Critical Bifurcation Value Research Articles

Effect of fractional order on the stability and the localisation of the critical Hopf bifurcation value in a fractional chaotic system

Effect of fractional order on the stability and the localisation of the critical Hopf bifurcation value in a fractional chaotic system

Stability and Hopf bifurcation of a modified Leslie–Gower predator–prey model with Smith growth rate and B–D functional response

This paper is concerned with a modified Leslie–Gower predator–prey diffusive dynamics system with Smith growth subject to homogeneous Neumann boundary condition, in which the functional response is Beddington–DeAngelis (Denote it by B–D) functional response term. Firstly, by applying the theory of stability and the Hopf bifurcation, we discuss the local stability and the existence of the Hopf bifurcation at the positive constant equilibrium solution of the ODE model, which the model undergoes the Hopf bifurcation when bifurcation parameter δ crosses the bifurcation critical value b0. Moreover, stability of the bifurcation periodic solution is analyzed. Secondly, the Turing instability and the direction of Hopf bifurcation of the corresponding to PDE system are investigated by using Normal form theory and Centre manifold theory. Finally, we study the numerical simulations of this system to illustrate the theoretical analysis.

Fractional order-induced bifurcations in a delayed neural network with three neurons.

This paper reports the novel results on fractional order-induced bifurcation of a tri-neuron fractional-order neural network (FONN) with delays and instantaneous self-connections by the intersection of implicit function curves to solve the bifurcation critical point. Firstly, it considers the distribution of the root of the characteristic equation in depth. Subsequently, it views fractional order as the bifurcation parameter and establishes the transversal condition and stability interval. The main novelties of this paper are to systematically analyze the order as a bifurcation parameter and concretely establish the order critical value through an implicit function array, which is a novel idea to solve the critical value. The derived results exhibit that once the value of the fractional order is greater than the bifurcation critical value, the stability of the system will be smashed and Hopf bifurcation will emerge. Ultimately, the validity of the developed key fruits is elucidated via two numerical experiments.

Stability and dynamics of spike-type solutions to delayed Gierer-Meinhardt equations

<p style='text-indent:20px;'>For a specific set of parameters, we analyze the stability of a one-spike equilibrium solution to the one-dimensional Gierer-Meinhardt reaction-diffusion model with delay in the components of the reaction-kinetics terms. Assuming slow activator diffusivity, we consider instabilities due to Hopf bifurcation in both the spike position and the spike profile for increasing values of the time-delay parameter <inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula>. Using method of matched asymptotic expansions it is shown that the model can be reduced to a system of ordinary differential equations representing the position of the slowly evolving spike solution. The reduced evolution equations for the one-spike solution undergoes a Hopf bifurcation in the spike position in two cases: when the negative feedback of the activator equation is delayed, and when delay is in both the negative feedback of the activator equation and the non-linear production term of the inhibitor equation. Instabilities in the spike profile are also considered, and it is shown that the spike solution is unstable as <inline-formula><tex-math id="M2">\begin{document}$ T $\end{document}</tex-math></inline-formula> is increased beyond a critical Hopf bifurcation value <inline-formula><tex-math id="M3">\begin{document}$ T_H $\end{document}</tex-math></inline-formula>, and this occurs for the same cases as in the spike position analysis. In all cases, the instability in the profile is triggered before the positional instability. If however the degradation of activator is delayed, we find stable positional oscillations can occur in this system.</p>

Stability and bifurcation control analysis of a delayed fractional-order eco-epidemiological system.

Considering the factor of artificial intervention in biological control, a delayed fractional eco-epidemiological system with an extended feedback controller is proposed. By using the digestion delay as bifurcation parameter, the stability and Hopf bifurcation are investigated, and the branching conditions are given. The system undergoes Hopf bifurcation, when the parameter tau passes through the critical value. In addition, it can be pointed out that the negative feedback gain and the feedback delay could affect the bifurcation critical value of the system. Therefore, the Hopf bifurcation can also be induced by taking the feedback delay as a bifurcation parameter. Finally, by plotting the solution curve of the system, the significance of the controller to the stability of the eco-epidemiological system is verified.

