Period-doubling Bifurcation Phenomena Research Articles

Qualitative behaviors of a four-dimensional Lorenz system

In this paper, the qualitative behaviors of an important four-dimensional Lorenz system with wild pseudohyperbolic attractor that proposed in (Gonchenko et al 2021 Nonlinearity 34 2018–47) are considered. Here, we prove that the four-dimensional Lorenz system with varying parameters is global bounded according to Lyapunov’s direct method. Furthermore, we provide a collection of global absorbing sets, where in addition we obtain the rate of the trajectories going from the exterior to the global absorbing set. In particular, we solve the critical case k→0+ that cannot be resolved by using the previous methods. The fundamental qualitative behaviors are analyzed theoretically and numerically. We present bifurcation diagrams to further explore the complicated dynamical behaviors of this system. The period-doubling bifurcation phenomenon is found. To illustrate the efficiency of our method, we present numerical simulations to show the validity of our research results. Finally, we present some applications of our research results in this paper.

Multi-scale contact induced period-doubling vibrations in rotor systems: Numerical and experimental studies

Bolted disk structure has the advantages of easy production and assembly and it is widely adopted in the design of gas turbine rotors. However, the discontinuity of bolted structure and the contact effects between the disks produce complex nonlinear properties to the rotor system, resulting in difficulties in solving the vibration dynamics of the rotors. In this paper, in order to reflect the contact effect between the rotor disks more accurately, a multi-scale contact mechanics model considering the whole deformation process of the material is deduced based on the skewed distribution theory. The contact effect between disks is considered by introducing an equivalent material layer. Newmark-β method is applied to obtain the dynamic response of the rotor system under different preloads. A multi-disk rotor-bearing test bench is designed to conduct dynamic tests to verify the accuracy of the contact modeling method and obtained dynamic results. Numerical and experimental results show that adjusting the preloads has the effect to change the bifurcation state of the rotor system and the increase in the preloads leads to the period-doubling bifurcation phenomenon moving toward low speed, and leads to the early appearance of some combined frequency components (such as 2fn-fr, where fn is the working frequency and fr is the oil film whirl frequency). This research provides a theoretical reference for the modeling, dynamic behavior prediction as well as fault diagnosis of rod-fastened gas turbine rotor systems.

A Simple memristive chaotic system with complex dynamics and ITS image encryption application

In this paper, a new memristive chaotic system with a relatively simple structure and complex dynamic behavior is proposed by introducing an absolute memristor into the VB19 chaotic system as a positive feedback term. The analysis shows that the new system is a fractional dimensional dissipative system and is unstable at the line equilibrium point. The system’s dynamic characteristics are studied using a bifurcation diagram and Lyapunov exponent’s spectrum. The numerical simulation shows that the period-doubling bifurcation phenomenon occurs when changing the parameters and system change from period to chaos. When the initial condition changes, the dynamic evolution of the system indicates that the coexistence of multiple attractors and chaotic signals can be controlled by local amplitude. In addition, a new image encryption algorithm based on the system is designed and the performance of this algorithm is tested through numerical analyses. The system’s analog circuit is built using electronic components and also realized by the microprocessor. The circuit realization results consistent with the numerical simulation results are obtained to verify the physical existence of the system.

Complex dynamic analysis of risk-averse newsvendor models with buyback guarantee financing

In the framework of the newsvendor model, both single- and multi-period models are proposed to investigate the ordering strategy of a risk-averse retailer under buyback guarantee financing (BGF) and stochastic demand. In the single-period model, the classical newsvendor is extended by introducing risk aversion and capital constraints simultaneously. It is found that the optimal ordering quantity is a monotonic function in some parameters, such as increasing in the retailer's risk aversion and salvage value, while decreasing in the initial capital and wholesale price. The multi-period model is a dynamical model described by the bounded rationality decision adjustment rules. We then investigate the complex dynamics they admit, such as period-doubling bifurcation and chaotic phenomena, and have found that the dynamical models experience more complex behaviour than their static counterpart. Furthermore, in the dynamical case, we could see periodic, bifurcating or chaotic changes in ordering quantity. We also investigate the influence of BGF on ordering quantity, which may lead to a risk-averse retailer to order aggressively. Finally, some numerical simulations are given to illustrate the theoretical results and some suggestions are also provided.

