Bifurcation Result Research Articles

Stability and Hopf Bifurcation of a Stochastic Cournot Duopoly Game in a Blockchain Cloud Services Market Driven by Brownian Motion

The blockchain technology has been rapidly popularizing and gaining a quick mileage in many domains owing to its innate nature, so there must be several tremendous market opportunities and uncertainties for the blockchain cloud service. We propose a Cournot duopoly game in a blockchain cloud services market for cyber-physical-social systems (CPSS). Introducing Brownian noise disturbance to the deterministic Cournot duopoly game, we construct a stochastic Cournot duopoly game in a blockchain cloud service market with uncertainty. We analyze the stochastic stability of the Cournot duopoly game by the Lyapunov exponent method, the Lyapunov function, and the singular boundary theory. Then we investigate the stochastic Hopf bifurcation of the Cournot duopoly game by the invariant measure theory. At last, we numerically validate these results by the probability density. These results of stability and Hopf bifurcation illustrate complex behaviors of the stochastic Cournot duopoly game in the blockchain cloud services market.

A flow force reformulation for steady irrotational water waves

The classical irrotational water wave problem described by Euler equations with a nonlinear free surface boundary condition and influenced by gravity over a flat bed is considered. Exploiting the monotonicity property of the flow force lines with depth, the unknown boundary problem is transformed into a problem with fixed domain. A flow force function formulation of the irrotational water wave problem is developed. Variational approach is used to prove the existence of small amplitude irrotational traveling wave solutions. Local bifurcation results have been proved relying on Crandall-Rabinowitz bifurcation theorem.

Dirichlet problem for a diffusive logistic population model with two delays

In this paper, we investigate a diffusive logistic equation with non-zero Dirichlet boundary condition and two delays. We first exclude the existence of positive heterogeneous steady states, which implies the uniqueness of constant positive steady state. Then, we analyze the local stability and local Hopf bifurcation at the positive steady state. We show that multiple delays can induce multiple stability switches. Furthermore, we prove global stability of the positive steady state under certain conditions and obtain global Hopf bifurcation results. Our theoretical results are illustrated with numerical simulations.

CONCEPTUAL FRAMEWORK OF THE FOREIGN POLICY STRATEGY OF THE KINGDOM OF NORWAY

The article analyzes the conceptual foundations of the foreign policy strategy of theKingdomofNorway. It is noted that the influence of the state and the ability to achieve its goals in the foreign policy arena are largely associated with the country's strategic priorities. The approaches to the definition of the concept of “strategy” are described, which describe the behavioral motives for choosing strategic objectives and priorities. When choosing a strategy, it is necessary to assess the value of available resources and choose the most important goal, the achievement of which will be directed to further efforts, taking into account their usefulness for the country. The article explores the evolution of the Norwegian foreign policy discourse. It is noted that Norway’s foreign policy discourse as a sovereign state is based on a combination of two areas - Norway’s vulnerable position in the international arena as a small country and the need for political elites to reach consensus on foreign policy. The development of foreign policy ofNorwaytook place within the framework of three paradigms - neutralism, moralism and internationalism, which were formed due to the specifics of the identity ofNorwayas a small country. Scientists note that the coexistence of the three paradigms in the foreign policy ofNorwaywas the result of the so-called bifurcation between the security policy and diplomacy of values. It is noted that the main interests of Norway since the 90s began to concentrate in the field of transnational economic activity, promoting the idea of global governance; Norway's new foreign policy priority has become an “engagement policy,” which was based on foreign policy efforts to overcome development problems, provide humanitarian assistance, maintain peace, and mitigate the effects of climate change. It is concluded thatNorwayimplements a foreign policy strategy based on the principle of “coherence,” which implies the relationship betweenNorway’s security policy and world political stability in general.