Nonlinear singular traveling waves in a slightly compressible thermo-hyperelastic cylindrical shell

Nonlinear dynamic researches of rubber-like materials have always been the focus of attention. However, it is usually based on the completely incompressible hyperelastic constitutive model. Moreover, the temperature effect is ignored. Here, the constitutive relation of a slightly compressible thermo-hyperelastic material is adopted, and strongly nonlinear traveling waves in a thermo-hyperelastic cylindrical shell are investigated. Under the assumption of axisymmetric deformations, the coupled partial differential equations describing the radial and axial motions of the shell are derived. Based on the bifurcation theory of dynamical systems, the qualitative properties of the system are discussed in detail. The influences of inner and outer boundary temperatures on the presence of bounded traveling waves are investigated. Furthermore, the effects of compressibility and material parameters on the presence of periodic waves, solitary waves, non-smooth solitary cusp waves, and periodic cusp waves are studied, and the corresponding critical bifurcation values are determined. Finally, numerical validations of the analytical results are carried out. Due to these interesting phenomena, the necessity of studying the inner and outer boundary temperatures and compressibility for nonlinear traveling waves in a cylindrical shell is demonstrated.

Global orbit of a complicated nonlinear system with the global dynamic frequency method

Global orbits connect the saddle points in an infinite period through the homoclinic and heteroclinic types of manifolds. Different from the periodic movement analysis, it requires special strategies to obtain expression of the orbit and detect the associated profound dynamic behaviors, such as chaos. In this paper, a global dynamic frequency method is applied to detect the homoclinic and heteroclinic bifurcation of the complicated nonlinear systems. The so-called dynamic frequency refers to the newly introduced frequency that varies with time t, unlike the usual static variable. This new method obtains the critical bifurcation value as well as the analytic expression of the orbit by using a standard five-step hyperbolic function-balancing procedure, which represents the influence of the higher harmonic terms on the global orbit and leads to a significant reduction of calculation workload. Moreover, a new homoclinic manifold analysis maps the periodic excitation onto the target global manifold that transfers the chaos discussion of non-autonomous systems into the orbit computation of the general autonomous system. That strategy unifies the global bifurcation analysis into a standard orbit approximation procedure. The numerical simulation results are shown to compare with the predictions.

Stochastic Hopf\u2013Hopf bifurcation of two-species discrete coupling logistic system with symbiotic interaction

In this paper, stochastic Hopf–Hopf bifurcation of the discrete coupling logistic system with symbiotic interaction is investigated. Firstly, orthogonal polynomial approximation of discrete random function in the Hilbert spaces is applied to reduce the discrete coupling logistic system with random parameter to the deterministic equivalent system. Then, it is concluded that Hopf–Hopf bifurcation exists in the equivalent deterministic system according to the principle of algebraic criteria. Numerical simulations show that the bifurcation critical value varies with the intensity of random parameter, and Hopf–Hopf bifurcation and period-doubling bifurcation behavior exist. In particular, Hopf–Hopf bifurcation can be drift with the change of random intensity, and frequency locking phenomenon occurs in the stochastic system.

Different types of solitary waves in a thermo-hyperelastic neo-Hookean cylindrical shell

Based on the hyperelastic material models, the propagation of nonlinear travelling waves in cylindrical rods has been studied. Here, different types of nonlinear travelling waves in a thermo-hyperelastic neo-Hookean cylindrical shell are determined. By using the variational principle, a complex differential dynamical system describing the axisymmetric motion of the shell is derived. The effect of the temperature fields is considered. Moreover, the structure is extended from a one-dimensional rod to a three-dimensional cylindrical shell. Then in terms of the bifurcation theory, a detailed qualitative analysis of the system is carried out, and for the bifurcation parameters, their corresponding critical bifurcation values are determined. Combining with the orbits in phase diagrams under different parameters, solitary waves and periodic waves are found in the shell. It is worth pointing out that the solitary waves with the valley form may appear in the radial direction of the shell. The implicit analytical solutions and profiles of these travelling waves are given. There is a promising prospect for the propagation of strongly nonlinear travelling waves to detect structural defects and determine material parameters.

Periodic Solutions of a Delayed Eco-Epidemiological Model with Infection-Age Structure and Holling Type II Functional Response

This paper is devoted to the study of a new delayed eco-epidemiological model with infection-age structure and Holling type II functional response. Firstly, the disease transmission rate function among the predator population is treated as the piecewise function concerning the incubation period [Formula: see text] of the epidemic disease and the model is rewritten as an abstract nondensely defined Cauchy problem. Besides, the prerequisite which guarantees the presence of the coexistence equilibrium is achieved. Secondly, via utilizing the theory of integrated semigroup and the Hopf bifurcation theorem for semilinear equations with nondense domain, it is found that the model exhibits a Hopf bifurcation near the coexistence equilibrium, which suggests that this model has a nontrivial periodic solution that bifurcates from the coexistence equilibrium as the bifurcation parameter [Formula: see text] crosses the bifurcation critical value [Formula: see text]. That is, there is a continuous periodic oscillation phenomenon. Finally, some numerical simulations are shown to support and extend the analytical results and visualize the interesting phenomenon.

