Complete Bifurcation Analysis Research Articles

Effects of the random walk and the maturation period in a diffusive predator–prey system with two discrete delays

This study aims to present a complete Hopf bifurcation analysis of a model describing the relationship between prey and predator. A ratio-dependent reaction–diffusion system with two discrete time delays operating under Neumann boundary conditions governs the model that represents this competition. The bifurcation parameter for the analysis is a delay parameter that reflects the amount of time needed for the predator to be able to hunt. Bilazeroğlu and Merdan’s algorithm (Bilazeroğlu et al., 2021), which is developed by using the center manifold theorem and normal form theory, is used to establish the existence of Hopf bifurcations and also the stability of the bifurcating periodic solutions. The same procedure is used to illustrate some specific bifurcation properties, such as direction, stability, and period. Furthermore, by examining a model with constant coefficients, we also analyze how diffusion and the amount of time needed for prey to mature impact the model’s dynamics. To support the obtained analytical results, we also run some numerical simulations. The results indicate that the dynamic of the mathematical model is significantly influenced by diffusion, the amount of time needed for the predator to gain the capacity to hunt, and the amount of time required for prey to reach maturity that the predator can hunt.

Bifurcation and Path-Following Analysis of Periodic Orbits of a Fermi Oscillator Model

In this research, we offer a bifurcation analysis to describe impacting, stick, and grazing between a particle and the piston to better understand the nonlinear dynamics of a Fermi oscillator. The principles of hybrid dynamical systems will be utilized to explain the moving process in such a Fermi oscillator. The motion complexity and stick mechanism of such an oscillator are demonstrated using periodic and chaotic motions. We introduce the hybrid dynamical system and numerical continuation detection tool COCO. Physical and mathematical models are used to construct the obtained bounded mathematical model. The frequency, amplitude in oscillating base displacement, and the gap between the stationary boundary and the piston’s equilibrium position are the major parameters. We employ path-following analysis to illustrate the bifurcation that leads to solution destabilization. From the viewpoint of eigenvalue analysis, the essence of period-doubling and Fold bifurcation is revealed. Numerical continuation methods are used to perform a complete one-parameter bifurcation analysis of the Fermi oscillator. The presence of codimension-1 bifurcations of limit cycles, like period-doubling, fold, and grazing bifurcations, is shown in this work. Then, the one-parameter continuation analysis is tracked in two-parameter bifurcation diagrams that include a second important factor of interest. Bifurcations cause all solutions to lose stability, according to our findings. The acquired results provide a better understanding of the Fermi oscillator mechanism and demonstrate that we may control the system dynamics by modifying the parameters.

Nonlinear vibrations of a bi-material beam under thermal and mechanical loadings

Nonlinear vibration of a two layers (bi-material) beam under thermal and mechanical loadings is studied in the paper. The model of the beam is based on the extended Timoshenko beam theory taking into account geometric nonlinearities. The derived nonlinear partial differential equations are reduced to ordinary differential equations for the first three vibration modes. The reduced model is compared with results based on modal experimental analysis and the finite element method and a very good agreement was found. The complete bifurcation analysis is carried out for the reduced 3 DOF system for the clamped–clamped boundary conditions. The obtained resonance curves demonstrated the hardening phenomenon with instability zones in a case of large vibrations. The elevated temperature leaded to buckling, stronger modes interaction, an increase of instability zones occurring on the resonance curves and to chaotic oscillations coexisting with the periodic response.

Linking bifurcation analysis of Holling–Tanner model with generalist predator to a changing environment