Bifurcation Mechanism and Stabilization of V2C Controlled Buck Converter

The V 2 C controlled buck converter by constant-frequency pulse-width modulation in continuous conduction mode gives rise to a great variety of instability behaviors, depending on the circuit parameter values. In this paper, the discrete-time model of the regulator is built by taking a current sampling resistor into consideration. The resulting Monodromy matrix is used to investigate the bifurcation phenomenon and stabilization property. The converter shows a series of period-doubling bifurcation phenomena accordingly as the feedback amplification coefficient increases. At the same time, a couple of Floquet multipliers pass through the unit circle along the negative real axis. Therefore, it reveals the mechanism of a series of period-doubling bifurcations happened in the system from the perspective of stability. Based on Floquet theory, a sine voltage compensation method is proposed to stabilize the bifurcation and chaotic behaviors. The simulations and experimental results proved the theoretical analysis.

Analysis of limit cycles and stochastic responses of a real-power vibration isolation system under delayed feedback control

In this paper, the dynamical properties of a real-power vibration isolation system with delayed feedback control subjected to deterministic and stochastic excitations are considered. According to the free vibration analysis, it is found that a large number of limit cycles may be existed for certain time delay and feedback gain. Then, the relationship of amplitude and frequency is derived for the undamped system. For the system with harmonic excitation, multi-valued phenomena are observed due to the existence of the limit cycles. In this respect, with the change of time delay, in every period the response is similar to time delay island, and the number of islands is different under different excitation frequency. Additionally, for analyzing the complex dynamic properties, the vibration isolation system with Gauss white noise excitation is explored by the largest Lyapunov exponent and the stationary probability density. The symmetrical period-doubling bifurcation phenomenon is found and verified. Finally, by using Monte Carlo simulation, the stationary probability density is explored from original system. The change of time delays can induce the occurrence of stochastic bifurcation and the response from two peaks becomes triple peaks.

Nonlinear Vibrations of Buried Rectangular Plate

We perform an investigation on the vibration response of a simply supported plate buried in glass particles, focusing on the nonlinear dynamic behaviors of the plate. Various excitation strategies, including constant-amplitude variable-frequency sweep and constant-frequency variable-amplitude sweep are used during the testing process. We employ the analysis methods of power spectroscopy, phase diagramming, and Poincare mapping, which reveal many complicated nonlinear behaviors in the dynamic strain responses of an elastic plate in granular media, such as the jump phenomena, period-doubling bifurcation, and chaos. The results indicate that the added mass, damping, and stiffness effects of the granular medium on the plate are the source of the nonlinear dynamic behaviors in the oscillating plate. These nonlinear behaviors are related to the burial depth of the plate (the thickness of the granular layer above plate), force amplitude, and particle size. Smaller particles and a suitable burial depth cause more obvious jump and period-doubling bifurcation phenomena to occur. Jump phenomena show both soft and hard properties near various resonant frequencies. With an increase in the excitation frequency, the nonlinear added stiffness effect of the granular layer makes a transition from strong negative stiffness to weak positive stiffness.

Nonlinear dynamics of charged particle slipping on rough surface with periodic force

This paper deals with two types of period-doubling bifurcation phenomena in a particle system, where the charged particle slips on rough surface with periodic force. The charged particle system is characterized by a two-dimensional nonlinear mapping, i.e., the Fermi–Ulam model (FUM) with dissipation. Based on the dissipative FUM, theoretical investigations are performed to identify the two types of period-doubling bifurcations with the assist of eigenvalue analysis approach and further to reveal their underlying mechanisms of the particle system as the strength of the periodic force for the particle system increases. Finally, bifurcation diagram and the maximum Lyapunov exponent are employed to analyze the dynamical evolution of bifurcation behaviors in the particle system quantitatively. These results are helpful for the in-depth understanding of transport mechanism of charged particle system.

