Period-doubling Bifurcations Research Articles

A discrete fractional order cournot duopoly game model with relative profit delegation: Stability, bifurcation, chaos, 0-1 testing and control

Due to the memory effect, fractional order dynamical systems provide more realistic results compared with ordinary counterparts. In this study, we consider a Cournot-duopoly game model with relative profit delegation in the sense of Caputo fractional derivative. To describe richer dynamical behavior such as chaos in the model, a discrete dynamical system is needed. As a result of the discretization method based on the use of piecewise constant arguments, we obtain a two dimensional system of difference equations. The stability conditions of all equilibrium points of the discrete dynamical system are given comprehensively. The existence of the flip bifurcation in the system has been demonstrated theoretically. Lyapunov exponents and 0–1 test chaos imply that chaotic structures are formed as a result of this bifurcation. In addition, we present the chaos control technique such as Pyragas method to eliminate chaos in the model. All theoretical results dealing with the stability, bifurcation and chaos in the model are stimulated by numerical simulations.

Ballistic to diffusive transition for swimmers in a periodic vortex array.

We study the transport of rigid ellipsoidal swimmers in a periodic vortex array via numerical simulation and dynamical systems analysis. Via ensemble simulations, we show the counterintuitive result that slower swimming speeds can generate fast ballistic transport, while faster swimming speeds generate chaotic and diffusive transport, which is inherently slower in the long run. To explain this, we use the symmetry of the flow to construct a time-reversible Poincaré return map on a two-dimensional surface of sectionin phase space. For sufficiently small swimming speeds, we find stable periodic orbits on the surface of sectionsurrounded by invariant tori, similar to Kolmogorov-Arnold-Moser curves. Trajectories within these tori are ballistic. As the swimming speed is increased, the periodic orbits undergo a sequence of period-doubling bifurcations that destroys the ballistic tori. These bifurcations exactly match the ballistic to diffusive transition from the ensemble simulations. Additional ensemble simulations are used to test the robustness of these results to noise. The ballistic behavior is destroyed as the strength of rotational diffusion increases. However, we estimate that the ballistic tori might still be seen in experiments.

Cascades of heterodimensional cycles via period doubling

A heterodimensional cycle is formed by the intersection of stable and unstable manifolds of two saddle periodic orbits that have unstable manifolds of different dimensions: connecting orbits exist from one periodic orbit to the other, and vice versa. The difference in dimensions of the invariant manifolds can only be achieved in vector fields of dimension at least four. At least one of the connecting orbits of the heterodimensional cycle will necessarily be structurally unstable, meaning that is does not persist under small perturbations. Nevertheless, the theory states that the existence of a heterodimensional cycle is generally a robust phenomenon: any sufficiently close vector field (in the C1-topology) also has a heterodimensional cycle.We investigate a particular four-dimensional vector field that is known to have a heterodimensional cycle. We continue this cycle as a codimension-one invariant set in a two-parameter plane. Our investigations make extensive use of advanced numerical methods that prove to be an important tool for uncovering the dynamics and providing insight into the underlying geometric structure. We study changes in the family of connecting orbits as two parameters vary and Floquet multipliers of the periodic orbits in the heterodimensional cycle change. In particular the Floquet multipliers of one of the periodic orbits change from real positive to real negative prior to a period-doubling bifurcation. We then focus on the transitions that occur near this period-doubling bifurcation and find that it generates new families of heterodimensional cycles with different geometric properties. Our careful numerical study suggests that further two-parameter continuation of the ‘period-doubled heterodimensional cycles’ gives rise to an abundance of heterodimensional cycles of different types in the limit of a period-doubling cascade.Our results for this particular example vector field make a contribution to the emerging bifurcation theory of heterodimensional cycles. In particular, the bifurcation scenario we present can be viewed as a specific mechanism behind so-called stabilisation of a heterodimensional cycle via the embedding of one of its constituent periodic orbits into a more complex invariant set.

Dynamics and control of two-dimensional discrete-time biological model incorporating weak Allee's effect.

