Doubling Bifurcation Research Articles

Bifurcation analysis in a discrete time square root response function of predator-prey system with fractional order

Stability and bifurcation analysis for a two spices discrete fractional order system by introducing square root response function of the form is examined in the closed first quadrant . The model parameters h, α, β, μ, σ, η are biologically feasible positive real numbers. Piecewise constant arguments method is applied to obtain the discrete fractional order system, particularly to study the rich dynamical behavior of the proposed system. Because the system has square root response function, the variation matrix of trivial and axial equilibrium states are in-determinant. In order to study the stability of the trivial and semi trivial equilibrium states, the change of variables x(j) = X 2(j) and y(j) = Y 2 (j) is considered. Moreover, we determine the criteria for stability of the interior equilibrium state using jury conditions. The numerical performance is shown for various parameter values and the time series and phase line diagrams are obtained. Bifurcation theory is applied to check whether the system undergoes periodic doubling bifurcations at its semi trivial and interior equilibrium states. Also we explore the periodic halving bifurcation which occurs for the fractional order as a bifurcation parameter. Numerical examples are presented to show the validity and feasibility of the obtained stability criterion in both species, including periodic doubling, periodic - 2, 4, 8, 16, periodic windows and Non periodic orbit (i.e., chaos).

Periodic solutions, chaos and bi-stability in the state-dependent delayed homogeneous Additive Increase and Multiplicative Decrease/Random Early Detection congestion control systems

The combination of Additive Increase and Multiplicative Decrease (AIMD) congestion control and Random Early Detection (RED) queue as a whole congestion control system plays a key role in the overwhelming success of the Internet. Thus it is important to investigate the periodic oscillations and complicated dynamics of the state-dependent delayed homogeneous Additive Increase and Multiplicative Decrease/Random Early Detection (AIMD/RED) congestion control system and its modified version fully in this paper. Firstly employing the semi-analytical method called as the harmonic balance method with alternating frequency/time (HB-AFT) domain technique, the approximate analytical expressions of periodic solutions of the generalized homogeneous-flow Additive Increase and Multiplicative Decrease/Random Early Detection (AIMD/RED) system with state-dependent round-trip delay are considered. We compare them with the results of numerical simulations by WinPP, they agree very well with each other. It demonstrates that the method employed here is versatile, valid, simple and effective. Then to the end of improving its modeling and performance, we modify the above model by taking an easy approximate dropping function. Furthermore, for the modified delayed homogeneous system, the approximate analytical expressions of periodic solutions are obtained accurately, and some complex dynamics are also presented. Four kinds of bi-stability, i.e., the coexistence of chaos and Period-3 solution, that of Period-1 and Period-2 solutions, that of Period-2 and Period-2 solutions, that of Period-4 and Period-2 solutions are disclosed. And a route to chaos, i.e., Period Doubling bifurcation to chaos, and the window of Period-3 to chaos are also discovered. The periodic oscillation can reduce the link utilization, induce the TCP stream synchronization services and further congestion. Chaotic oscillation may result in collapse. Therefore, all complex dynamical phenomena found in this paper are harmful and should be avoided. The obtained results can be very helpful for the researchers to have a better understanding of the mechanism of the network congestion control system, and they can select the parameters properly to improve the network stability and performance.

Predator–prey interaction system with mutually interfering predator: role of feedback control

