Types Of Bifurcations Research Articles

Nonlinear dynamic investigations and global analysis of a Cournot duopoly game with two different objectives

Considering a nonlinear price function a duopoly game with quantities setting is introduced. The two competitors in this game seek maximization of two different objectives. The first competitor want to detect the optimum of his/her production by maximizing an average of social welfare and profit while the second competitor wants to maximize his/her profit only. Due to the lack of market information, each competitor behaves rationally and so the bounded rationality mechanism is adopted in order to build the model describing the game. Studying the evolution of the game requires to investigate the model in discrete time periods. So a two-dimensional map is introduced to analyze the game’s evolution. For this map, we calculate its equilibrium points and study their stability. Through local and global dynamic analysis we prove that the Nash equilibrium point loses its stability because of flip bifurcation only. Other dynamic characteristics for the map such as contact bifurcation and multi-stability are analyzed. The obtained results show that the manifold of game’s map can be investigated based on a one-dimensional map whose analytical form looks like the famous logistic map. Through the critical curves analysis we prove that the phase plane of game’s map is divided into three zones that are Zi,i=0,2,4 and hence the map is noninvertible. Furthermore, an analysis of two types of contact bifurcation are discussed through simulation.

Quantum properties near the instability boundary in optomechanical system

The properties of the system near the instability boundary are very sensitive to external disturbances, which is important for amplifying some physical effects or improving the sensing accuracy. In this paper, the quantum properties near the instability boundary in a simple optomechanical system have been studied by numerical simulation. Calculations show that the transitional region connecting the Gaussian states and the ring states when crossing the boundary is sometimes different from the region centered on the boundary line, but it is more essential. The change of the mechanical Wigner function in the transitional region directly reflects its bifurcation behavior in classical dynamics. Besides, quantum properties, such as mechanical second-order coherence function and optomechanical entanglement, can be used to judge the corresponding bifurcation types and estimate the parameter width and position of the transitional region. The non-Gaussian transitional states exhibit strong entanglement robustness, and the transitional region as a boundary ribbon can be expected to replace the original classical instability boundary line in future applications.

Complex dynamics of hair bundle of auditory nervous system (I): spontaneous oscillations and two cases of steady states.

The hair bundles of inner hair cells in the auditory nervous exhibit spontaneous oscillations, which is the prerequisite for an important auditory function to enhance the sensitivity of inner ear to weak sounds, otoacoustic emission. In the present paper, the dynamics of spontaneous oscillations and relationships to steady state are acquired in a two-dimensional model with fast variable X (displacement of hair bundles) and slow variable . The spontaneous oscillations are derived from negative stiffness modulated by two biological factors (S and D) and are identified to appear in multiple two-dimensional parameter planes. In (S, D) plane, comprehensive bifurcations including 4 types of codimension-2 bifurcation and 5 types of codimension-1 bifurcation related to the spontaneous oscillations are acquired. The spontaneous oscillations are surrounded by supercritical and subcritical Hopf bifurcation curves, and outside of the curves are two cases of steady state. Case-1 and Case-2 steady states exhibit Z-shaped (coexistence of X) and N-shaped (coexistence of ) X-nullclines, respectively. In (S, D) plane, left and right to the spontaneous oscillations are two subcases of Case-1, which exhibit the stable equilibrium point locating on the upper and lower branches of X-nullcline, respectively, resembling that of the neuron. Lower to the spontaneous oscillations are 3 subcases of Case-2 from left to right, which manifest stable equilibrium point locating on left, middle, and right branches of X-nullcline, respectively, differing from that of the neuron. The phase plane for steady state is divided into four parts by nullclines, which manifest different vector fields. The phase trajectory of transient behavior beginning from a phase point in the four regions to the stable equilibrium point exhibits different dynamics determined by the vector fields, which is the basis to identify dynamical mechanism of complex forced oscillations induced by external signal. The results present comprehensive viewpoint and deep understanding for dynamics of the spontaneous oscillations and steady states of hair bundles, which can be used to well explain the experimental observations and to modulate functions of spontaneous oscillations.