Dissipative periodic and chaotic patterns to the KdV–Burgers and Gardner equations

We investigate the KdV-Burgers and Gardner equations with dissipation and external perturbation terms by the approach of dynamical systems and Shil'nikov's analysis. The stability of the equilibrium point is considered, and Hopf bifurcations are investigated after a certain scaling that reduces the parameter space of a three-mode dynamical system which now depends only on two parameters. The Hopf curve divides the two-dimensional space into two regions. On the left region the equilibrium point is stable leading to dissapative periodic orbits. While changing the bifurcation parameter given by the velocity of the traveling waves, the equilibrium point becomes unstable and a unique stable limit cycle bifurcates from the origin. This limit cycle is the result of a supercritical Hopf bifurcation which is proved using the Lyapunov coefficient together with the Routh-Hurwitz criterion. On the right side of the Hopf curve, in the case of the KdV-Burgers, we find homoclinic chaos by using Shil'nikov's theorem which requires the construction of a homoclinic orbit, while for the Gardner equation the supercritical Hopf bifurcation leads only to a stable periodic orbit.

Unilateral Bifurcation for Several-Parameter Eigenvalue Problem with Homogeneous Operator

We establish the unilateral global bifurcation result for the following nonlinear operator equation $$u = L\left( \lambda \right)u + H\left( {\lambda,u} \right),\,\left( {\lambda,u} \right) \in \mathbb{R}{^m} \times X$$ where m is a positive integer, X is a Banach space, L(·) is a positively homogeneous completely continuous operator and H: ℝm·X → X is completely continuous with H = o (||u||) near u = 0 uniformly on bounded λ sets.

Local and global Hopf bifurcation in a neutral population model with age structure

In this paper, an age‐structured population model with the form of neutral functional differential equation is studied. We discuss the stability of the positive equilibrium by analyzing the characteristic equation. Local Hopf bifurcation results are also obtained by choosing the mature delay as bifurcation parameter. On the center manifold, the normal form of the Hopf bifurcation is derived, and explicit formulae for determining the criticality of bifurcation are theoretically given. Moreover, the global continuation of Hopf bifurcating periodic solutions is investigated by using the global Hopf bifurcation theory of neutral equations. Finally, some numerical examples are carried out to support the main results.

Global Components of Positive Bounded Variation Solutions of a One-Dimensional Indefinite Quasilinear Neumann Problem

AbstractThis paper investigates the topological structure of the set of the positive solutions of the one-dimensional quasilinear indefinite Neumann problem{-(u′1+u′2)′=λ⁢a⁢(x)⁢f⁢(u) in ⁢(0,1),u′⁢(0)=0,u′⁢(1)=0,\begin{dcases}-\Bigg{(}\frac{u^{\prime}}{\sqrt{1+{u^{\prime}}^{2}}}\Bigg{)}^{% \prime}=\lambda a(x)f(u)\quad\text{in }(0,1),\\ u^{\prime}(0)=0,\quad u^{\prime}(1)=0,\end{dcases}whereλ∈ℝ{\lambda\in\mathbb{R}}is a parameter,a∈L∞⁢(0,1){a\in L^{\infty}(0,1)}changes sign, andf∈C1⁢(ℝ){f\in C^{1}(\mathbb{R})}is positive in(0,+∞){(0,+\infty)}. The attention is focused on the casef⁢(0)=0{f(0)=0}andf′⁢(0)=1{f^{\prime}(0)=1}, where we can prove, likely for the first time in the literature, a bifurcation result for this problem in the space of bounded variation functions. Namely, the existence of global connected components of the set of the positive solutions, emanating from the line of the trivial solutions at the two principal eigenvalues of the linearized problem around 0, is established. The solutions in these components are regular, as long as they are small, while they may develop jump singularities at the nodes of the weight functiona, as they become larger, thus showing the possible coexistence along the same component of regular and singular solutions.

Novel bifurcation results for a delayed fractional-order quaternion-valued neural network

This paper reports the innovative results on the stability and bifurcation for a delayed fractional-order quaternion-valued neural network(FOQVNN). Delay-stimulated bifurcation criteria of the developed FOQVNN are attained. Then, the bifurcation diagrams are perfectly exhibited to authenticate the veracity of the bifurcation results. Besides, the stability zone is more larger of the addressed FOQVNN in comparison with its counterpart if other parameters are intercalated. It further witnesses that the amplitudes of bifurcation oscillation get bigger with the augmentation of time delay. It discloses that the bifurcation phenomena engender earlier as the order incrementally magnifies. The exactness and merits of the achieved analytic results are eventually substantiated by a simulation example.