Hopf bifurcation of an infection-age structured eco-epidemiological model with saturation incidence

A novel infection-age structured eco-epidemiological model with saturation incidence is constructed. The model is regarded as an abstract non-densely defined Cauchy problem, and the condition of the existence of the interior equilibrium is derived. By employing the method of integrated semigroup and the Hopf bifurcation theory for semilinear equations with non-dense domain, we obtain that the model undergoes a Hopf bifurcation around the interior equilibrium which manifests that this model has a non-trivial periodic orbit that bifurcates from the interior equilibrium when bifurcation parameter τ crosses the bifurcation critical value τ0. In other words, the sustained periodic oscillation phenomenon appears. To support and extend our theoretically analytic results, numerical simulations are performed.

Hopf bifurcation of an age-structured prey–predator model with Holling type II functional response incorporating a prey refuge

In this paper, an age-structured prey–predator model with Holling type II functional response incorporating a prey refuge is constructed. Through applying the method of integrated semigroup and the Hopf bifurcation theory for semilinear equations with non-dense domain, we obtain that the model undergoes a Hopf bifurcation at the interior equilibrium which shows that this model has a non-trivial periodic orbit that bifurcates from the interior equilibrium when bifurcation parameter τ crosses the bifurcation critical value τ0. Numerical simulations are given to verify the theoretical analysis. The results manifest that the prey refuge has a stabilizing effect, namely, the prey refuge is a significant factor to maintain the balance between the prey population and the predator population.

Cross-diffusion induced Turing instability for a competition model with saturation effect

Competitive behavior, like predation, widely exists in the world which has influences on the dynamics of ecosystems. Though a good knowledge for the patten formation and selection of predator–prey models with diffusion is known in the literature, little is known for that of a diffused competition model. As a result, a competition model with saturation effect and self- and cross-diffusion is proposed and analyzed in this paper. Firstly, by making use of nullcline analysis, we derive the condition of existence and stability of positive equilibrium and prove global stability. Then using the linear stability analysis, the Turing bifurcation critical value and the condition of the occurrence of Turing pattern are obtained when control parameter is selected. Finally, we deduce the amplitude equations around the Turing bifurcation point by using the standard multiple scale analysis. Meanwhile, a series of numerical simulations are given to expand our theoretical analysis. This work shows that cross-diffusion plays a key role in the formation of spatial patterns for competitive model with saturation effect.

Chaos and Hopf bifurcation control in a fractional-order memristor-based chaotic system with time delay

In this paper, a time-delayed feedback controller is proposed in order to control chaos and Hopf bifurcation in a fractional-order memristor-based chaotic system with time delay. The associated characteristic equation is established by regarding the time delay as a bifurcation parameter. A set of conditions which ensure the existence of the Hopf bifurcation are gained by analyzing the corresponding characteristic equation. Then, we discuss the influence of feedback gain on the critical value of fractional order and time delay in the controlled system. Theoretical analysis shows that the controller is effective in delaying the Hopf bifurcation critical value via decreasing the feedback gain. Finally, some numerical simulations are presented to prove the validity of our theoretical analysis and confirm that the time-delayed feedback controller is valid in controlling chaos and Hopf bifurcation in the fractional-order memristor-based system.

Analytical study of global bifurcations, stabilization and chaos synchronization of jerk system with multiple attractors

The aim of this paper is to extend the recent analytical study of local bifurcations of a chaotic jerk model with multiple attractors to global bifurcations and examine the chaos synchronization problem for the case of multiple attractors and unknown system’s parameters. In particular, the different types of bifurcations of limit cycles exist in the model are explored analytically. The range of values in three-dimensional space of parameters, corresponding to each type of bifurcation, is found. A combination of time domain and frequency domain techniques, including multiple scales perturbation method and harmonic balance method, is employed in order to achieve this goal. More specifically, the study reveals that applying both multiple scales method and describing function method in a hybrid scheme enhances the accuracy of estimated critical bifurcation values as well as overcomes the problem of multiple scales method in capturing the correct values for bifurcation. Moreover, the chaos synchronization can be achieved in spite of the existence of multiple attractors. Finally, stabilization of fixed points and some periodic orbits of the system are studied using time-delayed feedback control scheme. Numerical simulations are presented so as to verify theoretical results.

