor‐based chaotic system is analyzed. Nonlinear dynamic behavior is presented by the bifurcation diagram and Lyapunov exponents. Finally, the hardware circuit of this Knowm‐memristor‐based chaotic system is designed and the experimental results are presented for confirmation.

A 4D conservative chaotic system: dynamics and realization

This paper proposes a novel four-dimensional conservative chaotic system (4D CCS) with a simple algebraic representation, comprising only two quadratic nonlinear terms. The dynamic characteristics of the 4D CCS are investigated by Poincaré mappings, Lyapunov exponents (LE), bifurcation diagrams, equilibrium points and spectral entropy (SE) complexity algorithm. Variations in parameters, initial values, and Hamiltonian energy lead to alternations between quasi-periodic and chaotic flows in the 4D CCS. The maximum Lyapunov exponent of the 4D CCS can reach a high value of 366300 under adjusting appropriate parameters and initial values. The pseudorandom sequences generated by the 4D CCS successfully pass the NIST test. Additionally, both the electronic circuit and FPGA implementation of the 4D CCS are carried out, with the experimental results aligning closely with the simulation results.

Analyzing and Controlling chaos phenomena in fractional chaotic supply chain models

Supply Chain Management (SCM) is a critical business function that involves the planning, coordination, and control of the flow of goods, information, and finances as they move from the manufacturer to the wholesaler to the retailer and finally to the end customer. SCM is a holistic approach to managing the entire process of delivering products or services to consumers. In this study, we will enhance the findings as outlined in Anne et al. (2009). While certain attributes of these systems will have been investigated, numerous aspects of these systems will still require further scrutiny. This calls for additional research studies on these systems. This paper examines a Fractional-Order Supply Chain Management (FOSCM) model utilizing the Adomian Decomposition Method (ADM) and explores qualitative aspects through an approach that addresses existence and uniqueness. By using Arzelà–Ascoli's principle, this system proves that the Caputo FOSCM model has at least one solution. Furthermore, we investigate the dynamics of the system by using the Lyapunov Exponent (LE), Bifurcation Diagram (BD), Complexity Analysis (CA) and 0–1 test. Finally, we introduce the control for FOSCM model using the Linear Feedback Control (LFC) method. We verify the correctness of our analysis by using numerical simulations.

Hysteresis behavior and generalized Hopf bifurcation in a three-degrees-of-freedom aeroelastic system with concentrated nonlinearities

A three-degrees-of-freedom aeroelastic system with concentrated structural nonlinearities is considered. The aerodynamic loads in the model are simulated by using unsteady aerodynamic forces. First, we evaluate the linear stability region of the aerodynamic system employing the Lienard–Chipart criterion so that the stability boundaries of the center of gravity position and the uncoupled pitch natural frequency are obtained. Next, we present the distribution of the first Lyapunov coefficient l1 in the parameter space by using the Poincaré projection method to determine the type of Hopf bifurcation (subcritical or supercritical). We identify three types of hysteresis loops associated with Hopf bifurcations: the first type contains one stable equilibrium point and one stable limit cycle, the second type consists of one stable equilibrium point and two stable limit cycles, and the third type comprises two stable limit cycles. Nonlinear analysis shows that the variations in hysteresis loop types are mainly caused by the nonlinear flutter critical points, namely the saddle–node bifurcation points of limit cycles. Furthermore, we determine a degenerate Hopf bifurcation type known as generalized Hopf bifurcation (Bautin bifurcation) and its corresponding second Lyapunov coefficient, l2. Subsequently, we conduct nonlinear flutter analyses for positive and negative values of l2 by using bifurcation diagrams and phase portraits. The results show that when the uncoupled pitch natural frequency is used as the control parameter, the occurring generalized Hopf bifurcations are all supercritical (l2<0). However, when the center of gravity position is used as the control parameter, the generalized Hopf bifurcations that occur have both supercritical and subcritical (l2>0) modes.

Stability and bifurcations in a model of chemostat with two inter‐connected inhibitions and a negative feedback loop

This paper deals with a model of chemostat with two cross‐feeding species and involving two inhibitions. The excess of hydrogen inhibits the growth of acetogenic bacteria which, in turn, inhibit the growth of methanogenic hydrogenotrophic bacteria. The model is described by a system of four ordinary differential equations. We established the conditions of existence and stability of equilibria with respect to the operating parameters which are the dilution rate and the input substrate concentrations of the species of the model, then we illustrate the operating and bifurcation diagrams. This technique makes it easy to interpret different regions of operating diagrams. Inhibitions let some regions of stability rise and others vanish.

Nonlinear dynamics and pattern formation in a space–time discrete diffusive intraguild predation model

In this paper, the spatiotemporal dynamics and pattern formation of a space–time discrete intraguild predation model with self-diffusion are investigated. The model is obtained by applying a coupled map lattice (CML) method. First, using linear stability analysis, the existence and stability conditions for fixed points are determined. Second, using the center manifold theorem and the bifurcation theory, the occurrence of flip, Neimark-Sacker, and Turing bifurcations are discussed. It is shown that the patterns obtained are results of Turing, flip, and Neimark-Sacker instabilities. Numerical simulations are performed to verify the theoretical analysis and to reveal complex and rich dynamics of the model, such as times series, maximal Lyapunov exponent, bifurcation diagrams, and phase portraits. Interesting patterns like spiral pattern, polygonal pattern, and the combinations of patterns of spiral waves and stripes are formed. The CML model’s results help to understand how a spatially extended, discrete intraguild predation model forms complex patterns. Notably, the continuous reaction–diffusion counterpart of the model under study is incapable of experiencing Turing instability.