In this study, we investigate the global dynamics of non-autonomous and autonomous systems based on the Leslie–Gower type model using the Beddington–DeAngelis functional response (BDFR) with time-independent and time-dependent model parameters. Unpredictable disturbances are introduced in the forms of feedback control variables. BDFR explains the feeding rate of the predator as functions of both the predator and prey densities. The global stability of the unique positive equilibrium solution of the autonomous model is determined by defining an appropriate Lyapunov function. The condition obtained for the global stability of the interior equilibrium ensures that the global stability is free from control variables, which is also a significant issue in the ecological balance control procedure. The autonomous system exhibits complex dynamics via bifurcation scenarios, such as period doubling bifurcation. We prove the existence of a globally stable almost periodic solution of the associated non-autonomous model. The different coefficients of the system are taken as almost periodic functions by generalizing periodic assumptions. The permanence of the non-autonomous system is established by defining upper and lower averages of a function. Our results also explain how the important hypothesis in ecology known as the “intermediate disturbance hypothesis” applies in predator–prey interactions. We show that moderate feedback intensity can make both the ordinary differential equation system and partial differential equation system more robust. The results obtained provide new insights into the protection of populations, where moderate feedback intensity can promote the coexistence of species and adjusting the intensity of the feedback in appropriate regions can control the population biomass while maintaining the stability of the system. Finally, the results obtained from extensive numerical simulations support the analytical results as well as the usefulness of the present study in terms of ecological balance and bio-control problems in agro-ecosystems.

Dynamics of Convective Thermal Explosion in Porous Media

In this paper, we study complex dynamics of the interaction between natural convection and thermal explosion in porous media. This process is modeled with the nonlinear heat equation coupled with the nonstationary Darcy equation under the Boussinesq approximation for a fluid-saturated porous medium in a rectangular domain. Numerical simulations with the Radial Basis Functions Method (RBFM) reveal complex dynamics of solutions and transitions to chaos after a sequence of period doubling bifurcations. Several periodic windows alternate with chaotic regimes due to intermittence or crisis. After the last chaotic regime, a final periodic solution precedes transition to thermal explosion.

Instabilities, bifurcations, and transition to chaos in electrophoresis of charge-selective microparticle

Electro-hydrodynamic instabilities near a cation-exchange microgranule in an electrolyte solution under an external electric field are studied numerically. Despite the smallness of the particle and practically zero Reynolds numbers, in the vicinity of the particle, several sophisticated flow regimes can be realized, including chaotic ones. The obtained results are analyzed from the viewpoint of hydrodynamic stability and bifurcation theory. It is shown that a steady-state uniform solution is a non-unique one; an extra solution with a characteristic microvortex, caused by non-linear coupling of the hydrodynamics and electrostatics, in the region of incoming ions is found. Implementation of one of these solutions is subject to the initial conditions. For sufficiently strong fields, the steady-state solutions lose their stability via the Hopf bifurcation and limit cycles are born: a system of waves grows and propagates from the left pole, θ = 180°, toward the angle θ = θ0 ≈ 60°. Further bifurcations for these solutions are different. With the increase in the amplitude of the external field, the first cycle undergoes multiple period doubling bifurcation, which leads to the chaotic behavior. The second cycle transforms into a homoclinic orbit with the eventual chaotic mode via Shilnikov’s bifurcation. Santiago’s instability [Chen et al., “Convective and absolute electrokinetic instability with conductivity gradients,” J. Fluid Mech. 524, 263 (2005)], the third kind of instability, was then highlighted: an electroneutral extended jet of high salt concentration is formed at the right pole (region of outgoing ions, θ = 0°). For a large enough electric field, this jet becomes unstable; the perturbations are regular for a small supercriticality, and they acquire a chaotic character for a large supercriticality. The loss of stability of the concentration jet significantly affects the hydrodynamics in this area. In particular, the Dukhin–Mishchuk vortex, anchored to the microgranule at θ ≈ 60°, under the influence of the jet oscillations loses its stationarity and separates from the microgranule, forming a chain of vortices moving off the granule. This phenomenon strongly reminds the Kármán vortices behind a sphere but has another physical mechanism to implement. Besides the fundamental importance of the results, the instabilities found in the present work can be a key factor limiting the robust performance of complex electrokinetic bio-analytical systems. On the other hand, these instabilities can be exploited for rapid mixing and flow control of nanoscale and microscale devices.

Feedback control of calcium driven alternans in cardiac myocytes.