Numerical study on the discharge characteristics and nonlinear behaviors of atmospheric pressure coaxial electrode dielectric barrier discharges**Project supported by the National Natural Science Foundation of China (Grant Nos. 11447244 and 11405208), the Science Foundation of Hengyang Normal University, China (Grant No. 14B41), and the Hunan Provincial Applied Basic Research Base of Optoelectronic Information Technology, China (Grant No. GDXX010).

The discharge characteristics and temporal nonlinear behaviors of the atmospheric pressure coaxial electrode dielectric barrier discharges are studied by using a one-dimensional fluid model. It is shown that the discharge is always asymmetrical between the positive pulses and negative pulses. The gas gap severely affects this asymmetry. But it is hard to acquire a symmetrical discharge by changing the gas gap. This asymmetry is proportional to the asymmetric extent of electrode structure, namely the ratio of the outer electrode radius to the inner electrode radius. When this ratio is close to unity, a symmetrical discharge can be obtained. With the increase of frequency, the discharge can exhibit a series of nonlinear behaviors such as period-doubling bifurcation, secondary bifurcation and chaotic phenomena. In the period-doubling bifurcation sequence the period-n discharge becomes more and more unstable with the increase of n. The period-doubling bifurcation can also be obtained by altering the discharge gas gap. The mechanisms of two bifurcations are further studied. It is found that the residual quasineutral plasma from the previous discharges and corresponding electric field distribution can weaken the subsequent discharge, and leads to the occurrence of bifurcation.

Bifurcation study of neuron firing activity of the modified Hindmarsh–Rose model

In this paper, the effects of different parameters on the dynamic behavior of the nonlinear dynamical system are investigated based on modified Hindmarsh---Rose neural nonlinear dynamical system model. We have calculated and analyzed dynamic characteristics of the model under different parameters by using single parameter bifurcation diagram, time response diagram and two parameter bifurcation diagram. The results show that the period-adding bifurcation (with or without chaos), period-doubling bifurcation and intermittent chaos phenomenon (periodic and intermittent chaotic) can be observed more clearly and directly from the two parameter bifurcation diagram, and the optimal parameters matching interval can also be found easily.

A Brief History of the Most Remarkable Numbers $$\pi $$ π , $$g$$ g and $$\delta $$ δ in Mathematical Sciences with Applications

This paper deals with a brief history of the most remarkable numbers $$\pi , g \text {and} \delta $$ in mathematical sciences with their many examples of applications. Several series, products, continued fractions and integral representations of $$\pi $$ are discussed with examples. The celebrated Newton method of approximation of $$\pi $$ to many decimal places is included. The appearance of $$\pi $$ in many problems, formulas, elliptic integrals and in probability and statistics is presented with examples of applications including the Tchebycheff problem of prime numbers, the Buffon needle problem and the Euler quadratic polynomial. The golden number $$g$$ and its applications to algebra and geometry are briefly discussed. The Feigenbaum universal constant, $$\delta $$ is discovered in 1978 and it is found to occur in many period doubling bifurcation phenomena in the celebrated logistic map and the Lorenz differential equation system with chaotic (or aperiodic) solutions. Included is a numerically computed Lorenz attractor which resembles a butterfly or figure eight. The Lorenz attractor is a strange attractor because it has a non-integer (or fractal) dimension. The major focus of this article is to provide basic pedagogical information through historical approach to mathematics teaching and learning of the fundamental knowledge and skills required for students and teachers at all levels so that they can understand the concepts of mathematics, and mathematics education in science and technology and pursue further research.

Dynamic Cournot duopoly games with nonlinear demand function

In this paper, dynamic duopolistic Cournot models are investigated with discrete time scales under the assumption of unknown inverse demand function and linear cost. With this motivation, we consider different types of models: bounded rational duopoly, Puu’s duopoly, bounded rational duopoly with delay, and bounded rational multi-team model. In these models, the firms use two important adjustment mechanism, the bounded rationality and Puu’s approach, to update their quantity in each period. The locally asymptotic stability of the fixed point of each model is investigated and complex dynamic characteristics including period doubling bifurcation, strang attractors and chaotic phenomena are also discussed. Numerical simulations are carried out to show such complex behavior of the four models and to point out the impact of the models’ parameters on the stability of the fixed points.

