Doubling Bifurcation Research Articles

Global studies on a continuous planar piecewise linear differential system with three zones

This paper is concerned with the global dynamics of a continuous planar piecewise linear differential system with three zones, where the dynamic of the one of the exterior linear zones is saddle and the remaining one is anti-saddle. We give all global phase portraits in the Poincaré disc and the complete bifurcation diagram including boundary equilibrium bifurcation curves, degenerate boundary equilibrium bifurcation curves, homoclinic bifurcation curves and double limit cycle bifurcation curves. Its application in a second-order memristor oscillator is shown. Finally, some numerical phase portraits are demonstrated to illustrate our theoretical results.

Rubbing vibration characteristics of double-rotor system under wave load

Structural characteristics of a certain type of gas turbine double rotor and its hull are analyzed. The wave of a regular form is considered, and the wave load is simplified to the form of sinusoidal force. The rubbing force and nonlinear supporting force of the bearing are deduced as per the Hertz contact theory and Coulomb friction law. The mechanical model of the double rotor system under the combined action of the wave load and rotor friction impact is established, which is solved by the fourth-order Runge-Kutta method. The bifurcation diagram, phase diagram, Poincare section diagram, axis trajectory diagrams, effects of rotor eccentricity, and friction coefficient on the nonlinear dynamic characteristics of the system under rub impact fault are studied. The obtained results show that with an increase in eccentricity, the system enters into a chaotic state through the period doubling and paroxysmal bifurcations, and successively experiences nonlinear behaviors such as periodic 2 bifurcation, periodic 4 bifurcation, and quasi periodic and chaos. In the double rotor system, the nonlinear dynamic behavior of the single period and semi Nermark-Sacker bifurcation occurs with increasing friction coefficient. The present findings can provide a theoretical basis for condition monitoring, fault diagnosis, and design optimization of this type of gas turbine double rotor system.

Double grazing bifurcations of the non-smooth railway wheelset systems

There are numerous non-smooth factors in railway vehicle systems, such as flange impact, dry friction, creep force, and so on. Such non-smooth factors heavily affect the dynamical behavior of the railway systems. In this paper, we investigate and mathematically analyze the double grazing bifurcations of the railway wheelset systems with flange contact. Two types of models of flange impact are considered, one is a rigid impact model and the other is a soft impact model. First, we derive Poincaré maps near the grazing trajectory by the Poincaré-section discontinuity mapping (PDM) approach for the two impact models. Then, we analyze and compare the near grazing dynamics of the two models. It is shown that in the rigid impact model the stable periodic motion of the railway wheelset system translates into a chaotic motion after the grazing bifurcation, while in the soft impact model a pitchfork bifurcation occurs and the system tends to the chaotic state through a period doubling bifurcation. Our results also extend the applicability of the PDM of one constraint surface to that of two constraint surfaces for autonomous systems.

Extremely rich dynamics of coupled heterogeneous neurons through a Josephson junction synapse

Artificial neural networks are generally used to emulate some biological activities of the brain. Those neurons in the network are connected to others through synapses. As a result, several works have been devoted to the design of various artificial synapses, including the Josephson Junction Synapse (JJS). In this contribution, a model of the Hindmarsh–Rose neuron coupled with a FitzHugh–Nagumo neuron through a JJS is considered. That JJS enables us to simulate the effects of the magnetic field by providing additional phase error between the junctions. The Hamilton function related to the energy released by the coupled neurons during the transition between electrical activities is determined using the Helmholtz theorem. The equilibrium points of the proposed model are investigated, and their stability justifies the self-excited dynamics of the model. During studies, sets of phenomena such as Hopf bifurcation, double Hopf bifurcation, periodic spikings, periodic and chaotic burstings, (resp. firings) are found. More importantly, several coexisting activities are found in the coupled neurons as well as hyperchaotic firing activities rarely reported in such classes of neurons. Finally, PSpice simulations are used to further support results obtained from the mathematical model of coupled neurons through a JJS.