Cardiac alternans is a beat-to-beat alternation of the action potential duration (APD), which has been implicated as a possible cause of ventricular fibrillation. Previous studies have shown that alternans can originate via a period doubling bifurcation caused by the nonlinear dependence of the APD on the previous diastolic interval. In this case, it has been demonstrated that alternans can be eliminated by applying feedback control on the pacing cycle length. However, studies have shown that alternans can also originate due to unstable calcium (Ca) cycling in cardiac myocytes. In this study, we explore the effectiveness of APD feedback control to suppress alternans when the underlying instability is due to unstable Ca cycling. In particular, we explore the role of the bi-directional coupling between Ca and voltage and determine the effectiveness of feedback control under a wide range of conditions. We also analyze the applicability of feedback control on a coupled two cell system and show that APD control induces spatially out-of-phase alternans. We analyze the onset and the necessary conditions for the emergence of these out-of-phase patterns and assess the effectiveness of feedback control to suppress Ca driven alternans in a multi-cellular system.

Effects of Inert Fuel Diluents on Thermoacoustic Instabilities in Gas Turbine Combustion

The effects of inert diluents in the fuel mixture of a model swirl stabilized gas turbine combustor, under thermoacoustically unstable limit cycle operation, were studied experimentally. The measurements included particle image velocimetry (PIV), high-speed CH* chemiluminescent imaging, and dynamic pressure signals. The paper focuses on the dynamic phenomenon of the period doubling bifurcation, which came about when the equivalence ratio of undiluted flames was enriched from 0.55 to 0.60 under constant Reynolds number (). The bifurcation featured an emergence of an aerodynamically related timescale in addition to the fundamental timescale that was induced from an unstable acoustic eigenmode. The aerodynamic timescale is introduced by azimuthal convection of a high-heat-release-rate region and is linked in the literature with a precessing motion of the recirculation zone. It was found that on increasing the diluent molar fraction up to 50%, the amplitude and the frequency of the limit cycle fundamental acoustic mode decreased. A mechanism to interpret this suppression is that increasing the diluent molar fraction of the fuel makes the flame more susceptible to quenching because the extinction strain rate of the mixture is decreased. The paper argues that the existence of inert diluents in the fuel can significantly alter the dynamic state of the combustor, because the anchoring locations of the flame greatly depend on the composition-sensitive extinction strain rate of the mixture.

The Impact of Service and Channel Integration on the Stability and Complexity of the Supply Chain

This paper constructs a supply chain consisting of a manufacturer and a retailer. Considering channel integration and service cooperation, two dynamic Stackelberg game models are established: one without unit profit allocation M and the other one with unit profit allocation Mε. In two dynamic models, we analyze the influence of relevant parameters on the stability and complexity of the dynamic system and system profit by nonlinear system theory and numerical simulation. We find that the higher adjustment parameters can cause the system to lose stability, showing double period bifurcation or wave-shape chaos. The stable region becomes larger with increase in service value and value of unit profit sharing. Besides, when the system is in chaotic state, we find that the profit of the system will fluctuate or even decline sharply; however, keeping the parameters in a certain range is helpful in maintaining the system stability and is conducive to decision-makers to obtain steady profits. In order to control the chaos phenomenon, the state feedback method is employed to control the chaotic system well. This study provides some valuable significance to supply chain managers in channel integration and service cooperation.

Damped superlinear Duffing equation with strong singularity of repulsive type

A damped Duffing equation with a singularity is considered in this paper, where the elastic restoring force g has a singularity at origin and satisfies superlinear condition at infinity. By applying the twist theorem of nonarea-preserving map, we obtain the equation has at least one period-mT solution, where the minimal period is mT. It is also shown by bifurcation analysis that for a explicit form, the equation undergoes fold bifurcation, period doubling bifurcation and Hopf bifurcation, which leads to different solutions, including harmonic solutions, subharmonic solutions and quasiperiodic solutions. At last, we give the phase portraits and correspond Poincare section of the solutions.