Discrete iterative-map modeling and dynamical analysis of digital voltage-mode controlled buck converter with dual-edge modulation

The operation principle of digital voltage-mode controlled buck converter with dual-edge modulation is analyzed in this paper. Based on the state equation of buck converter and six possible evolutions in one switching cycle, the discrete iterative-map model of digital voltage-mode controlled buck converter with dual-edge modulation is established. Ignoring the quantization error of analog-digital converter and on the basis of its discrete iterative-map model, the nonlinear dynamical behavior of digital voltage-mode controlled buck converter with dual-edge modulation is investigated in detail. Taking the input voltage and the load resistance as bifurcation parameters, the output voltage bifurcation diagram and the inductor current bifurcation diagram are plotted. Through analyzing the bifurcation diagrams, it is indicated that there are two kinds of similar but different Hopf bifurcation phenomena. By use of Poincar section, time-domain simulation waveforms and phase portraits, two different Hopf bifurcations and low-frequency oscillation phenomena are compared and studied. Observing the inductor current and capacitor voltage waveforms respectively, it is obviously found that their oscillation frequencies and amplitudes are different, the shapes of two Poincar$ sections and phase portraits are also different. In order to verify the correctness of the simulation and theoretical analysis, the eigenvalues of Jacobian matrix of the discrete iterative map model are introduced and solved in two kinds of stable evolutions. Through analyzing variation of eigenvalues of Jacobi matrix with input voltage, the existence and difference of two kinds of Hopf bifurcation phenomena are proved theoretically. Moreover, it is observed in this paper that the odd period-doubling bifurcation phenomenon exists in digital voltage-mode controlled buck converter with dual-edge modulation for the first time, where the operation state of the buck converter turns from period-one into period-three. Its authenticity is verified by using the time-domain simulation waveforms and phase portraits. In order to approach to the actual circuit, the equivalent series resistances of capacitor and inductor are considered. The actual circuit is simulated by using the software Psim. A comparison shows that there are little differences between the theoretical simulation and the actual circuit simulation. So the theoretical simulation can be used to analyze the performances of the actual circuit. The research results in this paper have guiding significance and practical value for designing the digital voltage-mode controlled buck converter with dual-edge modulation.

Analysis of effect of random perturbation on dynamic response of gear transmission system

In order to investigate the effects of random perturbation of a low-frequency excitation caused by torque fluctuations, gear damping ratio, gear backlash, meshing frequency and meshing stiffness, the random dynamic model of a single pair of three-degree-of-freedom spur gear transmission system is established. With gear meshing frequency changing, the dynamic characteristics of the gear transmission system were analyzed by bifurcation diagram, phase diagram, time course diagram and Poincaré map of the system. The effects of random perturbation caused by a low-frequency excitation caused by torque fluctuations, gear damping ratio, gear backlash, meshing frequency and meshing stiffness were comparative analyzed. Numerical simulation shows that the gear transmission system with nonlinear clearance exists rich period-doubling bifurcation phenomenon. With the increasing of the gear meshing frequency, gear transmission system will be from the chaotic motion to periodic motion by inverse period-doubling bifurcation. The effect of the meshing frequency random perturbation on the gear transmission system movement is largest. On the contrary, the effect of the meshing stiffness random perturbation on the system is minimum.

Complexity analysis of dual-channel game model with different managers’ business objectives

This paper considers dual-channel game model with bounded rationality, using the theory of bifurcations of dynamical system. The business objectives of retailers are assumed to be different, which is closer to reality than previous studies. We study the local stable region of Nash equilibrium point and find that business objectives can expand the stable region and play an important role in price strategy. One interesting finding is that a fiercer competition tends to stabilize the Nash equilibrium. Simulation shows the complex behavior of two dimensional dynamic system, we find period doubling bifurcation and chaos phenomenon. We measure performances of the model in different period by using the index of average profit. The results show that unstable behavior in economic system is often an unfavorable outcome. So this paper discusses the application of adaptive adjustment mechanism when the model exhibits chaotic behavior and then allows the retailers to eliminate the negative effects.

