Bifurcations Of Dynamical Systems Research Articles

Complex Dynamics for Airlines' Price Competition with Differentiation Strategy

To investigate the complex dynamics for airlines' price competition, this paper proposes a price competition model of airlines with differentiation strategy and with the theory of bifurcation of dynamical systems. The existence and stability of equilibrium points of the model are discussed according to dynamic stability criteria. The complex dynamics of this model in different market parameters are shown though numerical simulation. The simulation results show that the speed of price adjustment has a significant impact on the stability of the model, while the speed of price adjustment is larger than critical value, and the phenomenon of bifurcation and chaos will appear on the dynamic system. Compared with the Nash equilibrium profits, all airlines' profits are decreased obviously when chaos is occurred. Differentiation has an important impact on airlines' price and profits, it's helpful to have more competition advantage that to keep and strengthen the differentiation advantage against the competitors.

Refined inertias of tree sign-patterns

The refined inertia (n+,n ,nz,2np) of a real matrix is the ordered 4-tuple that subdivides the number n0 of eigenvalues with zero real part in the inertia (n+,n ,n0) into those that are exactly zero (nz) and those that are nonzero (2np). For n � 2, the set of refined inertias Hn = {(0,n,0,0),(0,n 2,0,2),(2,n 2,0,0)} is important for the onset of Hopf bifurcation in dynamical systems. Tree sign patterns of order n that require or allow the refined inertias Hn are considered. For n = 4, necessary and sufficient conditions are proved for a tree s pattern (necessarily a path or a star) to require H4. For n � 3, a family of n × n star sign patterns that allows Hn is given, and it is proved that if a star sign pattern requires Hn, then it must have exactly one zero diagonal entry associated with a leaf in its digraph.

Chaotic Dynamics and Bifurcations in Impact Systems

Bifurcations of dynamical systems described byseveral second order differential equations and by an impact condition are studied. It is shown that the variation of parameters when the number of impacts of a periodic solution increases, leads to the occurrence of a hyperbolic chaotic invariant set.

Chaotic dynamics and chaos control in differentiated Bertrand model with heterogeneous players

PurposeThe purpose of this paper is to investigate the chaotic dynamics behaviours and chaos control of differentiated Bertrand model.Design/methodology/approachThe paper analyzes a nonlinear differentiated Bertrand duopoly game by using the theory of bifurcations of dynamical system, where players have heterogeneous expectations and nonlinear cost function: two types of players are considered – bounded rational and naive expectation. The equilibrium point and local stability of the duopoly game are investigated.FindingsThe paper demonstrates that as some parameters of the game are varied, the stability of Nash equilibrium is lost through period doubling bifurcation. The chaotic features are justified numerically via computing Lyapunov exponents, and sensitive dependence on initial conditions. By using the delay feedback control method, the authors have made the system form chaotic state to stable state.Originality/valueThis paper fulfils an identified need to study the diversity of expectations and how to lead to rich dynamics and complexity.

Sign patterns that require or allow particular refined inertias

The refined inertia ri(A) of a real n×n matrix A is the ordered 4-tuple (n+,n-,nz,2np) where n+ (resp. n-,nz,2np) is the number of positive (resp. negative, zero, nonzero pure imaginary) eigenvalues of A. Let Hn={(0,n,0,0),(0,n-2,0,2),(2,n-2,0,0)}. An n×n sign pattern Sn requires Hn if Hn={ri(A)|A has sign pattern Sn} and allows Hn if Hn∈{ri(A)|A has sign pattern Sn}. Sign patterns that require or allow Hn are investigated for small values of n and for patterns with all diagonal entries negative. Examples are given relating these concepts to Hopf bifurcation in dynamical systems.

Transition sets of bifurcations of dynamical systems with two state variables with constraints

Bifurcation of periodic solutions widely exists in nonlinear dynamical systems. In this paper, categories of bifurcations of systems with two state variables with different types of constraints are discussed, where some new types of transition sets are added. Additionally, the bifurcation properties of two-dimensional systems without constraints are compared with the ones with constraints. The results obtained in this paper can be used by engineers for the choice of the structural parameters of the systems.

