Types Of Bifurcations Research Articles

An Analytic Investigation of Hopf Bifurcation Location Control for the Rulkov Map Model

From the point of view of nonlinear dynamical systems, some neurological disorders can be indicated by bifurcations because bifurcations change the firing patterns of neurons; therefore, it is essential to control the bifurcations. We can avoid undesirable dynamical behaviors such as the behaviors of the Rulkov map model by controlling bifurcation which, then, can assist in modeling neuronal diseases. In this paper, we investigate the existence of Hopf bifurcation and analytically identify the type of bifurcation for the Rulkov map model; then, we apply a dynamic feedback controller using a washout filter to control the onset of Hopf bifurcation. Also, we can control the behaviors of the neurons, such as spiking or spiking-bursting behavior of neurons, and create the Hopf bifurcation for some parameters. The results analytically obtained in this paper can be applied to control some epileptic seizures.

The Singularity of Expanded Jacobian Matrix in Incremental Harmonic Balance Method Directly Locates Bifurcation Points of Steady-State Responses

Abstract As a semi-analytical approach, the incremental harmonic balance (IHB) method is widely implemented for solving steady-state (including both periodic and quasi-periodic) responses through an iteration process. The iteration is carried out through a Jacobian matrix (JM) and a residual vector, both updated in each iteration. Though the JM is known to be singular at certain bifurcation points, the singularity is still an open question and could play a pivotal role in real applications. In this study, we define and calculate an expanded JM (EJM) by applying an expanded solution expression in the IHB iteration. The singularity of the EJM at several different bifurcation points is proved in a general manner, according to the bifurcation theory for equilibria in nonlinear dynamical systems. Given the possible bifurcation type, furthermore, the singularity is applied to locate the corresponding bifurcation point directly and precisely. Considered are the cases of the period-doubling, symmetry breaking, and Neimark-Sacker bifurcations of periodic and/or quasi-periodic responses.

Bistability and tristability in a deterministic prey–predator model: Transitions and emergent patterns in its stochastic counterpart

The paper focuses on studying the global dynamics of a simple prey–predator model. The model incorporates cooperative hunting behavior among predators and takes into account the effect of harvesting on the predator population. The interaction between the prey and predator follows the Crowley–Martin functional response. The study involves analyzing the equilibrium points of the system and investigating their stability properties through mathematical analysis. Various types of bifurcations, including Hopf bifurcation, saddle–node bifurcation, and transcritical bifurcation, are numerically demonstrated in the figures, highlighting the dynamic behavior of the model. One of the intriguing findings of the study is the occurrence of bistability and tristability in the model. To incorporate stochasticity into the model, white noise is added to the deterministic system. This allows for the examination of transitions between different steady states in the stochastic system. We have conducted an analysis of species persistence and extinction in relation to the presence of noise. The paper presents the stochastic sensitivity function (SSF) technique and the use of confidence ellipses to assess the likelihood of such transitions occurring in the system. Overall, the study provides insights into the complex dynamics of prey–predator interactions, considering factors such as cooperation, harvesting, and stochasticity. The results contribute to our understanding of population dynamics and the potential effects of environmental and human-induced perturbations on ecosystem stability.

Multifractality and scale-free network topology in a noise-perturbed laminar jet

We present experimental evidence of multifractality and scale-free network topology in a noise-perturbed laminar jet operated in a globally stable regime, prior to the critical point of a supercritical Hopf bifurcation and prior to the saddle-node point of a subcritical Hopf bifurcation. For both types of bifurcation, we find that (i) the degree of multifractality peaks at intermediate noise intensities, (ii) the conditions for peak multifractality produce a complex network whose node degree distribution obeys an inverse power-law scaling with an exponent of $2 < \gamma < 3$ , indicating scale-free topology and (iii) the Hurst exponent and the global clustering coefficient can serve as early warning indicators of global instability under specific operating and forcing conditions. By characterising the noise-induced dynamics of a canonical shear flow, we demonstrate that the multifractal and scale-free network dynamics commonly observed in turbulent flows can also be observed in laminar flows under certain stochastic forcing conditions.