Two-parameter non-smooth bifurcations of period-one motions in a plastic impacting oscillator

A 2-DOF plastic impacting oscillator is considered. The discontinuity boundaries of the system phase space are defined as well as the local maps and Poincaré map. Four cases of sliding bifurcation of limit cycles in impact system are described schematically. The period-one motion modes and occurrence regions of the system are identified in the two-parameter space by numerical simulation. One-parameter bifurcation diagrams and trajectories are computed to explain the evolutions of adjacent periodic motions and the non-smooth bifurcation behaviors including grazing and four types of sliding bifurcation. Due to the discontinuous grazing bifurcation there appear a narrow hysteresis domain in the two-parameter evolution between neighboring non-impacting and impacting motions. Grazing, switching–sliding and multi-sliding bifurcations of periodic sticking motions are all manifested as rising phenomenon of the time trajectory, but the different locations of the rises in the sticking phase correspond to the different types of non-smooth bifurcations. The grazing induced dynamics and the characteristics of rising events are discussed. Notably, the codimension-2 bifurcation points are detected and investigated by numerical continuation algorithm.

Broken symmetry and dynamics of a memristive diodes bridge-based Shinriki oscillator

We investigate the impact of a broken symmetry on the dynamics of the well-known Shinriki oscillator. The broken symmetry is caused by the memristive diodes bridge with an asymmetric pinched hysteresis loop current–voltage characteristic designed by selecting two pairs of semiconductor diodes with different electrical properties. We examine how the broken symmetry affects the topology of attractors, the nature of fixed points, the bifurcation structures, the number and types of coexisting solutions, and the topology of the basins of attraction as well. These features are highlighted by utilizing plots of Lyapunov exponents, bifurcation diagrams, basins of attraction and phase portraits. As sample results, up to four coexisting asymmetric chaotic and periodic attractors are reported following changes in both initial conditions and parameters. Moreover, some sets of parameters are revealed where the system develops the striking feature of coexisting bubbles of bifurcation. Breadboard experiments are carried out to support the theoretical investigations. Although the breaking of symmetry may be seen as a common practice to discover new nonlinear events, the results obtained in this work may help for a better understanding of the impact of a real memristor on the global behavior of chaotic oscillators including a memristor as nonlinear device.

Subcritical and supercritical bifurcations in axisymmetric viscoelastic pipe flows

Axisymmetric viscoelastic pipe flow of Oldroyd-B fluids has been recently found to be linearly unstable by Garget al.(Phys. Rev. Lett., vol. 121, 2018, 024502). From a nonlinear point of view, this means that the flow can transition to turbulence supercritically, in contrast to the subcritical Newtonian pipe flows. Experimental evidence of subcritical and supercritical bifurcations of viscoelastic pipe flows have been reported, but these nonlinear phenomena have not been examined theoretically. In this work, we study the weakly nonlinear stability of this flow by performing a multiple-scale expansion of the disturbance around linear critical conditions. The perturbed parameter is the Reynolds number with the others being unperturbed. A third-order Ginzburg–Landau equation is derived with its coefficient indicating the bifurcation type of the flow. After exploring a large parameter space, we found that polymer concentration plays an important role: at high polymer concentrations (or small solvent-to-solution viscosity ratio$\beta \lessapprox 0.785$), the nonlinearity stabilizes the flow, indicating that the flow will bifurcate supercritically, while at low polymer concentrations ($\beta \gtrapprox 0.785$), the flow bifurcation is subcritical. The results agree qualitatively with experimental observations where critical$\beta \approx 0.855$. The pipe flow of upper convected Maxwell fluids can be linearly unstable and its bifurcation type is also supercritical. At a fixed value of$\beta$, the Landau coefficient scales with the inverse of the Weissenberg number ($Wi$) when$Wi$is sufficiently large. The present analysis provides a theoretical understanding of the recent studies on the supercritical and subcritical routes to the elasto-inertial turbulence in viscoelastic pipe flows.

Bifurcations of a plane parallel flow with Kolmogorov forcing

We study the primary bifurcations of a plane parallel flow in a channel with Kolmogorov forcing. We find a new type of bifurcation with both the oscillation frequency and the amplitude of the growing mode being zero at the threshold. We call this a stationary drift bifurcation. The laminar steady flow can display different types of bifurcation depending on the forcing wave number of the base flow. This is in contrast to the case of doubly periodic boundary conditions for which the primary bifurcation is stationary.

