Different Types Of Bifurcations Research Articles

Dynamics of a discrete one‐predator two‐prey system with Michaelis–Menten‐type prey harvesting and prey refuge

In this paper, we propose and study a discretized one‐predator two‐prey system along with prey refuge and Michaelis–Menten‐type prey harvesting. The interaction among the species is considered as Holling type III functional response. Firstly, existence and local stability of all the fixed points are derived under certain parametric conditions. Furthermore, a special consideration is made to global asymptotic stability of the interior fixed point. Then, we have shown that the system undergoes different types of bifurcations including transcritical bifurcation, flip bifurcation, and Neimark–Sacker bifurcation by using center manifold theorem, bifurcation theory, and normal form method. Also, Feigenbaum's constant of the system is calculated. It is observed that both harvesting and refuge have a stabilizing effect on the system, and the stabilizing effect of harvesting dominates the stabilizing effect of refuge. Of most interest is the finding of coexisting attractors and multistability. In particular, optimal harvesting policy has been obtained by extension of Pontryagin's maximum principle to discrete system. Finally, some intriguing numerical simulations are provided to verify our analytic findings and rich dynamics of the three species system.

Morphometric Features of Different Types of Bifurcations in the Splenic Intraorgan Arterial System in Individuals of Different Gender and Age

INTRODUCTION: A promising direction that has recently emerged in morphology is investigation of the arterial bed of various human organs as fractal or quasi-fractal systems. Conceptual models have been developed permitting a quantitative description of the vascular bed features. This approach will help to create a morphometric standard of the intraorgan blood flow, which will be useful in objective diagnosis of probable deviations from the normal structure.
 AIM: To identify the morphometric features of various types of bifurcations of the splenic intraorgan arterial bed in individuals of different gender and age.
 MATERIALS AND METHODS: The characteristics of the splenic intraorgan arterial bed have been studied in 67 individuals who died suddenly and from accidental causes at the age of 21 to 60 years. The arterial bed was represented as connected graphs with vertices corresponding to the bifurcation points of arteries, and edges to the arterial segments. The diameter and length of an arterial segment were measured on corrosion preparations. Based on the morphometry data, the following parameters were determined: generation number, division level, form factor FF1, branching factor ƞ and asymmetry factor γ. Statistical analysis was carried out using the R language.
 RESULTS: The total number of examined arterial bifurcations was 6,840. The examined bifurcations were located at 20 division levels and presented 8 generations. In the structure of the vascular bed, bifurcations of neutral kind (0) predominated with the relative quantity 51%. The least numerous was type 2 bifurcations — 9%. The intermediate position was taken by open (1) bifurcations accounting for about 40% in the vascular bed structure.
 CONCLUSIONS: The intraorgan arterial bed of the spleen is a quasi-fractal system consisting of three types of bifurcations — open, neutral and closed. A relative number of different types of bifurcations differs depending on gender and age and is also related to the generation number and division level.

Axle characteristics and vehicle limit cycle behaviour

The paper discusses the limit cycle performance of the nonlinear 2DOF single track vehicle model. That means that periodic solutions arise under constant vehicle speed and steering angle, under excessive handling conditions, with axle characteristics playing a dominant role. These characteristics are modelled using the Magic Formula tyre model, with the model parameters varied to investigate the sensitivity of peak friction and slip stiffness on limit cycle occurrence. In the transition to limit cycling, different types of bifurcation are found, being mainly of homoclinic and Hopf type, and sometimes saddle-node bifurcation, or bifurcation involving unstable limit cycles. Dealing with models being strongly nonlinear, the research includes qualitative analysis tools being appropriate for such models, such as phase plane analysis and the application of the stability diagram. The paper discusses two conjectures on the necessary and sufficient conditions on axle characteristics for the existence of limit cycle behaviour, and the relationship between steering angle and vehicle speed at the transition to limit cycling. Limit cycle performance and the impact on global stability are elements of the dynamic vehicle performance under high acceleration conditions, and therefore very relevant for excessive handling situations and our understanding of active safety under critical circumstances.

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.

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.