AbstractBifurcation theory has been highly popular in the analysis of mathematical models. However, stability and bifurcation analyses are only for asymptotic dynamics while applied scientists care more about transient dynamics. In this paper, we first rigorously analyze Holling–Tanner model with generalist predators who have alternative food sources, and then discuss transient dynamics via a changing environment. For a constant environment, we provide a complete bifurcation analysis with high codimension. It is shown that the highest codimension of a nilpotent cusp is 3, and the model can undergo degenerate Bogdanov–Takens bifurcation of codimension 3. Moreover, by using resultant elimination to solve the semialgebraic varieties of Lyapunov coefficients, we show that a center‐type equilibrium is a weak focus with order at most 2, and the model can exhibit Hopf bifurcation of codimension 2. Our results indicate that generalist predators can cause not only richer dynamics and bifurcations, but also the extinction of prey for some positive initial densities. Numerical simulations, including the coexistence of a limit cycle and a homoclinic cycle, tristability, two limit cycles, are presented to illustrate the theoretical results. In a changing environment, the populations start along one stable state but can track unstable states or oscillations when the system crosses a bifurcation point, and then tend to another stable state or oscillations. This tracking on transient dynamics predicts regime shifts under environmental changes. When environmental conditions vary, the populations can track unstable states in the constant environment. The rate of environmental change determines how long the system tracks an unstable state although finally the solution under environmental change is attracted to a stable steady state or limit cycle. Finally, we focus on a periodic environment and find that the populations converge to a periodic solution or an invariant torus depending on both the initial environmental capacity and the amplitude of periodic fluctuation.

No Chaos in Dixon’s System

The so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behavior, if its two parameters take their values in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon’s system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into 16 different regions in each of which the system exhibits qualitatively the same behavior. In particular, we prove that in some regions two elliptic sectors with infinitely many homoclinic orbits exist.

Topological phase transition in the periodically forced Kuramoto model

A complete bifurcation analysis of explicit dynamical equations for the periodically forced Kuramoto model was performed in [L. M. Childs and S. H. Strogatz. Chaos 18 , 043128 (2008)], identifying all bifurcations within the model. We show that the phase diagram predicted by this analysis is incomplete. Our numerical analysis of the equations reveals that the model can also undergo an abrupt phase transition from oscillations to wobbly rotations of the order parameter under increasing field frequency or decreasing field strength. This transition was not revealed by bifurcation analysis because it is not caused by a bifurcation, and can neither be classified as first nor second-order since it does not display critical phenomena characteristic of either transition. We discuss the topological origin of this transition and show that it is determined by a singular point in the order-parameter space.

The organizing centre for the flow around rapidly spinning cylinders

Abstract

Dynamics of Wideband Time-Delayed Optoelectronic Oscillators With Nonlinear Filters

We present a theoretical and experimental study of an optoelectronic oscillator featuring a cubic-nonlinear filter in the feedback loop. In this architecture, the nonlinearity introduces an additional timescale that leads to the emergence of complex behavior and multiscale dynamics, ultimately leading to chaos as the gain is increased. A complete bifurcation analysis is performed and successfully compared to experimental measurements. We expect this class of systems to emulate novel functionalities for analog signal processing based on time-delayed optoelectronic oscillators.

Dynamical analysis of a two species amensalism model with Beddington–DeAngelis functional response and Allee effect on the second species

A two species amensalism model with Beddington–DeAngelis functional response is proposed and studied in this paper. The existence and stability of possible equilibria are investigated. Under some additional assumptions, there are two stable equilibria which implies this system is not asymptotically stable. Based on the stability analysis of equilibria, closed orbits and the saddle connection, we give some comprehensive bifurcation and global dynamics of the system. Next, we further incorporate the Allee effect into the second species and provide a complete qualitative and bifurcation analysis of the system with Allee effect. Numerical simulations show that the system with an Allee effect must take a longer time to reach its stable steady-state solution than that without Allee effect. There is a good agreement between the present results and the numeric simulations.