Qualitative properties, bifurcations and chaos of a discrete predator–prey system with weak Allee effect on the predator

In this work, the qualitative properties of the fixed points in the non-hyperbolic and degenerate cases, codimension-one bifurcations and Marotto’s chaos of a discrete predator–prey system with weak Allee effect on the predator are investigated. Utilizing the center manifold theorem and the reduction principle, the qualitative property of each fixed point in the non-hyperbolic case is explored. Based on the approximate flow theory, the qualitative property of the boundary fixed point in the degenerate case is investigated. Making use of the center manifold theorem and the bifurcation theory, all the potential codimension-one bifurcation types of the co-existence fixed point are explored, including flip bifurcation and Neimark–Sacker bifurcation. For each type of these bifurcations, not only the concise non-degenerate condition expressed by the system parameters is given, but also the analytical expression of the system orbit caused by the bifurcation is derived. Based on the non-negativity of the bifurcation critical values, the non-resonant and non-degenerate conditions for the occurrence of these bifurcations, the system parameter plane is divided into several regions using tools such as the polynomial complete discriminant system theory, real root separation theory, dichotomy method, implicit function theorem and Cardan’s formula. In each region, the direction and stability of these bifurcations are studied, which helps to theoretically explain the impact of Allee effect on the system. By proving that the co-existence fixed point is a snap-back repeller under appropriate conditions, it is obtained that the system is chaotic. Numerical simulations are completely consistent with all the theoretical analyses. The research results show that the impact of Allee effect on the population system depends not only on the assumption and construction of the system, but also on the system parameter and initial value of the system.

Vibro-impact dynamics of an experimental rig with two-sided constraint and bidirectional drift

This paper demonstrates a theoretical and experimental study on a universal vibro-impact system with bidirectional drift. With a dedicated experimental rig that is able to verify numerical simulations, the system has revealed various bifurcations from a piecewise-smooth dynamical model involving impact and dry friction. This experimental rig with a two-sided constraint is designed with adjustable configurations which can be used for experimental testing of versatile piecewise-smooth dynamical systems, e.g., the impact oscillator, the drifting oscillator and the capsule system. It is able to be customised to alter different system parameters, such as friction, number of constraints, mass ratio, impact gap and constraint stiffness, for developing and testing various control strategies. In this paper we particularly studied several types of bifurcations from our preliminary experimental results and discussed the dynamical responses of the system under different excitation frequencies and amplitudes. It was found that with a higher excitation frequency and amplitude, the vibro-impact dynamics on both ends of the constraints could result in a large amount of uncertainties, thus bi-stable and chaotic motions are likely to be observed. While being unstable, the system runs with a relatively high energy efficiency in this configuration.

Stability and HOPF Bifurcation Analysis of Periodic Solutions of a Duffing Equation

In this study, the stability and Hopf bifurcation analysis of periodic solutions of Duffing equations were considered. Also other types of bifurcation like the Saddle-node, Transcritical and Pitchkfork were also studied. The eigen value, Jacobian and Floquet theory were used to analyse both the stability and Hopf bifurcations of the periodic solutions of the equilibrium points. The results showed that equilibrium points have at most three -periodic solutions under a strong damped conditions due to the cubic nonlinearities. The bifurcation points showed; one critical, another subcritical and the third point showed that the homoclinic was present when the damping coefficient is zero. Furthermore, the presence of strange attractors varied with the driving force and damping. The MATCAD software was used to illustrate the numerical behaviour of the solution which extend some results in the literature.

Nonlinear dynamics of a three-dimensional discrete-time delay neural network

This paper examines a three-dimensional delayed discrete neural network model analytically and numerically to determine the existence of different types of bifurcations of the involved fixed points. The model exhibits different bifurcations such as pitchforks, flips, Neimark–Sackers, and flip-Neimark–Sackers. The critical coefficients are used to determine the structure of each bifurcation. The curves are calculated and plotted for each bifurcation when the parameters are changed. Further, these bifurcations are theoretically analyzed and numerically verified. From the obtained results, we observed that by drawing the curves associated with each bifurcation, the numerical simulations are consistent with the analytical results.