A double-side electrically-actuated arch microbeam for pressure sensing applications

In this study, we consider a pressure sensor whose main component is a clamped-clamped shallow arched microbeam. Two fixed electrodes are used to actuate the microbeam in the transverse direction. The lower electrode is powered by a combination of DC and AC voltages sources to excite the microbeam (drive mode). The upper electrode is polarized with a DC voltage and used to detect the change in the capacitance resulting from the transverse vibrations of the microbeam (sense mode). We formulate a fully-coupled multi-physics model of the electrically-actuated shallow arch microbeam combining the nonlinear Euler-Bernoulli beam theory with the nonlinear Reynolds equation governing the surrounding fluid domains (drive and sense zones). The model captures the inherent nonlinear physical aspects including the mid-plane stretching, the squeeze film damping, and fringing field effect. We validate the developed model by quantitatively comparing our nonlinear frequency-response against existing experimental results. We conduct static analysis to identify the occurrence of snap-through and pull-in instability for different initial midpoint rises. The coupled eigenvalue problem is also solved to compute the damped natural frequencies along with their corresponding beam and fluid mode shapes. The obtained natural frequencies are in good agreement with those reported in the literature. High sensitivity of the natural frequency to pressure variations is observed when decreasing the gap distance separating the microbeam from the fixed electrodes and reducing the initial midpoint rise of the curved microbeam.We also investigate the effect of breaking the symmetry of the microstructure on its dynamic response by introducing a slight perturbation in the mass distribution. The stability analysis reveals the occurrence of a new period doubling bifurcation point. Of interest, the location of this bifurcation point is observed to be sensitive to the pressure of the surrounding fluid medium. As such, we propose to exploit this nonlinear feature for design enhancement of the pressure sensor.

Quasi-periodic invariant 2-tori in a delayed BAM neural network

In this paper, we consider a four-neuron bi-directional associative memory (BAM, for short) neural network with two delays. We choose connection weights and the sum of delays as bifurcation parameters and derive the critical values where a double Hopf bifurcation may occur by analyzing the associated characteristic equation which is a fourth-degree polynomial exponential equation. Meanwhile, we obtain some parameter conditions on the existence of invariant 2-tori of the truncated normal form near the bifurcation point by the center manifold theorem and normal form method. Despite the fact that the higher-degree terms may destroy the invariant 2-tori of the truncated normal form, we prove that the neural network model has quasi-periodic invariant 2-tori for most of the parameter set where the truncated normal form possesses invariant 2-tori in a sufficiently small neighborhood of the bifurcation point. Numerical examples and simulations are given to support the theoretical analysis.

Controlling Chaos of the Ricker Population Model by OGY Method

The chaotic dynamics of the Ricker mapping are studied. Controlling chaos of the Ricker population model is searched by OGY control method. The dynamic behavior in the Ricker mapping is very complex in different values of a. When the value of a is changed from 0.455 to 90, the mapping goes through doubling bifurcation to Neimark-Sacker bifurcation. Several strange attractors coexist at last. Through the numerical simulation and analysis of the bifurcation and phase diagrams of the mapping, it is consistent with the theoretical analysis. Studies have shown that ecological balance can be achieved by appropriately adjusting the birth rate a, survival rate b.

Delay-induced novel dynamics in a hexagonal centrifugal governor system

Inherent time delays are often neglected in the modeling and dynamic analysis of centrifugal governor systems for the sake of simplicity, yet they can have a significant effect on the dynamic behavior of the governor systems. This paper investigates the effect of time-delay on the dynamics of a hexagonal centrifugal governor system through a comparative study on the stability and bifurcation of the equilibrium for the system with and without delay considered. It is found that the presence of time-delay can decrease the stability region of the equilibrium and generate many fine structures on the stability boundaries. New dynamic phenomena can be induced by the time-delay, including the 1:4 resonant and non-resonant double Hopf bifurcations. In addition, generic Hopf and Bautin bifurcations can be observed in the system for both the non-delay and delay cases. The unfolding of bifurcations which exhibits all possible behavior at the points of such complicated bifurcations is given by studying the normal form of the response amplitude obtained using the method of multiple scales. Numerical simulations are performed to validate the proposed theoretical analyses.