Dynamics of coral reef models in the presence of parrotfish

AbstractThe decline of coral reefs characterized by macroalgae increase has been a global threat. We consider a slightly modified version of an ordinary differential equation (ODE) model proposed in Blackwood, Hastings, and Mumby [Theor. Ecol. 5 (2012), pp. 105–114] that explicitly considers the role of parrotfish grazing on coral reef dynamics. We perform complete stability, bifurcation, and persistence analysis for this model. If the fishing effort (f) is in between two critical values and , then the system has a unique interior equilibrium, which is stable if and unstable if . If is less (more) than these critical values, then the system has up to two (zero) interior equilibria. Also, we develop a more realistic delay differential equation (DDE) model to incorporate the time delay and treating it as the bifurcation parameter, and we prove that Hopf bifurcation about the interior equilibria could occur at critical time delays, which illustrate the potential importance of the inherent time delay in a coral reef ecosystem.Recommendations for Resource Managers One serious threat to coral reefs is overfishing of grazing species, including high level of algal abundance. Fishing alters the entire dynamics of a reef (Hughes, Baird, & Bellwood, 2003), for which the coral cover was predicted to decline rapidly (Mumby, 2006). One major issue is to reverse and develop appropriate management to increase or maintain coral resilience. We have provided a detailed local and global analysis of model (Blackwood, Hastings, & Mumby, 2012) and obtained an ecologically meaningful attracting region, for which there is a chance of stable coexistence of coral–algal–fish state. The healthy reefs switch to unhealthy state, and the macroalgae–parrotfish state becomes stable as the fishing effort increases through some critical values. Also, for some critical time delays, a switch between healthy and unhealthy reef states occurs through a Hopf bifurcation, which can only appear in the delay differential equation (DDE) model. Eventually, for large enough time delay, oscillations appear and an unhealthy state occurs.

Multipulse dynamics of a passively mode-locked semiconductor laser with delayed optical feedback.

Passively mode-locked semiconductor lasers are compact, inexpensive sources of short light pulses of high repetition rates. In this work, we investigate the dynamics and bifurcations arising in such a device under the influence of time delayed optical feedback. This laser system is modelled by a system of delay differential equations, which includes delay terms associated with the laser cavity and feedback loop. We make use of specialised path continuation software for delay differential equations to analyse the regime of short feedback delays. Specifically, we consider how the dynamics and bifurcations depend on the pump current of the laser, the feedback strength, and the feedback delay time. We show that an important role is played by resonances between the mode-locking frequencies and the feedback delay time. We find feedback-induced harmonic mode locking and show that a mismatch between the fundamental frequency of the laser and that of the feedback cavity can lead to multi-pulse or quasiperiodic dynamics. The quasiperiodic dynamics exhibit a slow modulation, on the time scale of the gain recovery rate, which results from a beating with the frequency introduced in the associated torus bifurcations and leads to gain competition between multiple pulse trains within the laser cavity. Our results also have implications for the case of large feedback delay times, where a complete bifurcation analysis is not practical. Namely, for increasing delay, there is an ever-increasing degree of multistability between mode-locked solutions due to the frequency pulling effect.

The impact of cost uncertainty on Cournot oligopoly games

In this paper, Cournot oligopoly model is examined with an uncertain quadratic cost function. A random quadratic cost function is introduced in this model. The model consists of three competed firms on which each firm in the model wants to maximize its expected profit and also wants to minimize its uncertainty by minimizing the cost. To handle this multi-objective optimization problem, the expectation and worst-case approaches are adopted used to study uncertainty. The existence and uniqueness of the equilibrium are obtained. The asymptotic behavior of the equilibrium point is also investigated. Complete stability and bifurcation analysis are provided. The obtained theoretical results are verified by numerical simulation.

Bifurcation analysis of index infinity DAE parabolic models describing reactors and reacting flows

We show that most steady‐state models of chemical reactors and reacting flows in which convection effects are dominant and diffusion/conduction is neglected in the flow direction but included in the transverse directions, may change from parabolic type with a unique solution to index infinity differential‐algebraic equation (DAE) type with an infinite number of steady‐state solutions depending on the values of the reaction parameters. When a model is of index infinity, standard numerical methods may find only one of the solutions corresponding to latest possible ignition. We present complete bifurcation analysis of these models, a method for finding all solutions, determine the stability and, for some simpler cases, the domain of initial conditions attracted to these states. We also demonstrate that the various steady‐state solutions of the DAE systems are best found by integrating the transient hyperbolic versions of the models with appropriately selected capacitance terms and initial conditions. © 2016 American Institute of Chemical Engineers AIChE J, 63: 295–305, 2017

Finite-size effects and switching times for Moran process with mutation.