Spatiotemporal dynamics in a ratio-dependent predator-prey model with time delay near the Turing–Hopf bifurcation point

Spatiotemporal dynamics of a ratio-dependent predator-prey model with time delay and the Neumann boundary condition are considered. The existence of the codimension-two Turing–Hopf bifurcation point is proved, so that both the Turing and Hopf bifurcations will occur simultaneously. Then, through the global Hopf bifurcation results, the global Hopf bifurcation of the ratio-dependent predator-prey model is investigated when diffusion is absent. Further, to identify the spatiotemporal dynamics during the Turing–Hopf bifurcation, the method of the multiple time scale analysis is employed to derive the amplitude equations near the codimension-two bifurcation point, instead of employing the center manifold theory and the normal form reduction. By analyzing the amplitude equations, it is found that the original ratio-dependent predator-prey model may exhibit the spatial, temporal or the spatiotemporal patterns, such as the nonconstant steady state, spatially homogeneous periodic solutions as well as the spatially inhomogeneous periodic solutions. Numerical simulations are carried out to validate the theoretical results.

Enhanced bifurcation results for a delayed fractional neural network with heterogeneous orders

This paper addresses the stability and bifurcation of a delayed fractional neural network(FNN) with different orders. Firstly, system parameter acts as a bifurcation parameter, and bifurcation criterion are educed for such system. It discovers that the stability performance of the proposed system can be exalted by selecting modest system parameter. Secondly, the bifurcation diagrams are fully illustrated for checking the exactness of the procured bifurcation results. Thirdly, in terms of discreet calculation, it perceives that the onset of bifurcation can be advanced by single order or time delay if as they decrease. Finally, two numerical examples are exploited to validate the efficiency of the theoretical results.

Synchronization in a Kuramoto model with delay-dependent couplings

This paper aims to provide bifurcation analysis for a Kuramoto model with time-delay and random coupling strength. A delay differential equation governing the system is obtained on the Ott-Antonsen’s manifold, and the bifurcation analysis is proceeded by using the characteristic equation and the normal form method. The general case where the coupling strength is chosen as a function of delay is investigated. Afterwards, the synchronization of the model with three different distributions of time delay including degenerate distribution, two-point distribution and Gamma distribution, is discussed respectively. Particularly, the coupled system of which the coupling strength and the delays are divided into two groups is studied in detail and the bifurcation results are obtained both theoretically and numerically.

Unilateral global bifurcation for a class of quasilinear elliptic systems and applications

In this paper we establish a unilateral bifurcation result for a class of quasilinear elliptic system strongly coupled, extending the bifurcation theorem of [1]. To this aim, we use several results, such that Fredholm operator of index 0 theory, eigenvalues of elliptic operators and the Krein–Rutman theorem. Lastly, we apply this result to some particular systems arising from population dynamics and determine a region of existence of coexistence states.

Global Hopf bifurcation of a delayed phytoplankton-zooplankton system considering toxin producing effect and delay dependent coefficient.

In this paper, a delayed phytoplankton-zooplankton system with the coeffcient depending on delay is investigated. Firstly, it gives the nonnegative and boundedness of solutions of the delay differential equations. Secondly, it gives the asymptotical stability properties of equilibria in the absence of time delay. Then in the presence of time delay, the existence of local Hopf bifurcation is discussed when the delay changes. In addition to that, the stability of periodic solution and bifurcation direction are also obtained through the use of central manifold theory. Furthermore, he global continuity of the local Hopf bifurcation is discussed by using the global Hopf bifurcation result of FDE. At last, some numerical simulations are presented to show the rationality of theoretical analyses.