Complexity of a Duopoly Game in the Electricity Market with Delayed Bounded Rationality

According to a triopoly game model in the electricity market with bounded rational players, a new Cournot duopoly game model with delayed bounded rationality is established. The model is closer to the reality of the electricity market and worth spreading in oligopoly. By using the theory of bifurcations of dynamical systems, local stable region of Nash equilibrium point is obtained. Its complex dynamics is demonstrated by means of the largest Lyapunov exponent, bifurcation diagrams, phase portraits, and fractal dimensions. Since the output adjustment speed parameters are varied, the stability of Nash equilibrium gives rise to complex dynamics such as cycles of higher order and chaos. Furthermore, by using the straight‐line stabilization method, the chaos can be eliminated. This paper has an important theoretical and practical significance to the electricity market under the background of developing new energy.

Bifurcation Behaviour of the Travelling Wave Solutions of the Perturbed Nonlinear Schrödinger Equation with Kerr Law Nonlinearity

In this paper, we study the bifurcations and dynamic behaviour of the travelling wave solutions of the perturbed nonlinear Schrödinger equation (NLSE) with Kerr law nonlinearity by using the theory of bifurcations of dynamic systems. Under the given parametric conditions, all possible representations of explicit exact solitary wave solutions and periodic wave solutions are obtained

New applications of the renormalization group method in physics: a brief introduction

The renormalization group (RG) method developed by Ken Wilson more than four decades ago has revolutionized the way we think about problems involving a broad range of energy scales such as phase transitions, turbulence, continuum limits and bifurcations in dynamical systems. The Theme Issue provides articles reviewing recent progress made using the RG method in atomic, condensed matter, nuclear and particle physics. In the following, we introduce these articles in a way that emphasizes common themes and the universal aspects of the method.

Complexity of triopoly price game in Chinese cold rolled steel market

This paper considers a Bertrand model based on nonlinear demand functions which are closer to reality and different from previous studies. We apply the model into Chinese cold rolled steel market and study game process of triopoly. By using the theory of bifurcations of dynamical systems, local stable region of Nash equilibrium point is obtained. Simulations show complex dynamical behaviors of the system. The results illustrate that altering the relevant parameters of system can affect the stability of Nash equilibrium point and cause chaos to occur, and the complex dynamical behaviors will disappear by parameters control method. The results have an important theoretical and practical significance to Chinese cold rolled steel market.

Bifurcation of traveling wave solutions for the generalized compound KDV equations

By using the theory of bifurcations of dynamical systems to the generalized compound KDV equations, the existence of solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.

Bifurcations in dynamical systems with interior symmetry

We propose a definition of interior symmetry in the context of general dynamical systems. This concept appeared originally in the theory of coupled cell networks, as a generalization of the idea of symmetry of a network. The notion of interior symmetry introduced here can be seen as a special form of forced symmetry breaking of an equivariant system of differential equations. Indeed, we show that a dynamical system with interior symmetry can be written as the sum of an equivariant system and a ‘perturbation term’ which completely breaks the symmetry. Nonetheless, the resulting dynamical system still retains an important feature common to systems with symmetry, namely, the existence of flow-invariant subspaces. We define interior symmetry breaking bifurcations in analogy with the definition of symmetry breaking bifurcation from equivariant bifurcation theory and study the codimension one steady-state and Hopf bifurcations. Our main result is the full analogues of the well-known Equivariant Branching Lemma and the Equivariant Hopf Theorem from the bifurcation theory of equivariant dynamical systems in the context of interior symmetry breaking bifurcations.

A method for detecting false bifurcations in dynamical systems: application to neural-field models

In this article, we present a method for tracking changes in curvature of limit cycle solutions that arise due to inflection points. In keeping with previous literature, we term these changes false bifurcations, as they appear to be bifurcations when considering a Poincaré section that is tangent to the solution, but in actual fact the deformation of the solution occurs smoothly as a parameter is varied. These types of solutions arise commonly in electroencephalogram models of absence seizures and correspond to the formation of spikes in these models. Tracking these transitions in parameter space allows regions to be defined corresponding to different types of spike and wave dynamics, that may be of use in clinical neuroscience as a means to classify different subtypes of the more general syndrome.