The continuum of wood-induced channel bifurcations

Accumulations of wood in rivers can alter three-dimensional connectivity and facilitate channel bifurcations. Bifurcations divide the flow of water and sediment into secondary channels and are a key component of anastomosing rivers. While past studies illustrate the basic scenarios in which bifurcations can occur in anastomosing rivers, understanding of the mechanisms of bifurcations remains limited. We evaluate wood-induced bifurcations across thirteen anastomosing reaches in nine different streams and rivers in the U.S. Rocky Mountains to address conditions that favor different bifurcation types. We hypothesize that (1) wood-induced bifurcations exist as a continuum of different patterns in anastomosing rivers and (2) the position of a river segment along this continuum correlates with the ratio of erosive force to erosional resistance (F/R). We use field data to quantify F/R and compare varying F/R to bifurcation types across sites. Our results support these hypotheses and suggest that bifurcation types exist as a continuum based on F/R. At higher values of F/R, more channel avulsion is occurring and predominantly lateral bifurcations form. At lower values of F/R, banks are more resistant to erosive forces and wood-induced bifurcations are transitional or longitudinal with limited lateral extent. The relationship between F/R and bifurcation types is not linear, but it is progressive. Given the geomorphic and ecological functions associated with large wood and wood-induced channel bifurcations, it becomes important to understand the conditions under which wood accumulations can facilitate different types of bifurcations and the processes involved in these bifurcations. This understanding can inform river corridor restoration designed to enhance the formation of secondary channels, increase lateral and vertical connectivity, and promote an anastomosing planform.

Bifurcation analysis and complex dynamics of a Kopel triopoly model

In this paper, a novel type of generalized Kopel triopoly model is presented to reveal the complex dynamics and transitions between different dynamic behaviors. First, based on microeconomic theory, the construction process of the Kopel triopoly model is explained in detail. Second, the existence and stability of fixed points are derived and the corresponding transition processes are presented clearly for some fixed parameters. Bifurcation sets and the critical normal forms of different types of bifurcations are computed to detect possible dynamics. Finally, numerical simulations are conducted to derive representative orbits, chaotic indicators, Lyapunov exponents, bifurcation continuation, and one- (two-) dimensional parameter spaces for the triopoly model. For example, periodic structures are presented with the numbers of corresponding periods. Some of the derived periodic orbits and Lyapunov exponents are plotted to highlight potentially stable and unstable dynamic behaviors. The results demonstrate the complexity of the Kopel triopoly game and corresponding mechanisms.

Schooling behavior driven complexities in a fear-induced prey–predator system with harvesting under deterministic and stochastic environments

The well-being of humans is closely linked to the well-being of species in any ecosystem, but the relationship between humans and nature has changed over time as societies have become more industrialized. In order to ensure the future of our ecosystems, we need to protect our planet’s biodiversity. In this work, a prey–predator model with fear dropping prey’s birth as well as death rates and nonlinear harvesting, is investigated. In addition, we consider that the consumption rate of predators, i.e., the functional response, is dependent on schooling behavior of both species. We have investigated the local stability of the equilibrium points and different types of bifurcations, such as transcritical, saddle-node, Hopf and Bogdanov–Takens (BT). We find that consumption rate of predator, fear and harvesting effort give complex dynamics in the neighbourhood of BT-points. Harvesting effort has both stabilizing and destabilizing effects. There is bistability between coexistence and predator-free equilibrium points in the system. Further, we have studied the deterministic model in fluctuating environment. Simulation results of stochastic system includes time series solutions of one simulation run and corresponding phase portraits. Notably, several simulation runs are conducted to obtain time series solutions, histograms, and stationary distributions. Our findings exhibit that during stochastic processes, model species fluctuate around some average values of the deterministic steady-state for lower environmental disturbances. However, higher values of environmental disturbances lead the species to extinction.

Deterministic and stochastic analysis of a modified Leslie–Gower predator–prey model with Holling‐type II functional response

This paper aims to study the deterministic and stochastic features of a modified Leslie–Gower predator–prey model with Holling‐type II functional response. We first investigate the dynamical properties of the deterministic model, including existence and stability of the equilibrium, and different types of bifurcations. For the stochastic model, a phenomenon of noise‐induced state transition is found. By applying the stochastic sensitivity functions technique, we construct the confidence domain of stochastic attractor and then estimate the critical value of the intensity of noise generating this transition. Numerical simulations are performed to validate the analytical findings.