3/2 superharmonic resonance and 1/2 subharmonic resonance of functionally graded carbon nanotube reinforced composite beams

This paper deals with the 3/2 superharmonic resonance and 1/2 subharmonic resonance of functionally graded carbon nanotube reinforced composite (FG-CNTRC) beams. In the framework of Timoshenko beam theory and von Kármán type of geometric nonlinearity, the nonlinear equations of motion for FG-CNTRC beams are derived by Hamilton’s principle and discretized by the Galerkin method. The incremental harmonic balance (IHB) method is used to calculate the dynamic responses of FG-CNTRC beams subjected to transverse harmonic excitation. The stability of steady state solutions obtained from the IHB method is evaluated by Floquet theory. It is found that when the period-1 solutions of FG-CNTRC beams with asymmetric carbon nanotube distribution lose stabilities and the period doubling bifurcation occurs, the nonlinear system will generate 3/2 superharmonic resonance and 1/2 subharmonic resonance respectively in two specified frequency ratio intervals. The frequency response curves of 3/2 superharmonic resonance and 1/2 subharmonic resonance are constructed by the IHB method. The numerical results reveal the effects of material, geometry and excitation parameters on the 3/2 superharmonic and 1/2 subharmonic resonant responses of the system. Additionally, the two unstable regions where the 3/2 superharmonic and 1/2 subharmonic resonances occur are determined under different parameters.

Dynamical analysis of a cubic Liénard system with global parameters (III)**Supported by NSFC #11801079 and #11871355.

Continuing Chen and Chen (2015 Nonlinearity 28 3535) and (2016 Nonlinearity 29 1798) which deal with the cases of two equilibria and three equilibria respectively, in this paper we investigate the global dynamics of a cubic Liénard system with global parameters in the case of exact one equilibrium. After analyzing qualitative properties of all equilibria and judging the number of limit cycles, we give the bifurcation diagram and all global phase portraits. Our method in judging the number of limit cycles is to construct a parameter transformation such that in new parameter space the vector field is rotated about multiple parameters and, hence, is essential different from the methods used in previous publications. Associated with the results of last two publications, we get a positive answer to conjecture 3.2 of Khibnik et al (1998 Nonlinearity 11 1505) for general parameters about the existence of some function whose graph is exactly the surface of the double limit cycle bifurcation and therefore solve this conjecture completely.

Stability and Bifurcation: Discrete Differential Algebraic Planar Model with Square Root Response

The stability and existence of bifurcation analysis for a two spices discrete time model by introducing square root functional response and step size is examined in this work. Forward Euler scheme method is applied to formulate the discrete model from the continuous model, particularly to explore the rich dynamical behavior of the proposed model. Because the model has square root response function, trivial and axial equilibrium positions are singular. In order to discuss the stability of the trivial and axial equilibrium positions, a transformation is applied. Moreover, we explore the stability of the interior equilibrium position in a discrete two spices model using jury conditions. The numerical experiments are performed for distinct parameter values and also time series and phase line diagrams are presented. We also apply bifurcation theory to find whether the model of spices undergoes periodic doubling bifurcation at its axial and interior equilibrium positions. Numerical examples are provided and they exhibit rich dynamics in both species, including, period – 2, 4, 8 & 16 orbits, periodic windows and Non periodic orbit(chaos).

The effect of seasonal strength and abruptness on predator–prey dynamics

Coupled dynamical systems in ecology are known to respond to the seasonal forcing of their parameters with multiple dynamical behaviours, ranging from seasonal cycles to chaos. Seasonal forcing is predominantly modelled as a sine wave. However, the transition between seasons is often more sudden as illustrated by the effect of snow cover on predation success. A handful of studies have mentioned the robustness of their results to the shape of the forcing signal but did not report any detailed analyses. Therefore, whether and how the shape of seasonal forcing could affect the dynamics of coupled dynamical systems remains unclear, while abrupt seasonal transitions are widespread in ecological systems. To provide some answers, we conduct a numerical analysis of the dynamical response of predator–prey communities to the shape of the forcing signal by exploring the joint effect of two features of seasonal forcing: the magnitude of the signal, which is classically the only one studied, and the shape of the signal, abrupt or sinusoidal. We consider both linear and saturating functional responses, and focus on seasonal forcing of the predator’s discovery rate, which fluctuates with changing environmental conditions and prey’s ability to escape predation. Our numerical results highlight that a more abrupt seasonal forcing mostly alters the magnitude of population fluctuations and triggers period–doubling bifurcations, as well as the emergence of chaos, at lower forcing strength than for sine waves. Controlling the variance of the forcing signal mitigates this trend but does not fully suppress it, which suggests that the variance is not the only feature of the shape of seasonal forcing that acts on community dynamics. Although theoretical studies may predict correctly the sequence of bifurcations using sine waves as a representation of seasonality, there is a rationale for applied studies to implement as realistic seasonal forcing as possible to make precise predictions of community dynamics.