Stability Switches and Chaos in a Diffusive Toxic Phytoplankton–Zooplankton Model with Delay

To explore the effect of toxins produced by phytoplankton in algal blooms, a diffusive toxic phytoplankton–zooplankton system with delay is studied through detailed bifurcation analysis. The local stability of the positive equilibrium is studied by analyzing the characteristic equation. The critical values of delay at which Hopf bifurcations occur are obtained and ordered, and it is proved that the delay can induce stability switches, which means that the delay has a significant impact on both the formation and the termination of algal blooms. During the process of stability switching, double Hopf bifurcation arises. Taking the rate of toxin liberation and time delay as the bifurcation parameters, the normal form for double Hopf bifurcation is derived, from which the classification for dynamics and bifurcation sets near double Hopf bifurcation point is obtained. It is proved that the system has complex dynamics, such as periodic oscillations, quasi-periodic oscillations on two- or three-torus and even chaos. Numerical simulations are presented to support the theoretical results.

Complex dynamics in the improved Koren–Feingold cloud–rain system

In this paper, we study the improved aerosol–cloud–precipitation system. Since in our previous article, we guessed that the delayed term y(t−τ) in the denominator tends to cause singularity. So we remove the denominator by time transformation, the original cloud–rain model is transformed into a simpler polynomial system under certain conditions, namely, the improved Koren–Feingold cloud–rain system. Fortunately, the new model meets our expectations. The main work of the present paper is to study the mechanism of cloud and rain interaction by double Hopf bifurcation analysis via the methods of multiple scales. We find some complicated dynamical behaviors of the system via analytical method, such as stable quasi-periodic solutions and chaos. The numerical simulations agree well with those of theoretical analysis, which shows that we are improving the model in the right direction, meanwhile perhaps it suggests that the MMS employed in this paper provides us an effective and accurate method of analyzing double Hopf bifurcation. The above complex dynamical phenomena are of great significance for the researchers to understand the mechanism of aerosol–cloud–precipitation system. And it is also significant to ecosystem and climate control.

Dynamics of periodic solution to a electrostatic Micro-Electro-Mechanical system

A canonical mass–spring model of electrostatically actuated Micro-Electro-Mechanical System (MEMS) is studied. We investigate the existence and bifurcations of the periodic solution by means of the continuation theorem of Mańasevich and Mawhin with techniques of a priori estimate and bifurcation theory. The main results answer the open problem proposed by P. Torres in the known literature. It also reveals that the system undergoes saddle node and period doubling bifurcation leading to the multiplicity of periodic solution. Moreover, bistability of periodic solution generated by the combination of these bifurcations is detected for the first time, which provides some new insight into the so called pull-in instability in the system. At last, periodic orbits with different stability and corresponding time series are given to illustrate the results.

Stability Analysis and Coexisting Behaviors of a Delayed Multiplex Network Under Electromagnetic Radiation

This paper studies the dynamics of a nonlinear network with many interacting neural populations and time-delayed couplings under electromagnetic radiation. The influence of the electromagnetic radiation is described by using flux-controlled memristor. The delay-induced instability and bifurcated periodic oscillations of the memristive multiplex network are determined by decomposing and analyzing the characteristic equations. The bifurcation diagrams are shown and complicated dynamical behaviors are explored, such as multi-periodic orbits and period doubling bifurcations. These interesting phenomena are also observed from the circuit realization. It is found that both time delay and electromagnetic radiation have great effects on the performance of the network, such as synchronization transitions and multistability coexistence.

Dynamics and stability of two predators–one prey mathematical model with fading memory in one predator

This paper concentrates on dynamics and stability analysis of two predators–one prey mathematical model with competition between predators and fading memory in one predator. The investigation of the constructed model shows that there exist five equilibria, e.g. trivial extinction state of all populations, extinction of both predators state, extinction of first or second predator state and coexisting state. Investigating the eigenvalues of characteristic polynomial, conditions for the local stability around each equilibrium are also determined depending on the parameter space. Analytical formulations are complemented with numerical simulations, where time simulations and single parameter numerical continuation of each variable are performed with respect to model parameters and multiple sub-and super-critical Hopf bifurcations, period doubling bifurcation and transcritical bifurcation are detected for different values of memory related parameter. Our results show that fading memory and competition between predators have substantial impact on the existence and dynamics of all three populations and may shed lights on further understanding of interacting species in ecology.