Rate and memory effects in bifurcation-induced tipping.

A variation in the environment of a system, such as the temperature, the concentration of a chemical solution, or the appearance of a magnetic field, may lead to a drift in one of the parameters. If the parameter crosses a bifurcation point, the system can tip from one attractor to another (bifurcation-induced tipping). Typically, this stability exchange occurs at a parameter value beyond the bifurcation value. This is what we call, here, the shifted stability exchange. We perform a systematic study on how the shift is affected by the initial parameter value and its change rate. To that end, we present numerical simulations and partly analytical results for different types of bifurcations and different paradigmatic systems. We show that the nonautonomous dynamics can be split into two regimes. Depending on whether we exceed the numerical or experimental precision or not, the system may enter the nondeterministic or the deterministic regime. This is determined solely by the conditions of the drift. Finally, we deduce the scaling laws governing this phenomenon and we observe very similar behavior for different systems and different bifurcations in both regimes.

Stability and Bautin bifurcation of four-wheel-steering vehicle system with driver steering control.

In this paper, the stability and Bautin bifurcation of a four-wheel-steering (4WS) vehicle system, by considering driver steering control, are investigated. By using the central manifold theory and projection method, the first and second Lyapunov coefficients are calculated to predict the type of Hopf bifurcation of the vehicle system. The topological structure of Bautin bifurcation, a degenerate Hopf bifurcation of the 4WS vehicle system, is presented in parameter space, and it reveals the dynamics of the vehicle system of different choices of control parameters. The influences of system parameters on critical values of the bifurcation parameter are also analyzed. It is shown that with the increase in the frontal visibility distance of the driver control strategy coefficient and the cornering stiffness coefficients of rear wheels, the critical speed increases. Nevertheless, the critical speed decreases with the increase in the distance from the center of gravity of the vehicle to the front axles, Driver's perceptual time delay, and cornering stiffness coefficients of the front wheels.

Dynamical behaviors of an epidemic model with partial immunity having nonlinear incidence and saturated treatment in deterministic and stochastic environments

This manuscript deals with an epidemic model with partial immunity having nonlinear incidence and saturated treatment. Positivity and boundedness of the solutions have been established here. We discuss local stability of all equilibria. The proposed system experiences various types of bifurcations, namely Transcritical, Saddle–node, Hopf, and Bogdanov–Takens bifurcation of co-dimension 2. The system is reduced to a two-dimensional system using center manifold theorem to deduce normal form for Bogdanov–Takens bifurcation of co-dimension 2 when two eigenvalues at the endemic equilibrium point becomes zero. Whether deterministic model overestimates the condition for disease propagation, to observe this we also analysis stochastic model. We derive the condition for extinction, persistence in mean and stationary distribution. All theoretical findings are justified by numerical simulations. Finally, to check validity of the model, we fit it with real reported influenza data of Canada.

Numerical study on the nonlinear characteristics of shock induced two-dimensional panel flutter in inviscid flow

For an aeroelastic system of a two-dimensional elastic panel subjected to an impinging inviscid oblique shockwave, the nonlinear flutter characteristics are affected by many factors such as shock impingement location, cavity pressure and initial perturbation. The effects of the above factors on the variation of system bifurcation type and dynamic behaviors are investigated numerically. A low-fidelity computational method coupled with local piston theory and van Karman plate model, and a high-fidelity computational method coupled with Euler equations and finite element model are used for fluid-structure interaction simulations. Two sets of new findings are unveiled. First, either the variation of shock impingement location or cavity pressure can induce the aeroelastic system to transition between a subcritical bifurcation and a supercritical bifurcation. For some cases, the system bifurcation characteristics exhibit strong sensitivity to these two factors. Second, it is found that in addition to the limit cycle oscillation in the form of a combination of the second and third structural modes, multiple stable limit cycle oscillations due to the coupling of higher-order modes can be triggered by suitable initial perturbations. These limit cycle oscillations have high frequencies and amplitudes, which suggest a higher risk of structural fatigue failure.