We consider the Moran process with two populations competing under an iterated Prisoner's Dilemma in the presence of mutation, and concentrate on the case where there are multiple evolutionarily stable strategies. We perform a complete bifurcation analysis of the deterministic system which arises in the infinite population size. We also study the Master equation and obtain asymptotics for the invariant distribution and metastable switching times for the stochastic process in the case of large but finite population. We also show that the stochastic system has asymmetries in the form of a skew for parameter values where the deterministic limit is symmetric.

Complete Global and Bifurcation Analysis of a Stoichiometric Predator–Prey Model

Loladze et al. (Bull Math Biol 62:1137–1162, 2000) proposed a highly cited stoichiometric predator–prey system, which is nonsmooth, and thus it is extremely difficult to analyze its global dynamics. The main challenge comes from the phase plane fragmentation and parameter space partitioning in order to perform a detailed and complete global stability and bifurcation analysis. Li et al. (J Math Biol 63:901–932, 2011) firstly discussed its global dynamical behavior with Holling type I functional response and found that the system has no limit cycles, and the internal equilibrium is globally asymptotically stable if it exists. Secondly, for the system with Holling type II functional response, Li et al. (2011) fixed all parameters (with realistic values) except K to perform the bifurcation analysis and obtained some interesting phenomena, for instance, the appearance of bistability and many bifurcation types. The aim of this paper is to provide a complete global analysis for the system with Holling type II functional response without fixing any parameter. Our analysis shows that the model has far richer dynamics than those found in the previous paper (Li et al. 2011), for example, four types of bistability appear: besides the bistability between an internal equilibrium and a limit cycle as shown in Li et al. (2011), the other three bistabilities occur between an internal equilibrium and a boundary equilibrium, between two internal equilibria, or between a boundary equilibrium and a limit cycle. In addition, this paper rigorously provides all possible bifurcation passways of this stoichiometric model with Holling type II functional response.

Incoherent chimera and glassy states in coupled oscillators with frustrated interactions.

We suggest a site disorder model that describes the population of identical oscillators with quenched random interactions for both the coupling strength and coupling phase. We obtain the reduced equations for the suborder parameters, on the basis of Ott-Antonsen ansatz theory, and present a complete bifurcation analysis of the reduced system. New effects include the appearance of the incoherent chimera and glassy state, both of which are caused by heterogeneity of the coupling phases. In the incoherent chimera state, the system displays an exotic symmetry-breaking behavior in spite of the apparent structural symmetry where the oscillators for both of the two subpopulations are in a frustrated state, while the phase distribution for each subpopulation approaches a steady state that differs from each other. When the incoherent chimera undergoes Hopf bifurcation, the system displays a breathing incoherent chimera. The glassy state that occurs on a surface of three-dimensional parameter space exhibits a continuum of metastable states with zero value of the global order parameter. Explicit formulas are derived for the system's Hopf, saddle-node, and transcritical bifurcation curves, as well as the codimension-2 crossing points, including the Takens-Bogdanov point.