Pattern Formation and Oscillatory Dynamics in a Two-Dimensional Coupled Bulk-Surface Reaction-Diffusion System

On a two-dimensional circular domain, we analyze the formation of spatio-temporal patterns for a class of coupled bulk-surface reaction-diffusion models for which a passive diffusion process occurring in the interior bulk domain is linearly coupled to a nonlinear reaction-diffusion process on the domain boundary. For this coupled PDE system we construct a radially symmetric steady state solution and from a linearized stability analysis formulate criteria for which this base state can undergo either a Hopf bifurcation, a symmetry-breaking pitchfork (or Turing) bifurcation, or a codimension-two pitchfork-Hopf bifurcation. For each of these three types of bifurcations, a multiple time-scale asymptotic analysis is used to derive normal form amplitude equations characterizing the local branching behavior of spatio-temporal patterns in the weakly nonlinear regime. Among the novel aspects of this weakly nonlinear analysis are the two-dimensionality of the bulk domain, the systematic treatment of arbitrary reaction kinetics restricted to the boundary, the bifurcation parameters which arise in the boundary conditions, and the underlying spectral problem where both the differential operator and the boundary conditions involve the eigenvalue parameter. The normal form theory is illustrated for both Schnakenberg and Brusselator reaction kinetics, and the weakly nonlinear results are favorably compared with numerical bifurcation results and results from time-dependent PDE simulations of the coupled bulk-surface system. Overall, the results show the existence of either subcritical or supercritical Hopf and symmetry-breaking bifurcations, and mixed-mode oscillations characteristic of codimension-two bifurcations. Finally, the formation of global structures such as large amplitude rotating waves is briefly explored through PDE numerical simulations.

Dynamical Behaviour of a Nutrient-Plankton Model with Holling Type IV, Delay, and Harvesting

In this paper, a Holling type IV nutrient-plankton model with time delay and linear plankton harvesting is investigated. The existence and local stability of all equilibria of model without time delay are given. Regarding time delay as bifurcation parameter, such system around the interior equilibrium loses its local stability, and Hopf bifurcation occurs when time delay crosses its critical value. In addition, the properties of the bifurcating periodic solutions are investigated based on normal form theory and center manifold theorem. What is more, the global continuation of the local Hopf bifurcation is discussed by using a global Hopf bifurcation result. Furthermore, the optimal harvesting is obtained by the Pontryagin’s Maximum Principle. Finally, some numerical simulations are given to confirm our theoretical analysis.

Bifurcation of heteroclinic orbits via an index theory

Heteroclinic orbits for one-parameter families of nonautonomous vectorfields appear in a very natural way in many physical applications. Inspired by a recent bifurcation result for homoclinic trajectories of nonautonomous vectorfield proved by author in [13], we define a new $$\mathbf {Z}_2$$ -index and we construct a index theory for heteroclinic orbits of nonautonomous vectorfield. We prove an index theorem, by showing that, under some standard transversality assumptions, the $$\mathbf {Z}_2$$ -index is equal to the parity, a homotopy invariant for paths of Fredholm operators of index 0. As a direct consequence of the index theory developed in this paper, we get a new bifurcation result for heteroclinic orbits.

Bifurcation of displacement problem in nonlinear elastostatics

We study the following displacement problem in nonlinear elastostatics −I+∇u∇2u=λf(u)inΩ,cu=0on∂Ω.We obtain some global and local bifurcation results, which are related to an open problem.

Global structure of one-sign solutions for problem with mean curvature operator**Research supported by NNSF of China (No. 11871129), the Fundamental Research Funds for the Central Universities (No. DUT17LK05) and Xinghai Youqing funds from Dalian University of Technology.

We establish a unilateral global bifurcation result for the following problemwhere Ω is a bounded domain in . Based on this global bifurcation result, we also studied the global structure of one-sign solutions according to different asymptotic behaviors (sublinear/linear/superlinear/jumping) nonlinearity near zero.

A global bifurcation theorem for critical values in Banach spaces

We present a global bifurcation result for critical values of $$C^1$$ maps in Banach spaces. The approach is topological based on homotopy equivalence of pairs of topological spaces. For $$C^2$$ maps, we prove a particular global bifurcation result, based on the notion of spectral flow.