The study of dynamic process of the triopoly games in chinese 3G telecommunication market

At first, the paper described a brief history of communication in China. In 2008, the third generation (3G) telecommunication technologies began to develop. In Chinese 3G telecommunication market, there are three oligopolistic competitors. In order to study competing process of the triopoly, the paper considered a Bertrand model with bounded rational. A triopoly game is modeled by three nonlinear difference equations. By using the theory of bifurcations of dynamical systems, the existence and stability for the equilibria of this system were obtained. Numerical simulations were used to show bifurcations diagrams, the maximum Lyapunov exponent and sensitive dependence on initial conditions. We observed that increasing the speed of adjustment of bounded rational player might change the stability of Nash equilibrium point and cause bifurcation and chaos to occur. Thus, it will be difficult to make strategy and to forecast for each enterprise. It might cause the telecom markets to be irregular. The analysis and results in this paper are important for Chinese telecommunication markets, the same to mathematics.

On the dynamical behaviour of FitzHugh–Nagumo systems: Revisited

The purpose of this paper is to analyse a general form of the FitzHugh–Nagumo model as completely as possible. The main result is that no more than two limit cycles can be bifurcated from the unique fixed point via Hopf bifurcation, and there exist parameters such that this upper bound is attained. For these parameters, the stability of the inner and outer cycle, together with the unique fixed point is also established. The results are approached through Lyapunov coefficients and rely on a theorem by Andronov and Aleksandrovic [A.A. Andronov, A.A. Aleksandrovic, Theory of Bifurcations of Dynamical System on a Plane, Wiley, 1971]. Based on singular perturbation theory a sufficient condition for existence of a unique stable limit cycle is given under certain assumptions.

Bifurcations of travelling wave solutions for the combined [formula omitted]–[formula omitted] equation

By using the theory of bifurcations of dynamical systems to the combined kdv–mkdv equation, the existence of solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.

From networks of unstable attractors to heteroclinic switching

We present a dynamical system that naturally exhibits two unstable attractors that are completely enclosed by each other's basin volume. This counterintuitive phenomenon occurs in networks of pulse-coupled oscillators with delayed interactions. We analytically show that upon continuously removing a local noninvertibility of the system, the two unstable attractors become a set of two nonattracting saddle states that are heteroclinically connected. This transition equally occurs from larger networks of unstable attractors to heteroclinic structures and constitutes a new type of singular bifurcation in dynamical systems.

A Detection Algorithm for Bifurcations in Dynamical Systems using Reduced Order Models

Finite element or finite volume discretizations of distributed parameter systems (DPS) typically lead to high order finite dimensional systems. Model approximation is then an important first step towards the construction of optimal controllers. However, model reduction methods hardly take model uncertainties and parameter variations into account. As such, reduced order models are not well equipped when uncertain system parameters vary in time. This is particularly true when system behavior does not depend continuously on the parameters. It is shown in this paper that the performance of reduced order models inferred from Galerkin projections and proper orthogonal decompositions can deteriorate considerable when system parameters vary over bifurcation points. Motivated by these observations, we propose a detection mechanism based on reduced order models and proper orthogonal decompositions that allows to characterize the influence of parameter variations around a bifurcation value. for this, a hybrid model structure is proposed. The ideas are applied on the example of a tubular reactor. In particular, this paper discusses the difficulties in approximating the transition from extinction to ignited state in a tubular reactor.

Grazing bifurcation and chaotic oscillations of vibro-impact systems with one degree of freedom

The bifurcations of dynamical systems, described by a second-order differential equation with periodic coefficients and an impact condition, are investigated. It is shown that a continuous change in the coefficients of the system, during which the number of impacts of the periodic solution increases, leads to the occurrence of a chaotic invariant set.

Introducing the Analysis of Bifurcation in Dynamical Systems by Symbolic Computation

The aim of this paper is to introduce a few topics about nonlinear systems that are usual in electrical engineering but are frequently disregarded in undergraduate courses. More precisely, the main subject of this paper is to present the analysis of bifurcations in dynamical systems through the use of symbolic computation. The necessary conditions for the occurrence of Hopf, saddle-node, transcritical or pitchfork bifurcations in first order state space nonlinear equations depending upon a vector of parameters are expressed in terms of symbolic computation. With symbolic computation, the relationship between the state variables and the parameters that play a crucial role in the occurrence of such phenomena can be established. Firstly, the symbolic computation is applied to a third order dynamic Lorenz system in order to familiarise the students with the technique. Then, the symbolic routines are used in the analysis of the simplified model of a power system, bringing new insights and a deeper understanding about the occurrence of these phenomena in physical systems.