Bifurcation of solutions through a contact manifold in bidisperse models

This research focuses on a hyperbolic system that describes bidisperse suspensions, consisting of two types of small particles dispersed in a viscous fluid. The dependence of solutions on the relative position of contact manifolds in the phase space is examined. The wave curve method serves as the basis for the first and second analyses. The former involves the classification of elementary waves that emerge from the origin of the phase space. Analytical solutions to prototypical Riemann problems connecting the origin with any point in the state space are provided. The latter focuses on semi-analytical solutions for Riemann problems connecting any state in the phase space with the maximum packing concentration line, as observed in standard batch sedimentation tests. When the initial condition crosses the first contact manifold, a bifurcation occurs. As the initial condition approaches the second manifold, another structure appears to undergo bifurcation, although it does not represent an actual bifurcation according to the triple shock rule. The study reveals important insights into the behavior of solutions in relation to these contact manifolds. This research sheds light on the existence of emerging quasi-umbilic points within the system, which can potentially lead to new types of bifurcations as crucial elements of the elliptic/hyperbolic boundary in the system of partial differential equations. The implications of these findings and their significance are discussed.

A generically singular type of saddle–node bifurcation that occurs for relativistic shock waves

This paper is concerned with two-parameter families of planar dynamical systems in which a saddle–node bifurcation interacts with a singular perturbation. We identify bifurcation curves along which the non-saddle point switches from node to focus and establish the existence of heteroclinic connections in either case. As an application, the result is used to show the existence of oscillatory shock profiles in a model of relativistic fluid dynamics.

Hopf Bifurcations of Moore-Greitzer PDE Model with Additive Noise

The Moore-Greitzer partial differential equation (PDE) is a commonly used mathematical model for capturing flow and pressure changes in axial-flow jet engine compressors. Determined by compressor geometry, the deterministic model is characterized by three types of Hopf bifurcations as the throttle coefficient decreases, namely surge (mean flow oscillations), stall (inlet flow disturbances) or a combination of both. Instabilities place fundamental limits on jet-engine operating range and thus limit the design space. In contrast to the deterministic PDEs, the Hopf bifurcation in stochastic PDEs is not well understood. The goal of this particular work is to rigorously develop low-dimensional approximations using a multiscale analysis approach near the deterministic stall bifurcation points in the presence of additive noise acting on the fast modes. We also show that the reduced-dimensional approximations (SDEs) contain multiplicative noise. Instability margins in the presence of uncertainties can be thus approximated, which will eventually lead to lighter and more efficient jet engine design.

Dynamical analysis of a discrete-time SIR epidemic model

In this paper, a discrete-time seasonally forced SIR epidemic model is investigated for different types of bifurcations. Although, many researchers already suggested numerically that this model can exhibit chaotic dynamics but not much focus is given to the bifurcation theory of the model. We prove analytically and numerically the existence of different types of bifurcations in the model. First, the one and two parameters bifurcations of this model are investigated by computing their critical normal form coefficients. Secondly, the flip, Neimark–Sacker, and strong resonances bifurcations are detected for this model. The critical coefficients identify the scenario associated with each bifurcation. The complete complex dynamical behavior of the model is investigated. Some graphical representations of the model are presented to verify the obtained results.

Elastic Snap-Through Instabilities Are Governed by Geometric Symmetries.

Many elastic structures exhibit rapid shape transitions between two possible equilibrium states: umbrellas become inverted in strong wind and hopper popper toys jump when turned inside out. This snap through is a general motif for the storage and rapid release of elastic energy, and it is exploited by many biological and engineered systems from the Venus flytrap to mechanical metamaterials. Shape transitions are known to be related to the type of bifurcation the system undergoes, however, to date, there is no general understanding of the mechanisms that select these bifurcations. Here we analyze numerically and analytically two systems proposed in recent literature in which an elastic strip, initially in a buckled state, is driven through shape transitions by either rotating or translating its boundaries. We show that the two systems are mathematically equivalent, and identify three cases that illustrate the entire range of transitions described by previous authors. Importantly, using reduction order methods, we establish the nature of the underlying bifurcations and explain how these bifurcations can be predicted from geometric symmetries and symmetry-breaking mechanisms, thus providing universal design rules for elastic shape transitions.