Design and circuit implementations of multimemristive hyperchaotic system

To enhance the order and Kolmogorov entropy of the memristive hyperchaotic system and improve the realizability of the circuit, a sixth-order memristive hyperchaotic system that consists of two flux-controlled memristors and one charge-controlled memristor is designed. Combined with Lyapunov exponential spectrum, phase trajectory diagram and bifurcation diagram, dynamic characteristics of the system are analyzed, and the effects of system control parameters and initial state on the system dynamic behavior are explored. Equivalent circuit models of flux-controlled and charge-controlled memristors with cubic nonlinear characteristics are constructed by the basic operation modules, and the hardware simulation design of the memristive hyperchaotic system is completed. The experimental results show that the proposed system can exhibit kinds of attractors, such as periodic-loop, single-scroll, double-scroll and superposition by adjusting system control parameters, and the system can show the behavior of period doubling bifurcation entering chaos and anti-period doubling exiting chaos under the different system initial states, which corresponds to the multistable coexistence phenomena of the system, such as quasi-period and hyperchaos. Meanwhile, the sixth-order hyperchaotic system is implemented by hardware circuit simulation. The phase diagrams of the circuit simulation are consistent with that of the numerical simulation, which verifies the physical realizability of the hyperchaotic system. Based on the strong initial value sensitivity and large K entropy, the memristive hyperchaotic system can be applied to the image encryption, and the confidentiality and security of images can be effectively enhanced.

Dynamic characteristics of wheel–rail collision vibration for high-speed train under crosswind

ABSTRACT This paper focuses on the dynamic characteristics of wheel–rail collision vibration for high-speed train under crosswind. The wind load model is established by using fast Fourier transform harmonic synthesis method and Davenport coherence function, the lateral displacement of wheelset of high-speed train under crosswind is calculated, the dynamic model of wheel–rail collision vibration system of high-speed train under crosswind is established. The effects of vibration frequency, wheel–rail clearance and lateral damping on the characteristics of wheel–rail lateral clearance impact vibration system of high-speed train are discussed. Numerical simulation results show that the impact velocity of wheelset collision on left and right rails increases with the increase of vibration frequency. When ω > 3.3, the system changes from chaos state to single period stable state through inverse doubling bifurcation; when b < 0.0033, it is in a chaos state, and with the increase of b, the inverse doubling bifurcation occurs until the hunting motion of the wheel–rail system is in a stable state; when , the system is in a stable single period operation state, and with the continuous increase of , it begins to have hop bifurcation, enters a multi-period and chaos state, and finally it is in a stable single period operation state.

Complicated Dynamics of a Delayed Photonic Reservoir Computing System

In this paper, we consider the complicated dynamics of a delay-based photonic reservoir computing system. Since conventional computer architectures are approaching their limit, it is imperative to find new, efficient and fast ways of data processing. Photonic reservoir computing (RC) is a promising way which combines the computational capabilities of recurrent neural networks with high processing speed and energy efficiency of photonics. As the RC system is very promising, we analyze its dynamics so that we can make better use of it. In this paper, we mainly focus on its double Hopf bifurcation. We first analyze the existence of double Hopf bifurcation points. Then we use DDE-BIFTOOL to draw the bifurcation diagrams with respect to two bifurcation parameters, i.e. feedback strength [Formula: see text] and delay [Formula: see text], and give a clear picture of the double Hopf bifurcation points of the system. These figures show stability switches and the existence of double Hopf bifurcation points. Finally, we employ the method of multiple scales to obtain their normal forms, use the method of normal form to unfold and classify their local dynamics. The classification and unfolding of these double Hopf bifurcation points are obtained. Three types of double Hopf bifurcations are found. We verify the results by numerical simulations and find its complicated behavioral dynamics. For example, there exist stable equilibrium, stable periodic and quasi-periodic solutions in distinct regions. The discovered rich dynamical phenomena can help us to choose suitable values of parameters to achieve excellent performance of RC.

Double Hopf Bifurcation Analysis in the Memory-based Diffusion System

In this paper, we derive the algorithm for calculating the normal form of the double Hopf bifurcation that appears in a memory-based diffusion system via taking memory-based diffusion coefficient and the memory delay as the perturbation parameters. Using the obtained theoretical results, we study the dynamical classification near the double Hopf bifurcation point in a predator-prey system with Holling type II functional response. We show the existence of different kinds of stable spatially inhomogeneous periodic solutions, the transition from one kind to the other as well as the coexistence of two types of periodic solutions with different spatial profiles by varying the memory-based diffusion coefficient and the memory delay.