Prey-taxis destabilizes homogeneous stationary state in spatial Gause–Kolmogorov-type model for predator–prey system

We consider a continuous taxis-diffusion-reaction system of partial-differential equations describing spatiotemporal dynamics of a predator–prey system. The local kinetics of the system is defined by general Gause–Kolmogorov-type model. The predator ability to pursue the prey is modelled by the Patlak–Keller–Segel taxis model, assuming that movement velocities of predators are proportional to the gradients of specific cues emitted by prey (e.g., odour, pheromones, exometabolites). The linear stability analysis of the model showed that the non-trivial homogeneous stationary regime of the model becomes unstable with respect to small heterogeneous perturbations with increase of prey-taxis activity; an Andronov–Hopf bifurcation occurs in the system when the taxis coefficient of predator exceeds its critical bifurcation value that exists for all admissible values of model parameters. These findings generalize earlier results obtained for particular cases of the Gause–Kolmogorov-type model assuming logistic reproduction of the prey population and the Holling types I and II functional responses of the predator population. Numerical simulations with theta-logistic growth of the prey population and the Ivlev functional response of predators illustrate and support results of the analytical study.

Analytical determination of bifurcations of periodic solution in three-degree-of-freedom vibro-impact systems with clearance

The smooth bifurcation and non-smooth grazing bifurcation of periodic solution of three-degree-of-freedom vibro-impact systems with clearance are studied in this paper. Firstly, six-dimensional Poincaré maps are established through choosing suitable Poincaré section and solving periodic solutions of vibro-impact system. Then, as the analytic expressions of all eigenvalues of Jacobi matrix of six-dimensional map are unavailable, the numerical calculations to search for the critical bifurcation values point by point is a laborious job based on the classical critical criterion described by the properties of eigenvalues. To overcome the difficulty from the classical bifurcation criteria, the explicit critical criterion without using eigenvalues calculation of high-dimensional map is applied to determine bifurcation points of Co-dimension-one bifurcations and Co-dimension-two bifurcations, and then local dynamical behaviors of these bifurcations are further analyzed. Finally, the existence of the grazing periodic solution of the vibro-impact system and grazing bifurcation point are analyzed, the discontinuous grazing bifurcation behavior is studied based on the compound normal form map near the grazing point, the discontinuous jumping phenomenon and the co-existing multiple solutions near the grazing bifurcation point are revealed.

Hopf bifurcation and chaos control in a new chaotic system via hybrid control strategy

In this paper, we investigate the problem of Hopf bifurcation and chaos control in a new chaotic system. A hybrid control strategy using both state feedback and parameter control is proposed. Theoretical analysis shows that the Hopf bifurcation critical value can be changed via hybrid control. Meanwhile, this control strategy can also control the chaos state. The direction and stability of bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. Finally, numerical simulations are carried out to illustrate the effectiveness of the main theoretical results.

Mixed-mode oscillations in a nonlinear time delay oscillator with time varying parameters

In this study, the mechanism for the action of time-invariant delay on a non-autonomous system with slow parametric excitation is investigated. The complex mix-mode oscillations (MMOs) are presented when the parametric excitation item slowly passes through critical bifurcation values of this nonlinear time delay oscillator. We use bifurcation theory to clarify certain generation mechanism related to three complex spiking formations, i.e., ``symmetric sup-pitchfork bifurcation'', ``symmetric sup-pitchfork/sup-Hopf bifurcation'', and ``symmetric sup-pitchfork/sup-Hopf/homoclinic orbit bifurcation''. Such bifurcation behaviors result in various hysteresis loops between the spiking attractor and the quasi-stationary process, which are responsible for the generation of MMOs. We further identify that the occurrence and evolution of such complex MMOs depend on the magnitude of the delay. Specifically, with the increase of time delay, the two limit cycles bifurcated from Hopf bifurcations may merge into an enlarged cycle, which is caused by a saddle homoclinic orbit bifurcation. We can conclude that time delay plays a vital role in the generation of MMOs. Our findings enrich the routes to spiking process and deepen the understanding of MMOs in time delay systems.

Proof of a Conjecture for the One-Dimensional Perturbed Gelfand Problem from Combustion Theory

We study global bifurcation curves and the exact multiplicity of positive solutions for the two-point boundary value problem arising in combustion theory: $$\left\{\begin{array}{l}u^{\prime \prime }(x)+\lambda \exp \left(\frac{au}{a+u}\right) =0,\quad -1<x<1, \\ u(-1)=u(1)=0, \end{array}\right.$$ where \({\lambda > 0}\) is the Frank–Kamenetskii parameter and a > 0 is the activation energy parameter. We prove that there exists a critical bifurcation value a0\({\approx 4.069}\) such that, on the \({(\lambda,||u||_{\infty })}\)-plane, the bifurcation curve is S-shaped for \({a > a_{0}}\) and is monotone increasing for \({0 < a \leqq a_{0}}\). That is, we prove the long-standing conjecture for the one-dimensional perturbed Gelfand problem. We also study, in the \({(a,\lambda, \left \Vert u\right\Vert _{\infty})}\)-space, the shape and structure of the bifurcation surface.