How bifurcations affect functional connectivity in finite-size neural networks

Understanding how the functional connectivity of a neural network (i.e. the statistical dependencies among different neurons) depends upon its anatomical connectivity and how it is modulated by other network parameters is a central topic in neuroscience [1]. However, only few analytical investigations of functional connectivity in network models have been reported [2]. A difficulty for such analytical studies is that they require in principle characterizing the bifurcation points (i.e. the parameter values at which the network undergoes a sudden change in dynamics) and how functional connectivity changes close to such points. Given that biological networks have a finite number of neurons, it is also important to develop such analytical studies without relying only on large network approximations that neglect finite-size effects. To overcome these difficulties, here we developed a new approach to investigate the dynamics of finite-size networks composed by interconnected inhibitory and excitatory neurons. We applied this approach to a network of neurons described by voltage-based differential equations, with an input to each neuron provided by a deterministic current superimposed to a weak normally distributed stochastic component. To understand the interplay between the deterministic dynamics of the network and the effects of random fluctuations in a finite-size network, we proceeded in two steps. First, we neglected the stochastic component of the input and we performed a complete numerical and analytical bifurcation analysis while varying the deterministic inputs and the strength of the connections. We did this using the MatCont continuation package [3] and the analytical expressions of the Jacobian matrix eigenvalues. We found that our finite-size network displayed standard bifurcations such as saddle-node and Hopf manifolds. Specifically, these manifolds describe the input currents values for which equilibria collide and annihilate to each other, and for which self-sustained oscillations arise, respectively. Moreover, strong inhibition gave rise to additional branch point bifurcations: for highly negative synaptic weights, the system exhibited special multiple solutions characterized by heterogeneous membrane potentials. These particular solutions represent new finite-size effects not captured by large network approximations. Second, we inserted stochastic components into the input and studied the behaviour of the network under the effect of noise. Specifically, we used a new first-order perturbative method where the perturbative parameter is the standard deviation of the noise in the input. This technique allowed the analytical calculation of the cross-correlation between firing rates, and of the Fisher information about the external input carried by the network. Our analytical solutions showed that the neurons become independent when the deterministic component of the input is strong, while they become correlated for smaller current values lying on the saddle-node manifold. The latter is due to a counter-intuitive finite-size effect [4] that can be fully captured by our formalism. Furthermore, in our network strong correlations led to large Fisher information values, thus enhancing the encoding precision of the network with respect to the input variables. This also shows that reducing correlations among neurons activity does not imply an enhancement of the information encoding, as suggested in some studies [5].

Global Bifurcation Analysis of a Population Model with Stage Structure and Beverton–Holt Saturation Function

In the present paper, we perform a complete bifurcation analysis of a two-stage population model, in which the per capita birth rate and stage transition rate from juveniles to adults are density dependent and take the general Beverton–Holt functions. Our study reveals a rich bifurcation structure including codimension-one bifurcations such as saddle-node, Hopf, homoclinic bifurcations, and codimension-two bifurcations such as Bogdanov–Takens (BT), Bautin bifurcations, etc. Moreover, by employing the polynomial analysis and approximation techniques, the existences of equilibria, Hopf and BT bifurcations as well as the formulas for calculating their bifurcation sets have been provided. Finally, the complete bifurcation diagrams and associate phase portraits are obtained not only analytically but also confirmed and extended numerically.

Qualitative analysis of a predator–prey system with double Allee effect in prey

Growing biological evidence from various ecosystems (Berec et al., 2007) suggests that the Allee effect generated by two or more mechanisms can act simultaneously on a single population. Surprisingly, prey–predator system incorporating multiple Allee effect is relatively poorly studied in literature. In this paper, we consider a ratio dependent prey–predator system with a double Allee effect in prey population growth. We have taken into account the case of the strong and the weak Allee effect separately and study the stability and complete bifurcation analysis. It is shown that the model exhibits the bi-stability and there exists separatrix curve(s) in the phase plane implying that dynamics of the system is very sensitive to the variation of the initial conditions. Generic normal forms at double zero singularity are derived by a rigorous mathematical analysis to show the different types of bifurcation of the proposed model system including Bogdanov–Takens bifurcation, saddle–node bifurcation curve, a Hopf bifurcation curve and a homoclinic bifurcation curve. The complete analysis of possible topological structures including elliptic, parabolic, or hyperbolic orbits, and any combination of them in a neighborhood of the complicated singular point (0,0) in the interior of the first quadrant is explored by using a blow up transformation. These structures have important implications for the global behavior of the model. Numerical simulation results are given to support our theoretical results. Finally, the paper concludes with a discussion of the ecological implications of our analytic and numerical findings.

Bifurcation Analysis for Nonlinear Recurrence Relations with Threshold Control and2k-Periodic Coefficients

A nonlinear recurrence involving a piecewise constant McCulloch-Pitts function and2k-periodic coefficient sequences is investigated. By allowing the threshold parameter to vary from 0 to∞, we work out a complete bifurcation analysis for the asymptotic behaviors of the corresponding solutions. Among other things, we show that each solution tends towards one of four different limits. Furthermore, the accompanying initial regions for each type of solutions can be determined. It is hoped that our analysis will provide motivation for further results for recurrent McCulloch-Pitts type neural networks.