Emergence of hidden dynamics in different neuronal network architecture with injected electromagnetic induction

The diverse firing responses in a single neuron model as well as in a neuronal network play a major role in understanding the collective neuronal dynamics. In this paper, we consider an excitable slow-fast memristive model and study its various intrinsic dynamics by allowing a periodic external stimulus. The single model exhibits various types of spiking, bursting, mixed-mode oscillations (MMOs), and mixed-mode bursting oscillations (MMBOs) depending on the amplitude and frequency of the periodic injected current. Corresponding bifurcation analysis reveals the existence of supercritical and subcritical Hopf bifurcations in the system depending on the major predominant parameters that establish the scenarios of the neuronal responses. We have verified analytically the existence and stability of Hopf bifurcations. The memristive system can produce cascade of period doubling bifurcations for particular parameter regimes. Next, we investigate the emergence of multi-arm antispiral waves in the diffusively coupled system with proper analytical justification. We have also computed the group and phase velocities to discern the spiral and antispiral waves near the Hopf instability. A transition from target wave to multi-arm antispiral wave is observed in the diffusive system. Moreover, we use a random network architecture different from the diffusive network to study the dynamics of the coupled network for certain firing activities. It is observed that the network of heterogeneous desynchronized neurons with higher electrical couplings, can generate MMOs, MMBOs, and bursting for different external stimuli. However, the uncoupled systems cannot reveal such typical dynamics for particular parameters. Further, we introduce a reduced-order model to summarize the complete dynamics of the larger random network and report the findings.

Normal forms of double Hopf bifurcation for a reaction-diffusion system with delay and nonlocal spatial average and applications

In this paper, we are concerned with a reaction-diffusion model incorporating delay and nonlocal effects. The normal form of double Hopf bifurcation is derived. The diffusive model of pollen tube tip growth is discussed and numerical simulations show that spatially homogeneous and inhomogeneous periodic solutions can be both stable or connected by a heteroclinic orbit under certain conditions. In addition, the diffusive Lotka-Volterra model with delay and nonlocality is considered and spatially inhomogeneous quasi-periodic solution is obtained.

Prognostic Impact of New-Onset Atrial Fibrillation After Single or Double Stent Left Main Bifurcation PCI.

Incidence and prognostic value of new-onset atrial fibrillation after single versus double stent strategy in bifurcation left main disease has not been yet investigated. We retrospectively analyzed the procedural and medical data of patients referred to our center for complex left main bifurcation disease, treated using crossover provisional stenting, T or T-and-Protrusion, Culotte, and Nano-inverted-T techniques between January 1, 2008, and May 1, 2018. Multivariate Cox-regression analysis was used to assess the role of different stent strategies, adjusted for confounders, on the risk of new-onset atrial fibrillation during the follow-up period. Five hundred two patients (316 males, mean age 70.3 ± 12.8 years, mean Syntax score 31.6 ± 6.3) were evaluated. At a mean follow-up of 37.1 ± 10.8 months (range: 22.1- 39.3 months); Target lesion failure rate was 10.1%. Stent thrombosis and cardiovascular mor- tality were observed in 1.2% and 3.6% in of cases, respectively. New-onset atrial fibrillation occurred in 23 out of 502 patients (4.6%). Patients with new-onset atrial fibrillation resulted more frequently female, older, obese, and diabetic and more frequently experienced target lesion failure and cardiovascular death. New-onset atrial fibrillation-free survival favored single versus double stent technique and among double stent techniques nano-inverted-T tech- niques compared to the others. Single stent strategy had a lower risk of new-onset atrial fibril- lation compared to double stent technique on multivariate analysis (Hazard Ratio (HR): 1.14, 95% CI: 1.10-1.19, P < .001 vs. HR: 1.28, 95% CI: 1.23-1.32, P < .0001). New-onset atrial fibrillation in distal left main bifurcation disease treated with per- cutaneous coronary intervention had a low incidence but resulted more frequently after double than after single stenting technique and was associated with worse outcomes.