Complex dynamics of a discrete‐time seasonally forced SIR epidemic model

In this paper, a discrete‐time seasonally forced SIR epidemic model with a nonstandard discretization scheme is investigated for different types of bifurcations. Although many researchers have already suggested numerically that this model can exhibit chaotic dynamics, 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, one and two parameters bifurcations of this model are investigated by computing their critical normal form coefficients. Second, the flip, Neimark–Sacker, and strong resonance 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. The model is discretized by a novel technique, namely a nonstandard finite difference discretization scheme (NSFD). Some graphical representations of the model are presented to verify the obtained results.

Mixed-mode bursting oscillations in the neighborhood of a triple Hopf bifurcation point induced by parametric low-frequency excitation

The mechanism of bursting oscillations in high dimensional system is one of the key topics for the theoretical framework of slow–fast dynamics, especially when higher co-dimensional bifurcation involves the vector field. This paper focuses on a six-dimensional vector field with triple Hopf bifurcation at the origin to explore possible bursting oscillations as well as the mechanism. By introducing a parametric excitation with low-frequency to the normal form, several types of bursting attractors can be obtained, the mechanism of which is presented upon the bifurcation analysis of the generalized autonomous system. With the increase of exciting amplitude, different types of bifurcations may influence the structure of the movement, which may cause a 2-D torus to evolve to 2-D bursting oscillations with single-mode, two-mode and three-mode, respectively, with different projections on three sub-planes. Because of the existence of three independent sub-planes, the bifurcations on different sub-planes may lead to the synchronization and non-synchronization between the associated state variables. Furthermore, the inertia of the movement may lead to the disappearance of the effect of the stable attractor with short window in the generalized autonomous system.

Dinámicas de un modelo de depredación considerando respuesta funcional sigmoidea y alimento alternativo para los depredadores

Interrelationships between two species are a basic theme in Population Dynamics, particularly the interaction between predators and their prey. This importance is due to the fact that it allows a deeper understanding of the behavior of complex food webs. In this paper we extend the analysis of a modified Leslie-Gower predator-prey model by assuming that the functional response is sigmoid or Holling type III and the predator have an alternative food. We show that the system representing the model has up to three positive equilibrium points; we establish conditions to determine the nature of each equilibrium point. In addition, we show the existence of different types of bifurcations, including those of Hopf and the homoclinic. The analytical results are discussed from an ecological perspective.

Analysis of a Cournot-Bertrand Duoploy Game with Differentiated Products: Stability, Bifurcation and Control

this work, we investigate a Cournot-Bertrand duopoly model with product differentiation between two different firms, where one company uses price as a decision variable and the other uses quantity. The stability of the Nash equilibrium point is investigated, and the role of the parameters on the model stability is studied. It is shown that the system experiences three different types of bifurcations when the parameters go through the different boundary curves of the stability region. Furthermore, the model's bifurcation is controlled using a hybrid control strategy.

Bifurcations and chaos control in a discrete-time prey–predator model with Holling type-II functional response and prey refuge

In this paper, we study the complex dynamical behavior in a discrete prey–predator model with Holling type II functional response with prey refuge. It is exhibited that the system shows different types of bifurcations viz. flip bifurcation, transcritical bifurcation and Neimark–Sacker bifurcation(NSB) by using center manifold theorem and bifurcation theory. Moreover, the chaos occurred in the system has been controlled by deploying control strategies viz. state feedback, pole placement and hybrid control techniques. The certain conditions have been determined under which chaos and bifurcation can be stabilized. The extensive numerical simulation is performed to demonstrate the analytical findings.

Generalized Swing Equation and Transient Synchronous Stability With PLL-Based VSC

With widespread application of voltage source converter (VSC) as a key energy-conversion power electronic device by using the phase-locked loop (PLL) technique for synchronization, the system dynamics has become much complicated. In this paper, the nonlinear dynamics and transient stability of the PLL-based VSC system are investigated, within a unified framework of the (normalized) generalized swing equation. It is found that there are three different types of bifurcation, including the generalized saddle-node, Hopf, and homoclinic bifurcations. Within the coexistence parameter region, the basin boundary of the stable equilibrium point shows either a closed-loop or a fish-like pattern. With the help of the equal area criterion (EAC), the transient stabilities under different transient disturbances including short circuit, voltage dip, and power rise are analyzed. Because the equivalent damping of the VSC is state-dependent, the theoretical results based on the EAC are examined. Furthermore, based on the analytical results from the generalized swing equation, the principle for all major transient stability enhancement methods is uncovered. All these findings are well verified by extensive electromagnetic transient simulations. Therefore, the generalized swing equation provides a deeper physical insight and plays a crucial role in transient stability problems in power-electronic-dominated power systems, similar to the swing equation in traditional power systems.