Incorporating a weak Allee effect in a two-dimensional biological model in ℜ2, the study delves into the application of bifurcation theory, including center manifold and Ljapunov-Schmidt reduction, normal form theory, and universal unfolding, to analyze nonlinear stability issues across various engineering domains. The focus lies on the qualitative dynamics of a discrete-time system describing the interaction between prey and predator. Unlike its continuous counterpart, the discrete-time model exhibits heightened chaotic behavior. By exploring a biological Mmdel with linear functional prey response, the research elucidates the local asymptotic properties of equilibria. Additionally, employing bifurcation theory and the center manifold theorem, the analysis reveals that, for all α1 (i.e., intrinsic growth rate of prey), ð1˙ (i.e., parameter that scales the terms yn), and m (i.e., Allee effect constant), the model exhibits boundary fixed points A1 and A2, along with the unique positive fixed point A∗, given that the all parameters are positive. Additionally, stability theory is employed to explore the local dynamic characteristics, along with topological classifications, for the fixed points A1, A2, and A∗, considering the impact of the weak Allee effect on prey dynamics. A flip bifurcation is identified for the boundary fixed point A2, and a Neimark-Sacker bifurcation is observed in a small parameter neighborhood around the unique positive fixed point A∗=(mð1˙-1,α1-1-α1mð1˙-1). Furthermore, it implements two chaos control strategies, namely, state feedback and a hybrid approach. The effectiveness of these methods is demonstrated through numerical simulations, providing concrete illustrations of the theoretical findings. The model incorporates essential elements of population dynamics, considering interactions such as predation, competition, and environmental factors, along with a weak Allee effect influencing the prey population.

Analysis of Reverse Whirl Characteristics of a Rub-Impact Rotor System

The rub-impact fault between the rotor and stator of the aeroengine can lead to a reverse whirl phenomenon, which will induce the instability of the rotor system. This paper uses the bending–torsion coupled vibration signal to analyze the reverse whirl characteristics of the single-rotor system under single-disc and multidisc rub-impact faults. In the full-spectrum analysis of the bending direction, the reverse whirl frequency, which is the nonpower frequency component caused by rubbing, is found. This whirl frequency leads to the period-doubling bifurcation and the reverse whirl phenomenon of the system. In the analysis of the torsional direction, it is found that the torsional signal can better reflect the different types of rub-impact faults. Further research shows that the unbalanced phase between the rubbing discs is a crucial parameter influencing the bending–torsion coupled vibration and the whirl characteristics of the rotor system. An experimental platform with nonuniform clearance is designed and manufactured to reproduce the reverse whirl phenomenon. The reverse whirl motion near the second-order superharmonic resonance speed is designed to verify the mechanism of the reverse whirl.

Global dynamic analyzes of the discrete SIS models with application to daily reported cases

Emerging infectious diseases, such as COVID-19, manifest in outbreaks of varying magnitudes. For large-scale epidemics, continuous models are often employed for forecasting, while discrete models are preferred for smaller outbreaks. We propose a discrete susceptible-infected-susceptible model that integrates interaction between parasitism and hosts, as well as saturation recovery mechanisms, and undertake a thorough theoretical and numerical exploration of this model. Theoretically, the model incorporating nonlinear recovery demonstrates complex behavior, including backward bifurcations and the coexistence of dual equilibria. And the sufficient conditions that guarantee the global asymptotic stability of a disease-free equilibrium have been obtained. Considering the challenges posed by saturation recovery in theoretical analysis, we then consider the case of linear recovery. Bifurcation analysis for of the linear recovery model displays a variety of bifurcations at the endemic equilibrium, such as transcritical, flip, and Neimark–Sacker bifurcations. Numerical simulations reveal complex dynamic behavior, including backward and fold bifurcations, periodic windows, period-doubling cascades, and multistability. Moreover, the proposed model could be used to fit the daily COVID-19 reported cases for various regions, not only revealing the significant advantages of discrete models in fitting, evaluating, and predicting small-scale epidemics, but also playing an important role in evaluating the effectiveness of prevention and control strategies. Furthermore, sensitivity analyses for the key parameters underscore their significant impact on the effective reproduction number during the initial months of an outbreak, advocating for better medical resource allocation and the enforcement of social distancing measures to curb disease transmission.

Crookedness detection of thrust ball bearing shaft washer based on nonlinear vibration responses

Bearing misalignments are common defects in machinery. One of the most important types of that is the angular misalignment of the bearing rings. In this study, crookedness specific signature related to thrust ball bearing shaft washer is investigated. For this purpose, a vertical structural thrust ball bearing test rig was considered. In the following, effects of rotational shaft speed, crookedness value and axial external force on the shaft washer crookedness signature were studied at several relative positions of sensors. It is shown that crookedness of thrust ball bearing shaft washer has specific frequency and sensor phase difference pattern in the nonlinear vibration responses. Also, the experimental results are applied to validate results of a 4-dof nonlinear dynamical system related to the shaft washer crookedness of thrust ball bearing. In the next step, by comparing input and output amplitude and also existence of flip bifurcation, it is illustrated that this system is a nonlinear dynamical experimental system. It was shown that this nonlinearity makes crooked shaft washer pattern difficult to identify. Finally, by the phase portrait of the sensor response and the relative error value of crookedness signature frequencies, it is clear that the crookedness of shaft washer creates quasi-chaotic behavior supporting crookedness signature frequencies and phase pattern as a supplement property.