TRANSITION CURVES AND BIFURCATIONS OF A CLASS OF FRACTIONAL MATHIEU-TYPE EQUATIONS

A general version of the fractional Mathieu equation and the corresponding fractional Mathieu–Duffing equation are established for the first time and investigated via the harmonic balance method. The approximate expressions for the transition curves separating the regions of stability are derived. It is shown that a change in the fractional derivative order remarkably affects the shape and location of the transition curves in the n = 1 tongue. However, the shape of the transition curve does not change very much for different fractional orders for the n = 0 tongue. The steady state approximate responses of the corresponding fractional Mathieu–Duffing equation are obtained by means of harmonic balance, polynomial homotopy continuation and technique of linearization. The curves with respect to fractional order versus response amplitude, driving amplitude versus response amplitude with different fractional orders are shown. It can be found that the bifurcation point and stability of branch solutions is different under different fractional orders of system. When the fractional order increases to some value, the symmetric breaking, saddle-node bifurcation as well as period-doubling bifurcation phenomena are found and exhibited analytically by taking the driving amplitude as the bifurcation parameter.

A unified bifurcation control strategy for voltage source inverter

There exist fast scale and slow scale nonlinear dynamic behaviors in the voltage source inverter. Period doubling bifurcation and low-frequency oscillation instability phenomena are investigated. A unified bifurcation control strategy based on frequency-domain transfer function is proposed to suppress the fast-scale and slow-scale instabilities under the guarantee of the inverter's good dynamic and static performance. The stability boundary of the inverter is greatly expanded with the help of the controller. Harmonic balance method and small-signal averaged model are used to verify the effectiveness of the proposed control method when it is used to suppress the period-doubling bifurcation and the low-frequency oscillation, the stability domain border is also derived with the help of the analysis method. Theoretical analysis and simulation results show that the proposed control strategy is correct and feasible.

Study on period doubling bifurcation in current-mode SEPIC converter

Choosing the current of the front inductor as control object, the discrete time model of single-ended primary-inductor converter (SEPIC) converter working in the discontinuous conduction mode (DCM) is established in this paper, and the stability of the fixed point is analyzed. By studying the phase portrait, power spectrum and bifurcation diagram of the circuitry, a special period-doubling bifurcation phenomenon is found in this circuitry. The circuit experiences period, double period, quadruple period and then double period, quadruple period, finally enters into chaotic condition. The experimental results accord with the simulation results, which confirms that the special period-doubling bifurcation exists in the SEPIC converter working in the DCM.

STABILIZING A TWO-CELL DC-DC BUCK CONVERTER BY FIXED POINT INDUCED CONTROL

In this paper, we study nonlinear and bifurcation behavior of a two-cell DC-DC buck power electronic converter. The system shows nonsmooth period doubling bifurcation and chaotic phenomena in a certain zone of parameter space. This zone is located both analytically and from numerical simulations. One-dimensional, two-dimensional bifurcation diagrams and Lyapunov exponent spectrum are used to detect the different dynamic behaviors of the system. The Fixed Point Induced Control (FPIC) technique is applied to the system in order to widen the stability zone. The performance of the FPIC technique applied to the stabilization of a two-cell DC-DC buck converter is analyzed. With this technique, stabilization is achieved without changing the fixed point. The robustness in the presence of a noisy environment is checked by numerical simulations by considering different noise levels.

Nonlinear Characteristic of Two-Span Dissymmetric Elastic Supported Rotor System with Crack and Rub-Impact Faults

A new twenty-freedom non-linear dynamic model for two-span dissymmetric elastic rotor system was established by considering the crack fault, rub-impact fault and oil film force. The complex motion characteristics of this system were numerically studied and the effect of different parameters on the system is investigated. Moreover, the both cases with and without crack fault were discussed in detail. The results indicate that the system has obvious single-period, period-doubling bifurcation and chaos phenomena. There are marked influences of the crack faults on the nonlinear characteristics of the system, especially under some the rotating speed. The case considering crack faults has different chaos region from the case without crack fault. The chaos phenomenon appears at lower rotating speed, and when the chaos comes again, the corresponding rotating speed extends backwards. Synchronously, the influence of crack fault increases along with the depth of crack. The results may provide theoretical references for rotor fault diagnosis.