Bifurcation and Calabi-Bernstein type asymptotic property of solutions for the one-dimensional Minkowski-curvature equation

In this paper, we investigate the global structure of bifurcation branches of one-sign solutions and sign-changing solutions for one-dimensional Minkowski-curvature problems with a strong singular weight. Our interest of the nonlinearity is either linear or sublinear near zero. Growth conditions near ∞ are not necessary and the proofs are mainly employed by bifurcation theories based on Whyburn's limit argument and analysis techniques. We also show Calabi-Bernstein type asymptotic property of one-sign solutions by proving that one-sign solutions on two bifurcation branches converge to two linear functions.

Modeling the importance of temporary hospital beds on the dynamics of emerged infectious disease.

To explore the impact of available and temporarily arranged hospital beds on the prevention and control of an infectious disease, an epidemic model is proposed and investigated. The stability analysis of the associated equilibria is carried out, and a threshold quantity basic reproduction number ( R0) that governs the disease dynamics is derived and observed whether it depends both on available and temporarily arranged hospital beds. We have used the center manifold theory to derive the normal form and have shown that the proposed model undergoes different types of bifurcations including transcritical (backward and forward), Bogdanov-Takens, and Hopf-bifurcation. Bautin bifurcation is obtained at which the first Lyapunov coefficient vanishes. We have taken advantage of Sotomayor's theorem to establish the saddle-node bifurcation. Numerical simulations are performed to support the theoretical findings.

Homoclinic Renormalization Group Flows, or When Relevant Operators Become Irrelevant.

We study an N=1 supersymmetric quantum field theory with O(M)×O(N) symmetry. Working in 3-ε dimensions, we calculate the beta functions up to second loop order and analyze in detail the renormalization group (RG) flow and its fixed points. We allow N and M to assume general real values, which results in them functioning as bifurcation parameters. In studying the behavior of the model in the space of M and N, we demarcate the region where the RG flow is nonmonotonic and determine curves along which Hopf bifurcations take place. At a number of points in the space of M and N we find that the model exhibits an interesting phenomenon: at these points the RG flow possesses a fixed point located at real values of the coupling constants g_{i} but with a stability matrix (∂β_{i}/∂g_{j}) that is not diagonalizable and has a Jordan block of size two with zero eigenvalue. Such points correspond to logarithmic conformal field theories and represent Bogdanov-Takens bifurcations, a type of bifurcation known to give rise to a nearby homoclinic orbit-an RG flow that originates and terminates at the same fixed point. In the present example, we are able to employ analytic and numeric evidence to display the existence of the homoclinic RG flow.

Dynamics of a hybrid vibro-impact oscillator: canonical formalism

Hybrid vibro-impact (HVI) oscillations is a strongly nonlinear dynamical regime that involves both linear oscillations and collisions under periodic, impulsive, or stochastic excitation. This regime arises in various engineering systems, such as mechanical components under tight rigid constraints, seismic-induced sloshing in partially-filled liquid storage tanks, and more. The adaptive nonlinearity of the HVI oscillator is used by the HVI-nonlinear energy sink as an effective vibration mitigation solution for broad energy and frequency range. Due to the extreme nonlinearity of this regime, traditional analytical methods are inapplicable for the description of its transient dynamics. In the current work, we model the HVI oscillator by a forced particle in a truncated quadratic potential well with infinite depth. The slow flow dynamics of the system in the vicinity of primary resonance is described by canonical transformation to action-angle (AA) variables and the corresponding reduced resonance manifold (RM). Two types of bifurcation are examined. The former is associated with transition between linear oscillations and the HVI-regime and vice versa, and the latter with reaching a chosen maximal transient energy level. The transition boundaries on the forcing parameters plane associated with both bifurcation types are obtained analytically. The maximal transient energy level obtained for any given set of forcing parameters is described analytically as well. The energy jumps associated with the bifurcation of type I and crossing the corresponding transition boundary are obtained. Two underlying dynamical mechanisms that govern the occurrence of bifurcations are identified. They correspond to two distinct scenarios: in the first scenario, the energy of the slow flow gradually reaches the threshold energy level and is thus referred to as the ”maximum” mechanism. The second, potentially more dangerous scenario, involves abrupt transitions of the system’s energy response from a relatively small value to the threshold energy level. This pattern is related to the passage of the slow-flow phase trajectory through the saddle point of the RM, and thus is referred to as the “saddle” mechanism. Both mechanisms are universal for systems that undergo escape from a potential well. All theoretical results are in complete agreement with full-scale numerical simulations.