Analyzing Bifurcations and Optimal Control Strategies in SIRS Epidemic Models: Insights from Theory and COVID-19 Data

This study investigates the dynamic behavior of an SIRS epidemic model in discrete time, focusing primarily on mathematical analysis. We identify two equilibrium points, disease-free and endemic, with our main focus on the stability of the endemic state. Using data from the US Department of Health and optimizing the SIRS model, we estimate model parameters and analyze two types of bifurcations: Flip and Transcritical. Bifurcation diagrams and curves are presented, employing the Carcasses method. for the Flip bifurcation and an implicit function approach for the Transcritical bifurcation. Finally, we apply constrained optimal control to the infection and recruitment rates in the discrete SIRS model. Pontryagin’s maximum principle is employed to determine the optimal controls. Utilizing COVID-19 data from the USA, we showcase the effectiveness of the proposed control strategy in mitigating the pandemic’s spread.

Complex dynamical properties and chaos control for a discrete modified Leslie-Gower prey-predator system with Holling II functional response

In this study, the semi-discretization technique is employed to establish a discrete representation of a modified Leslie-Gower prey-predator system that includes a Holling II type functional response. The dynamics of this model are then analyzed through the application of center manifold theory and bifurcation theory. We present comprehensive results for the local stability of the fixed points across the entire parameter space. Additionally, we provide sufficient conditions for the occurrence of flip bifurcation and Neimark-Sacker bifurcation. Besides, the system has experienced a flip bifurcation to chaos controlled using the method of chaos control, viz., state feedback method, pole placement technique, and hybrid control strategy. Furthermore, we provide specific conditions to ensure that bifurcation and chaos can be stabilized. Finally, numerical simulations are conducted to validate theoretical analysis and illustrate several new complex dynamical behaviors between two species.

Single-photon induced instabilities in a cavity electromechanical device

Cavity-electromechanical systems are extensively used for sensing and controlling the vibrations of mechanical resonators down to their quantum limit. The nonlinear radiation-pressure interaction in these systems could result in an unstable response of the mechanical resonator showing features such as frequency-combs, period-doubling bifurcations and chaos. However, due to weak light-matter interaction, typically these effects appear at very high driving strengths. By using polariton modes formed by a strongly coupled flux-tunable transmon and a microwave cavity, here we demonstrate an electromechanical device and achieve a single-photon coupling rate g0/2π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left({g}_{0}/2\\pi \\right)$$\\end{document} of 160 kHz, which is nearly 4% of the mechanical frequency ωm. Due to large g0/ωm ratio, the device shows an unstable mechanical response resulting in frequency combs in sub-single photon limit. We systematically investigate the boundary of the unstable response and identify two important regimes governed by the optomechanical backaction and the nonlinearity of the electromagnetic mode. Such an improvement in the single-photon coupling rate and the observations of microwave frequency combs at single-photon levels may have applications in the quantum control of the motional states and critical parametric sensing. Our experiments strongly suggest the requirement of newer approaches to understand instabilities.

Spectral Signatures of Bifurcations

One way to characterize an orbit of a dynamical system is through its frequency content. Using a “spectral bifurcation diagram”, it has been shown earlier how the frequency content changes when a system undergoes a period-doubling cascade. In this paper, we extend the scope of this technique to obtain newer insights into various bifurcations like pitchfork bifurcation, border-collision bifurcation, period-adding cascade, and various torus bifurcations. We show that applying this method can enrich our understanding of bifurcations by providing vital information about generating or annihilating frequency components in a bifurcation.

