•Dynamics of non symmetric Kopel model were explored through fold, transcritical, pitchfork and Neimark–Sacker bifurcation.•The relations between different kinds of bifurcation are considered ...numerically when the parameters change.•Some schematic orbits are displayed to divide the domain into several kinds of regions with different dynamics.•2D bifurcation diagrams are provided to illustrate the periodic dynamics between agents and possible transitions.
In this paper, complex dynamics of Kopel model with nonsymmetric response between oligopolists are discussed. The research show that for the nonsymmetric model fixed point may undergo fold, transcritical, pitchfork and Neimark–Sacker bifurcation when satisfying some specific parameter combinations. The research also show us that effects from the rivals may incur more complex dynamics than self-adjust. During the process of numerical simulation, different combinations of parameter are chosen to show multiple-period phenomenon, the change of stability and the transition of different dynamical behaviors.
•Finance and economics are complex nonlinear systems, affected by various external factors.•We study the dynamics and complexity in a fractional-order financial system with time delays.•We observe ...fascinating transitions to deterministic chaos, including cascading period doubling.•Complexity is particularly high in response to variations of derivative orders.•Derivative orders are identified as key system parameters.
Finance and economics are complex nonlinear systems that are affected by various external factors, including of course human action, bilateral relations, conflicts, and policy. Time delays in a financial system take into account the amount of time that passes from a particular policy or decision being made to it actually taking effect. It is thus important to consider time delays as an integral part of modeling in this field. Moreover, many features of financial systems cannot be expressed sufficiently precisely by means of integer-order calculus. Fractional-order calculus alleviates these shortcomings. The aim of this paper is therefore to study the dynamics and complexity in a fractional-order financial system with time delays. We observe fascinating transitions to deterministic chaos, including cascading period doubling, as well as high levels of complexity. This is particularly true in response to variations of derivative orders, which are thus identified as key system parameters.
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.
•A new chaotic system with coexisting attractors is presented. A detailed investigation of the dynamic behaviors of the system is given.•The circuit realization of the new system is established. The ...coexisting attractors and chaotic attractor of the system are observed in oscilloscope.•The passive controller of the system is designed. It can suppress the chaos of the system and switch the states of the system between different attractors.•A chaotic image encryption algorithm is proposed according to the system. The performance of the algorithm is numerically analyzed.
This paper introduces an extended Lü system with coexisting attractors. The number and stability of equilibria are determined. The coexisting attractors of the system are displayed by the bifurcation diagrams, Lyapunov exponent spectrum, phase portraits. It is shown that the system has a pair of strange attractors, a pair of limit cycles, a pair of point attractors for different initial conditions. The circuit implementation of the chaotic attractor and coexisting attractors of the system are presented. The control problem of the system is studied as well. A controller is designed to stabilize the system to the origin and realize the switching between two chaotic attractors based on the passive control method. Moreover, a chaotic image encryption algorithm is proposed according to the system. The performance of the algorithm is numerically analyzed.
Four-dimensional chaotic systems are a very interesting topic for researchers, given their special features. This paper presents a novel fractional-order four-dimensional chaotic system with ...self-excited and hidden attractors, which includes only one constant term. The proposed system presents the phenomenon of multi-stability, which means that two or more different dynamics are generated from different initial conditions. It is one of few published works in the last five years belonging to the aforementioned category. Using Lyapunov exponents, the chaotic behavior of the dynamical system is characterized, and the sensitivity of the system to initial conditions is determined. Also, systematic studies of the hidden chaotic behavior in the proposed system are performed using phase portraits and bifurcation transition diagrams. Moreover, a design technique of a new fuzzy adaptive sliding mode control (FASMC) for synchronization of the fractional-order systems has been offered. This control technique combines an adaptive regulation scheme and a fuzzy logic controller with conventional sliding mode control for the synchronization of fractional-order systems. Applying Lyapunov stability theorem, the proposed control technique ensures that the master and slave chaotic systems are synchronized in the presence of dynamic uncertainties and external disturbances. The proposed control technique not only provides high performance in the presence of the dynamic uncertainties and external disturbances, but also avoids the phenomenon of chattering. Simulation results have been presented to illustrate the effectiveness of the presented control scheme.
•A novel fractional-order four-dimensional chaotic system with self-excited and hidden attractors is introduced.•We have characterized the chaos of the dynamical system and determined the sensitivity of the system.•Systematic studies of the hidden chaotic behavior in the proposed system are performed.•We have designed a fuzzy adaptive sliding mode control to synchronize the proposed system.