Complicated oscillations and non-resonant double Hopf bifurcation of multiple feedback delayed control system of the gut microbiota

In this paper, we consider the complex dynamics of a novel mathematical model for a feedback control system of the gut microbiota, which was proposed by Dong, et al. The main work of the present paper is to study the effect of antibiotics injection on the gut microbiota through some dynamical methods, such as double Hopf bifurcation analysis and so on. We first use DDE-BIFTOOL to find the non-resonant double Hopf bifurcation points of the system, and draw the bifurcation diagram with two bifurcation parameters, τ1 and τ2, i.e., respective measurement delays. Then we study small perturbations of two delay differential equations at these double Hopf bifurcation points, and the method of multiple scales is employed to obtain two common complex amplitude equations. By analyzing the amplitude equations, we can derive the classification and unfolding of these double Hopf bifurcation points. Finally, we verify the results by numerical simulations. We find more complicated dynamic behaviors of the system via analytical method. For example, there exists stable equilibrium, stable periodic solution or even the co-existing stable periodic solutions in respective region. And the numerical simulations are consistent with the analytic results, meanwhile it implies that the MMS is effective and accurate. All complex dynamical phenomena found in the present paper can be very helpful for the researchers to understand the mechanism of the system of gut microbiota. And it is also very significant to microbiology and engineering control. It reveals that the measurement delays can induce the complicated dynamics in this system and to the ends of excellent performance, we should take the proper values of these delays.

Double Hopf Bifurcation in Microbubble Oscillators with Delay Coupling

Using center manifold reduction methodswe investigate the double Hopf bifurcation in the dynamics of microbubble with delay couplingwith main attention focused on nonresonant double Hopf bifurcation. We obtain the normal form of the system in the vicinity of the double Hopf point and classify the bifurcations in a two-dimensional parameter space near the critical point. Some numerical simulations support the applicability of the theoretical results. In particularwe give the explanation for some physical phenomena of the system using the obtained mathematical results.

Double zero singularity and spatiotemporal patterns in a diffusive predator-prey model with nonlocal prey competition

A diffusive predator-prey model with nonlocal prey competition and the homogeneous Neumann boundary conditions is considered, to explore the effects of nonlocal reaction term. Firstly, conditions of the occurrence of Hopf, Turing, Turing-Turing and double zero bifurcations, are established. Then, several concise formulas of computing normal form at a double zero singularity for partial functional differential equations, are provided. Next, via analyzing normal form derived by utilizing these formulas, we find that diffusive predator-prey system admits interesting spatiotemporal dynamics near the double zero singularity, like tristable phenomenon that a stable spatially inhomogeneous periodic solution with the shape of \begin{document}$ \cos\omega_0 t\cos\frac{x}{l}- $\end{document} like which is unstable in model without nonlocal competition and also greatly different from these with the shape of \begin{document}$ \cos\omega_0 t+\cos\frac{x}{l}- $\end{document} like resulting from Turing-Hopf bifurcation, coexists with a pair of spatially inhomogeneous steady states with the shape of \begin{document}$ \cos\frac{x}{l}- $\end{document} like. At last, numerical simulations are shown to support theory analysis. These investigations indicate that nonlocal reaction term could stabilize spatially inhomogeneous periodic solutions with the shape of \begin{document}$ \cos\omega_0 t\cos\frac{kx}{l}- $\end{document} like for reaction-diffusion systems subject to the homogeneous Neumann boundary conditions.