Evolutionarily stable strategies to overcome Allee effect in predator-prey interaction.

Every successful species invasion is facilitated by both ecological and evolutionary mechanisms. The evolution of population's fitness related traits acts as functional adaptations to Allee effects. This trade-off increases predatory success at an expense of elevated death rate of potential predators. We address our queries employing an eco-evolutionary modeling approach that provides a means of circumventing inverse density-dependent effect. In the absence of evolution, the ecological system potentially exhibits multi-stable configurations under identical ecological conditions by allowing different bifurcation scenarios with the Allee effect. The model predicts a high risk of catastrophic extinction of interacting populations around different types of saddle-node bifurcations resulting from the increased Allee effect. We adopt the game-theoretic approach to derive the analytical conditions for the emergence of evolutionarily stable strategy (ESS) when the ecological system possesses asymptotically stable steady states as well as population cycles. We establish that ESSs occur at those values of adopted evolutionary strategies that are local optima of some functional forms of model parameters. Overall, our theoretical study provides important ecological insights in predicting successful biological invasions in the light of evolution.

Transitional cluster dynamics in a model for delay-coupled chemical oscillators.

Cluster synchronization is a fundamental phenomenon in systems of coupled oscillators. Here, we investigate clustering patterns that emerge in a unidirectional ring of four delay-coupled electrochemical oscillators. A voltage parameter in the experimental setup controls the onset of oscillations via a Hopf bifurcation. For a smaller voltage, the oscillators exhibit simple, so-called primary, clustering patterns, where all phase differences between each set of coupled oscillators are identical. However, upon increasing the voltage, secondary states, where phase differences differ, are detected, in addition to the primary states. Previous work on this system saw the development of a mathematical model that explained how the existence, stability, and common frequency of the experimentally observed cluster states could be accurately controlled by the delay time of the coupling. In this study, we revisit the mathematical model of the electrochemical oscillators in order to address open questions by means of bifurcation analysis. Our analysis reveals how the stable cluster states, corresponding to experimental observations, lose their stability via an assortment of bifurcation types. The analysis further reveals complex interconnectedness between branches of different cluster types. We find that each secondary state provides a continuous transition between certain primary states. These connections are explained by studying the phase space and parameter symmetries of the respective states. Furthermore, we show that it is only for a larger value of the voltage parameter that the branches of secondary states develop intervals of stability. For a smaller voltage, all the branches of secondary states are completely unstable and are, therefore, hidden to experimentalists.

Lyapunov-based robust optimal control for time-delay systems with application in milling process

In this paper, we propose an optimal delay-independent robust controller based on the Lyapunov–Krakovskii theorem for the milling process as a time-delay system in the presence of varying axial depth of cut and different parametric uncertainty, such as stiffness, damping, etc. Milling is widely used in the manufacturing processes for the production of complex-shaped workpieces with high accuracy. However, chatter is a self-excited vibration that may cause adverse effects, such as tool damage, poor surface quality, excessive noise, etc. The dynamic model of the milling process is considered a two-dimensional time-delay system. A nonlinear programming with linear matrix inequality constraints is solved in order to obtain the controller gain where the objective function is corresponding to the norm-2 of controller gain, and the constraints guarantee robust stability. Using the semi-discretization method, stability lobes are shown in both uncontrolled and controlled plants to illustrate the improvement of the stable region via the proposed controller. Bifurcation phenomena have been improved with this controller by postponing the adverse effects to the higher values of the axial depth of cut, reducing the amplitude of limit cycles, and changing the type of bifurcation. Finally, we will compare the proposed controller with an intelligent controller in order to show the efficiency of the proposed method. It is shown that the proposed controller has improved the integral absolute error index by about 3.65 times compared with the intelligent controller.