Early warning signals of complex critical transitions in deterministic dynamics

Early Warning Signals (EWS) have generated much excitement for their potential to anticipate transitions in various systems, ranging from climate change in ecology to disease staging in medicine. EWS hold particular promise for bifurcations, a transition mechanism in which a smooth, gradual change in a control parameter of the system results in a rapid change in system dynamics. The predominant reason to expect EWS is because many bifurcations are preceded by Critical Slowing Down (CSD): if assuming the system is subject to continuous, small, Gaussian noise, the system is slower to recover from perturbations closer to the transition. However, this focus on warning signs generated by stochasticity has overshadowed warning signs which may already be found in deterministic dynamics. This is especially true for higher-dimensional systems, where more complex attractors with intrinsic dynamics such as oscillations not only become possible—they are increasingly more likely. The present study focuses on univariate and multivariate EWS in deterministic dynamics to anticipate complex critical transitions, including the period-doubling cascade to chaos, chaos-chaos transitions, and the extinction of a chaotic attractor. In a four-dimensional continuous-time Lotka–Volterra model, EWS perform well for most bifurcations, even with lower data quality. The present study highlights three reasons why EWS may still work in the absence of CSD: changing attractor morphology (size, shape, and location in phase space), shifting power spectra (amplitude and frequency), and chaotic transitional characteristics (density across attractor). More complex attractors call for different warning detection methods to utilise warning signs already contained within purely deterministic dynamics.

Research on Stability and Bifurcation for Two-Dimensional Two-Parameter Squared Discrete Dynamical Systems

This study investigates a class of two-dimensional, two-parameter squared discrete dynamical systems. It determines the conditions for local stability at the fixed points for these proposed systems. Theoretical and numerical analyses are conducted to examine the bifurcation behavior of the proposed systems. Conditions for the existence of Naimark–Sacker bifurcation, transcritical bifurcation, and flip bifurcation are derived using center manifold theorem and bifurcation theory. Results of the theoretical analyses are validated by numerical simulation studies. Numerical simulations also reveal the complex bifurcation behaviors exhibited by the proposed systems and their advantage in image encryption.

A Generalized Beddington Host–Parasitoid Model with an Arbitrary Parasitism Escape Function

This research delves into the generalized Beddington host–parasitoid model, which includes an arbitrary parasitism escape function. Our analysis reveals three types of equilibria: extinction, boundary, and interior. Upon examining the parameters, we discover that the first two equilibria can be globally asymptotically stable. The boundary equilibrium undergoes period-doubling bifurcation with a stable two-cycle and a transcritical bifurcation, creating a threshold for parasitoids to invade. Furthermore, we determine the interior equilibrium’s local stability and analytically demonstrate the period-doubling and Neimark–Sacker bifurcations. We also prove the permanence of the system within a specific parameter space. The numerical simulations we conduct reveal a diverse range of dynamics for the system. Our research extends the results in [ Kapçak et al. , 2013 ] and applies to a broad class of the generalized Beddington host–parasitoid model.

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.

Investigation of conjunctivitis adenovirus spread in human eyes by using bifurcation tool and numerical treatment approach

In order to investigate the dynamics of the system, a mathematical model must be created to comprehend the dynamics of various prevalent diseases worldwide. The purpose of this investigation is to explore the early identification and treatment of conjunctivitis adenovirus by introducing vaccination methods for asymptomatic individuals. A mathematical model is constructed with the aim of strengthening the immune system. The ABC operator is then utilized to convert the model into a fractionally ordered one. The developed system is analyzed with analytical solutions by employing Sumudu transforms, including convergence analysis. The boundedness and uniqueness of the model are investigated using Banach space, which are key properties of such epidemic models. The uniqueness of the system is confirmed to ensure it has a unique solution. The stability of the newly constructed SEVIR system is investigated both qualitatively and statistically, and the system’s flip bifurcation has been verified. The developed system is examined through a Lyapunov function-based local and global stability study. The solution to the system is found using the Atangana-Toufik technique, a sophisticated method for reliable bounded solutions, employing various fractional values. Error analysis has also been conducted for the scheme. Simulations have been carried out to observe the real behavior and effects of the conjunctivitis virus, confirming that individuals with a strong immune response can recover without medication during the acute stage of infection. This helps to understand the real situation regarding the control of conjunctivitis adenovirus after early detection and treatment by introducing vaccination measures due to the strong immune response of the patients. Such investigations are useful for understanding the spread of the disease and for developing control strategies based on the justified outcomes.

Newton-Krylov continuation of amplitude-modulated rotating waves in sheared annular electroconvection.