Bifurcation analysis of modified density gradient continuum traffic model

This paper proposes a new density gradient continuous traffic flow model, and analyzes the linear stability of the model, as well as the bifurcation type of the model. Numerical simulation of the new model verifies the usability of the model. From the perspective of system stability, the bifurcation analysis method is used to analyze the nonlinear traffic phenomena on the expressway. The equilibrium solution of the model is discussed. On this basis, Hopf bifurcation, saddle bifurcation and Bogdanov–Takens bifurcation are obtained, and the existence conditions and fractional types of Hopf bifurcation and saddle bifurcation are obtained. The traffic flow characteristics of Hopf bifurcation and saddle node bifurcation are analyzed.

Revascularizaci�n percut�nea de las lesiones en bifurcaci�n

In this article, we analyse the changes occurred over the last 20 years in the percutaneous treatment of bifurcation lesions based on our own experience. We also analyse the more relevant papers recently published, as well as the strategies and guidelines recommended for the percutaneous management of bifurcation lesions. Technical aspects are relevant in this field and, for this reason, a significant section of this manuscript is dedicated to the technique used together with some tips and tricks also provided here. The technical advances made over the last years have significantly increased the success rate in the management of these complex lesions. At the same time, the technique of treatment has been standardized allowing the management of all type of bifurcations including distal left main disease.

Effect of temperature sensitive ion channels on the single and multilayer network behavior of an excitable media with electromagnetic induction

The dynamical behavior of the neurons directly depends on the transition from resting to spiking states. These transitions show different types of bifurcations and has different spiking periods. The transitions are also affected by the temperature exposure of the ionic channels. To understand such effects, we investigate the Morris-Lecar (ML) neuron model with temperature affected calcium, potassium and leak current channels. The presented ML model is considered with electromagnetic field coupling considering a simple cubic memristor flux relation. Firstly the basic dynamical properties of the ML model is analyzed considering the current temperature as the control parameter. The temperature affected ionic channels in the ML model leads to various types of oscillations from periodic spiking to chaotic bursting. These bifurcation patterns are well discussed with corresponding Lyapunov exponents. To study the wave propagation in the temperature dependent ML model (TDML), we have constructed two different types of network structure. In the first a simple lattice network is considered with the local nodes of the TDML neurons and the temperature effects on the wave propagation is studied individually for the three channels. In the second type of network, we have considered inter coupled three lattice layers of TDML neurons. This discussion is subdivided in to two cases and in the first case the layers are constructed such that the first, second and third layers having temperature affected calcium, potassium and leaky current channels respectively. In the second case we considered only one channel to have temperature effects and the others have no temperature affected ion channels. The wave propagation phenomenon in both the types of network is analyzed considering the current temperature as the control parameter.

Singularly perturbed boundary-focus bifurcations

We consider smooth systems limiting as ϵ → 0 to piecewise-smooth (PWS) systems with a boundary-focus (BF) bifurcation. After deriving a suitable local normal form, we study the dynamics for the smooth system with sufficiently small but non-zero ϵ , using a combination of geometric singular perturbation theory and blow-up . We show that the type of BF bifurcation in the PWS system determines the bifurcation structure for the smooth system within an ϵ −dependent domain which shrinks to zero as ϵ → 0 , identifying a supercritical Andronov-Hopf bifurcation in one case, and a supercritical Bogdanov-Takens bifurcation in two other cases. We also show that PWS cycles associated with BF bifurcations persist as relaxation oscillations in the smooth system, and prove existence of a family of stable limit cycles which connects the relaxation oscillations to regular cycles within the ϵ −dependent domain described above. Our results are applied to models for Gause predator-prey interaction and mechanical oscillation subject to friction.

Border collision bifurcations of chaotic attractors in one-dimensional maps with multiple discontinuities

In one-dimensional piecewise smooth maps with multiple borders, chaotic attractors may undergo border collision bifurcations, leading to a sudden change in their structure. We describe two types of such border collision bifurcations and explain the mechanisms causing the changes in the geometrical structure of the attractors, in particular, in the number of their bands (connected components).