This article is basically concerned with the stability and Hopf bifurcation problem of fractional-order three-triangle multi-delayed neural networks. Based on laplace transform, we obtain the ...characteristic equation of the considered fractional-order three-triangle multi-delayed neural networks. By discussing the distribution of the roots for the characteristic equation, the delay-independent stability condition and delay-induced bifurcation criterion are built. The research manifests that time delay is an important factor which affects the stability and the onset of Hopf bifurcation for fractional-order three-triangle multi-delayed neural networks. The computer simulation results and bifurcation figures are displayed to support the established main conclusions. The derived fruits of this article have great theoretical values in dominating neural networks.
We study the global bifurcation diagram of the positive solutions to the problem{Δu+λf(u)=0inB,u=0on∂B, where B is the unit ball in RN with N≥3. Under general supercritical growth conditions on f(u), ...we show that an unbounded bifurcation curve has no turning point, which indicates the existence of the singular extremal solution. In particular, our theory can be applied to the super-exponential cases of f(u), and we exhibit that a bifurcation curve for Δu+λf(u)=0 has the same qualitative property as a classical Gel'fand problem Δu+λeu=0 for N≥3 except N=10. Main technical tools are intrinsic transformations for semilinear elliptic equations and ODE techniques.
We are interested in the global bifurcation diagram of radial solutions for the Gelfand problem with the exponential nonlinearity and a positive radially symmetric weight in the unit ball. When the ...weight is constant, it is known that the bifurcation curve has infinitely many turning points if the dimension 3≤N≤9, and it has no turning points if N≥10.
In this paper, we show that the perturbation of the weight does not affect the bifurcation structure when 3≤N≤9. Moreover, we find a one-parameter family of radial singular solutions for a parametrized weight and study the Morse index of the singular solution. As a result, we prove that the perturbation affects the bifurcation structure in the critical dimension N=10. Moreover, we give a classification of the bifurcation diagrams in the critical dimension.
In this paper we propose a new vibration isolator system, which combines the advantages of a quasi-zero stiffness (QZS) isolator, a damper exhibiting fractional properties and an inerter, to reduce ...the vibrations of a multi-span continuous beam bridge excited by moving loads. The inerter produces a fictitious mass amplification effect to improve the controller performance and the fractional order takes into account the previous state of the viscoelastic material. After formulating the dynamics equation using beam theory, the amplitude response is determined analytically using the averaging method. The results obtained from the analytical study are validated using the direct numerical simulation method (Newton–Leipnik method). By comparing the isolation performance of the FQZS (fractional quasi-zero stiffness) isolator and the IFQZS (inerter fractional quasi-zero stiffness) isolator, it is shown that the addition of inertance can significantly suppress the tendency of the curve to slope to the right, allowing us to have a wider isolation frequency range on force transmissibility while improving the efficiency of the isolator. One also shows that increasing the fractional parameter contributes to a decrease in the vibration amplitude of the structure, the amplitude of the force transmitted and the area in which unstable solutions appear. However, beyond a certain value of the fractional parameter, we observe an increase in the latter. In order to further extend the study, bifurcation diagram, phase portrait, time history and power spectral density are explored.
•Beam bridge with inerter fractional quasi-zero stiffness (IFQZS) isolator.•Dynamics response when subjected to moving load.•Isolator take into account memory effect and fictitious mass amplification.•Inertance of inerter improves the efficiency of the isolator.•For some parameter of the IFQZS influence consequently the response of the system.
•A new simple chaotic system with various types of multiple coexisting attractors is presented. A detailed investigation of the dynamic behaviors of the system is given.•The multiple coexisting ...attractors of the proposed system are investigated. It shows that the system will generate six limit cycles, two chaotic attractors with four limit cycles, one chaotic attractors with four limit cycles corresponding to different initial values.•The circuit implementation of the proposed system is presented. The designed circuit successfully realized the chaotic attractors and coexisting attractors of the system.
This letter proposes a new 4D autonomous chaotic system characterized by the abundant coexisting attractors and a simple mathematical description. The new system which is constructed from the Sprott B system is dissipative, symmetric, chaotic and has two unstable equilibria. For a given set of parameters, butterfly attractors are emerged from the system. These butterfly attractors will be broken into a pair of symmetric strange attractors with the variation of the parameters. A variety of coexisting attractors are spotted in the system including six periodic attractors, four periodic attractors with two chaotic attractors, two periodic attractors with three chaotic attractors, two periodic attractors with two chaotic attractors, four periodic attractors, etc. Finally, the system is established via an electronic circuit which can physically confirm the complex dynamics of the system.
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