Studies on of phase error oscillations in a class of third-order optical phase locked loop and its effect on a slave optical phase locked loop

Dynamics of a third-order optical phase locked loop (OPLL) with a resonant type loop filter (LF) has been studied analytically and numerically. The range of stable synchronous operating zone of such OPLL has been estimated analytically in system and signal parameters and obtained predictions are in good agreement with numerical results. Beyond the stable region, it is found that chaotic oscillation of phase error occurs through a sequence of period doubling bifurcation. In addition to this, tracking of such periodic as well as chaotic dynamics of third-order OPLL by another OPLL in master-slave configuration has also been explored here. The tracking is possible by slave OPLL within a very short range of effective frequency offset by keeping other design parameters within the stable mode of operation. The synchronization of the master and slave OPLLs has been studied in generalized sense by considering auxiliary system. The study is very important and helpful for optical signal processing and also in chaos-based secure communication.

Bifurcation Analysis of Periodic Oscillation in a Hematopoietic Stem Cells Model with Time Delay Control

Underlying the state feedback control, the complex dynamical disease of the hematopoietic stem cells model based on Mackey’s mathematical description is analyzed. The bifurcating periodical oscillation solutions of the system are continued by applying numerical simulation method. The limit point cycle bifurcation and period doubling bifurcation are observed frequently in the continuation process. The attraction basins of the positive equilibrium solution shrink as the differentiate rate is ascending and the observed Mobiüs strain is simulated with boundary as the period-2 solution. The period doubling bifurcation leads to period-2, period-4, and period-8 solutions which are simulated. Starting from period doubling bifurcation point, the continuation of the bifurcating solution routes to homoclinic solution is finished. The simulation results improve the comprehension related to the spontaneous dynamical character manifested in the hematopoietic stem cells model.

Transitions of zonal flows in a two-layer quasi-geostrophic ocean model

We consider a 2-layer quasi-geostrophic ocean model where the upper layer is forced by a steady Kolmogorov wind stress in a periodic channel domain, which allows to mathematically study the nonlinear development of the resulting flow. The model supports a steady parallel shear flow as a response to the wind stress. As the maximal velocity of the shear flow (equivalently the maximal amplitude of the wind forcing) exceeds a critical threshold, the zonal jet destabilizes due to baroclinic instability and we numerically demonstrate that a first transition occurs. We obtain reduced equations of the system using the formalism of dynamic transition theory and establish two scenarios which completely describe this first transition. The generic scenario is that a conjugate pair of modes loses stability and a Hopf bifurcation occurs as a result. Under an appropriate set of parameters describing related midlatitude oceanic flows, we show that this first transition is continuous: a supercritical Hopf bifurcation occurs and a stable time periodic solution bifurcates. We also investigate the case of double Hopf bifurcations which occur when four modes of the linear stability problem simultaneously destabilize the zonal jet. In this case, we prove that, in the relevant parameter regime, the flow exhibits a continuous transition accompanied by a bifurcated attractor homeomorphic to \(S^3\). The topological structure of this attractor is analyzed in detail and is shown to depend on the system parameters. In particular, this attractor contains (stable or unstable) time-periodic solutions and a quasi-periodic solution.

Mechanism of the low-frequency wide-band within a nonlinear acoustic metamaterial

In this paper, the formation and broadening mechanism of the low-frequency bandgap are revealed by designing a type of nonlinear acoustic metamaterial (NAM) with gaps. The mechanism of the chaotic evolution through the harmonic coupling for the NAM structure is revealed in detail, which can effectively change spectral characteristics in the bifurcation sequence, and finally, lead to a low frequency and chaos. Then, a low-frequency broadband within the NAM structures with gaps is obtained by the finite element method (FEM) due to the mechanism of chaos. Most importantly, the width of the bandgap can be successfully enlarged by adjusting the small gap, because the energy of the elastic wave can be greatly decreased or redistributed with the system going from periodic motion to doubling bifurcation. This mechanism for achieving a low-frequency bandgap overcomes the shortcomings both in the traditional local resonance with too large additional mass and in the inertial amplification structures with narrow bandgaps. Therefore, this idea of achieving a low-frequency bandgap has great theoretical significance for controlling low-frequency sound waves.