We present an approach for studying the primary, secondary, and tertiary flow transitions in sheared annular electroconvection. In particular, we describe a Newton-Krylov method based on time integration for the computation of rotating waves and amplitude-modulated rotating waves, and for the continuation of these flows as a parameter of the system is varied. The method exploits the rotational nature of the flows and requires only a time-stepping code of the model differential equations, i.e., it does not require an explicit code for the discretization of the linearized equations. The linear stability of the solutions is computed to identify the parameter values at which the transitions occur. We apply the method to a model of electroconvection that simulates the flow of a liquid crystal film in the smectic A phase suspended between two annular electrodes and subjected to an electric potential difference and a radial shear. Due to the layered structure of the smectic A phase, the fluid can be treated as two-dimensional (2D) and is modeled using the 2D incompressible Navier-Stokes equationscoupled with an equationfor charge continuity. The system is a close analog to laboratory-scale geophysical fluid experiments and thus represents an ideal system in which to apply the method before its application to these other systems that exhibit similar flow transitions. In the model for electroconvection, we identify the parameter values at which the primary transition from steady axisymmetric flow to rotating waves occurs, as well as at which the secondary transition from the rotating waves to amplitude-modulated rotating waves occurs. In addition, we locate the tertiary transition, which corresponds to a transition from the amplitude-modulated waves to a three-frequency flow. Of particular interest is that the method also finds a period-doubling bifurcation from the amplitude-modulated rotating waves and a subsequent transition from the flow resulting from this bifurcation.

Coexisting phenomena and antimonotonic evolution in a memristive Shinriki circuit

This paper is aimed at investigating the dynamical behaviors of the Shinriki circuit with a grounded asymmetric diode-bridge-based memristor through the implicit mapping method, including coexisting phenomena and antimonotonicity. To elucidate these behaviors, the basic properties, including equilibrium points and symmetry, of a Shinriki circuit incorporating a grounded asymmetric diode-bridge-based memristor are first illustrated. The discretization scheme for the circuit is established through the implicit mapping method, and the relatively complete bifurcation trees of each periodic motion by tuning a variable resistor can be analyzed through the Newton-Raphson algorithm. The unique unstable periodic bifurcation routes are also revealed. The evolution of antimonotonicity triggered by period-doubling or saddle-node bifurcations, along with the formed unstable motions, are comprehensively discussed. Furthermore, the simulation results are verified via a Field Programmable Gate Array (FPGA), by which both the asymmetric stable and unstable coexisting periodic attractors can be captured. The results will contribute to explaining the dynamical behaviors of numerous memristive circuits from a novel semi-analytical way, thus opening the possibility of analyzing some phenomena, such as unstable bifurcation routes, that cannot be directly observed via the numerical method.

Controlling chaos and codimension-two bifurcation in a discrete fractional-order Brusselator model

This paper explores the qualitative behavior of a discrete fractional–order Brusselator model. We analyze the local dynamics of the model around its fixed point and determine its topological classification. We perform the bifurcation analysis for both codimension-one and codimension-two cases to examine the system behavior near critical parameter values. Using normal form theory and center manifold theorem (CMT), we prove that the model exhibits period-doubling bifurcation around its interior fixed point. We also study the existence and direction of Neimark–Sacker bifurcation using normal form theory. For codimension-two bifurcation, we show that the model undergoes 1:2, 1:3, and 1:4 resonances by applying normal form theory and suitable affine transformations. The system displays a rich variety of bifurcations, including quasi–periodicity, periodic orbits, chaotic behavior, and resonance bifurcation. Furthermore, the existence of chaos is discussed in the sense of Marotto, and a novel chaos control method is proposed for discrete Brusselator model using an extended pole–placement approach. This modified approach is more suitable for codimension-two bifurcation situations. Numerical simulations are used to illustrate the theoretical discussion.

Analytical analysis and bifurcation of pine wilt dynamical transmission with host vector and nonlinear incidence using sustainable fractional approach

To study the dynamical system, it is necessary to formulate a mathematical model to comprehend the dynamics of the diseases that are prevalent around the world by using fractional calculus. A mathematical model is developed with the hypothesis created by adding control and asymptomatic variables to observe the rate of change of pine wilt and the ABC operator is used to turn the model into a fractional ordered model for continuous monitoring. The Boundedness and uniqueness of the developed model are investigated for bounded findings by using Banach space, which are the key properties of such an epidemic model. A newly developed system is examined both qualitatively and quantitatively to determine its stable position, and the verification of flip bifurcation has been made for developed systems. Derived reproductive numbers using the next-generation technique as well as the sensitivity of each involved parameter are verified. The Atangana–Toufik scheme is employed to find the solution for the developed system using different fractional values, which are advanced tools for reliable bounded solutions. Simulations have been made to see the real behavior and effects of pine wilt disease with control and asymptomatic battels in the community. Also, identify the real situation of the spread as well as the control of pine wilt after employing control and asymptomatic battels due to treatment. Such a type of investigation will be useful in investigating the spread of disease as well as helpful in developing control strategies based on our justified outcomes.