On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

The main purpose of the paper is to reveal the mechanism of certain special phenomena in bursting oscillations such as the sudden increase of the spiking amplitude. When multiple equilibrium points coexist in a dynamical system, several types of stable attractors via different bifurcations from these points may be observed with the variation of parameters, which may interact with each other to form other types of bifurcations. Here we take the modified van der Pol–Duffing system as an example, in which periodic parametric excitation is introduced. When the exciting frequency is far less than the natural frequency, bursting oscillations may appear. By regarding the exciting term as a slow-varying parameter, the number of the equilibrium branches in the fast generalized autonomous subsystem varies from one to five with the variation of the slow-varying parameter. The equilibrium branches may undergo different types of bifurcations, such as Hopf and pitchfork bifurcations. The limit cycles, including the cycles via Hopf bifurcations and the cycles near the homo-clinic orbit, may interact with each other to form the fold limit cycle bifurcations. With the increase of the exciting amplitude, different stable attractors and bifurcations of the generalized autonomous fast subsystem involve the full system, leading to different types of bursting oscillations. Fold limit cycle bifurcations may cause the sudden change of the spiking amplitude, since at the bifurcation points, the trajectory may oscillate according to different stable limit cycles with obviously different amplitudes. At the pitchfork bifurcation point, two possible jumping ways may result in two coexisted asymmetric bursting attractors, which may expand in the phase space to interact with each other to form an enlarged symmetric bursting attractor with doubled period. The inertia of the movement along the stable equilibrium may cause the trajectory to pass across the related bifurcations, leading to the delay effect of the bifurcations. Not only the large exciting amplitude, but also the large value of the exciting frequency may increase inertia of the movement, since in both the two cases, the change rate of the slow-varying parameter may increase. Therefore, a relative small exciting frequency may be taken in order to show the possible influence of all the equilibrium branches and their bifurcations on the dynamics of the full system.

Dynamics of stage-structured prey–predator model with prey refuge and harvesting

ABSTRACT A stage-structured prey–predator model with prey refuge is assumed, where predators have two stages as immature and mature. We consider here the harvesting on prey only and analyze the model using Beddington-DeAngelis functional response. Local stability as well as global stability at an interior equilibrium point are investigated in the proposed model. We choose different types of bifurcation like Hopf bifurcation, saddle-node bifurcation, and transcritical bifurcation in the formulated model. Moreover, in this paper, we check the optimality of the harvesting function with the help of optimal control. We provide the numerical simulation of the proposed model for the confirmation of analytical results.

Mathematical analysis of a fractional-order epidemic model with nonlinear incidence function

<abstract><p>In this paper, we are interested in studying the spread of infectious disease using a fractional-order model with Caputo's fractional derivative operator. The considered model includes an infectious disease that includes two types of infected class, the first shows the presence of symptoms (symptomatic infected persons), and the second class does not show any symptoms (asymptomatic infected persons). Further, we considered a nonlinear incidence function, where it is obtained that the investigated fractional system shows some important results. In fact, different types of bifurcation are obtained, as saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation, where it is discussed in detail through the research. For the numerical part, a proper numerical scheme is used for the graphical representation of the solutions. The mathematical findings are checked numerically.</p></abstract>

Correlation lags give early warning signals of approaching bifurcations

Identifying approaching bifurcations and regime transitions from observations is an important challenge in time series analysis with practical applications in many fields of science. Well-known indicators are the increase in spatial and temporal correlations. However, the performance of these indicators depends on the system under study and on the type of approaching bifurcation, and no indicator provides a reliable warning for any system and bifurcation. Here we propose an indicator that simultaneously takes into account information about spatial and temporal correlations. By performing a bivariate correlation analysis of signals recorded in pairs of adjacent spatial points, and analyzing the distribution of lag times that maximize the cross-correlation, we find that the variance of the lag distribution displays an extreme value that is a consistent early warning indicator of the approaching bifurcation. We demonstrate the reliability of this indicator using different types of models that present different types of bifurcations, including local bifurcations (transcritical, saddle-node, supercritical and subcritical Hopf), and global bifurcations.