Linear and Nonlinear Stability Analyses of Double-Diffusive Convection in a Vertical Brinkman Porous Enclosure under Soret and Dufour Effects

Analytical and numerical investigations were performed to study the influence of the Soret and Dufour effects on double-diffusive convection in a vertical porous layer filled with a binary mixture and subject to horizontal thermal and solute gradients. In particular, the study was focused on the effect of Soret and Dufour diffusion on bifurcation types from the rest state toward steady convective state, and then toward oscillatory convective state. The Brinkman-extended Darcy model and the Boussinesq approximation were employed to model the convective flow within the porous layer. Following past laboratory experiments, the investigations dealt with the particular situation where the solutal and thermal buoyancy forces were equal but acting in opposite direction to favor the possible occurrence of the rest state condition. For this situation, the onset of convection could be either supercritical or subcritical and occurred at given thresholds and following various bifurcation routes. The analytical investigation was based on the parallel flow approximation, which was valid only for a tall porous layer. A numerical linear stability analysis of the diffusive and convective states was performed on the basis of the finite element method. The thresholds of supercritical, RTCsup, and overstable, RTCover, convection were computed. In addition, the stability of the established convective flow, predicted by the parallel flow approximation, was studied numerically to predict the onset of Hopf’s bifurcation, RTCHopf, which marked the transition point from steady toward unsteady convective flows; a route towards the chaos. To support the analytical analyses of the convective flows and the numerical stability methodology and results, nonlinear numerical solutions of the full governing equations were obtained using a second-order finite difference method. Overall, the Soret and Dufour effects were seen to affect significantly the thresholds of stationary, overstable and oscillatory convection. The Hopf bifurcation was marked by secondary convective flows consisting of superposed vertical layers of opposite traveling waves. A good agreement was found between the predictions of the parallel flow approximation, the numerical solution and the linear stability results.

A hyperchaotic cycloid map with attractor topology sensitive to system parameters.

We propose herein a novel discrete hyperchaotic map based on the mathematical model of a cycloid, which produces multistability and infinite equilibrium points. Numerical analysis is carried out by means of attractors, bifurcation diagrams, Lyapunov exponents, and spectral entropy complexity. Experimental results show that this cycloid map has rich dynamical characteristics including hyperchaos, various bifurcation types, and high complexity. Furthermore, the attractor topology of this map is extremely sensitive to the parameters of the map. The x--y plane of the attractor produces diverse shapes with the variation of parameters, and both the x--z and y--z planes produce a full map with good ergodicity. Moreover, the cycloid map has good resistance to parameter estimation, and digital signal processing implementation confirms its feasibility in digital circuits, indicating that the cycloid map may be used in potential applications.

Entrainment Dynamics Organised by Global Manifolds in a Circadian Pacemaker Model

Circadian rhythms are established by the entrainment of our intrinsic body clock to periodic forcing signals provided by the external environment, primarily variation in light intensity across the day/night cycle. Loss of entrainment can cause a multitude of physiological difficulties associated with misalignment of circadian rhythms, including insomnia, excessive daytime sleepiness, gastrointestinal disturbances, and general malaise. This can occur after travel to different time zones, known as jet lag; when changing shift work patterns; or if the period of an individual’s body clock is too far from the 24 h period of environmental cycles. We consider the loss of entrainment and the dynamics of re-entrainment in a two-dimensional variant of the Forger-Jewett-Kronauer model of the human circadian pacemaker forced by a 24 h light/dark cycle. We explore the loss of entrainment by continuing bifurcations of one-to-one entrained orbits under variation of forcing parameters and the intrinsic clock period. We show that the severity of the loss of entrainment is dependent on the type of bifurcation inducing the change of stability of the entrained orbit, which is in turn dependent on the environmental light intensity. We further show that for certain perturbations, the model predicts counter-intuitive rapid re-entrainment if the light intensity is sufficiently high. We explain this phenomenon via computation of invariant manifolds of fixed points of a 24 h stroboscopic map and show how the manifolds organise re-entrainment times following transitions between day and night shift work.

Generalized Hopf bifurcation analysis of a towed caster wheel system

This work concerns the generalized Hopf bifurcation analysis of a piecewise-smooth wheel system with higher order discontinuities. Center manifold reduction is used to reduce the towed caster wheel system to a planar dynamical system, in which the non-smoothness factor is considered. The analysis can be divided into two cases by the coefficient of quadratic term. If the coefficient of quadratic term is equal to zero, the type of generalized Hopf bifurcation can be determined by the stability of first-order fine focus. Otherwise, we have to construct a Poincaré map for illustrating the type of generalized Hopf bifurcation, which is the heart of our discussion. Besides, we put forward another convenient way to judge the bifurcation types due to the symmetrical characteristic of this wheel system. Numerical simulations show the feasibility of theoretical analysis.