Bifurcation,chaos, and their control in a wheelset model

In this paper, we present an improved wheelset motion model with two degrees of freedom and study the dynamic behaviors of the system including the symmetry, the existence and uniqueness of the solution, continuous dependence on initial conditions, and Hopf bifurcation. The dynamic characteristics of the wheelset motion system under a nonholonomic constraint are investigated. These results generalize and improve some known results about the wheelset motion system. Meanwhile, based on multiple equilibrium analysis, calculation of Lyapunov exponents and Poincaré section, the chaotic behaviors of the wheelset system are discussed, which indicates that there are more complex dynamic behaviors in the railway wheelset system with higher order terms of Taylor series of trigonometric functions. This paper has also realized the chaos control and bifurcation control for the wheelset motion system by adaptive feedback control method and linear feedback control. The results show that the chaotic wheelset system and bifurcation wheelset system are all well controlled, whether by controlling the yaw angle and the lateral displacement or only by controlling the yaw angle. Numerical simulations are carried out to further verify theoretical analyses.     

Neimark-Sacker bifurcation and chaos control in Hassell-Varley model

We investigate the complex behaviour of a modified Nicholson–Bailey model. The modification is proposed by Hassel and Varley taking into account that interaction between parasitoids is taken in such a way that the searching area per parasitoid is inversely proportional to the m-th power of the population density of parasitoids. Under certain parametric conditions the unique positive equilibrium point of system is locally asymptotically stable. Moreover, it is proved that system undergoes Neimark-Sacker bifurcation for small range of parameters by using standard mathematical techniques of bifurcation theory. In order to control Neimark-Sacker bifurcation, we apply simple feedback control strategy and pole-placement technique which is a modification of OGY method. Moreover, the hybrid control methodology is also implemented for chaos controlling. Numerical simulations are provided to illustrate theoretical discussion.     

Bifurcation analysis and chaos control in a host‐parasitoid model

We investigate the qualitative behavior of a host‐parasitoid model with a strong Allee effect on the host. More precisely, we discuss the boundedness, existence and uniqueness of positive equilibrium, local asymptotic stability of positive equilibrium and existence of Neimark–Sacker bifurcation for the given system by using bifurcation theory. In order to control Neimark–Sacker bifurcation, we apply pole‐placement technique that is a modification of OGY method. Moreover, the hybrid control methodology is implemented in order to control Neimark–Sacker bifurcation. Numerical simulations are provided to illustrate theoretical discussion. Copyright © 2017 John Wiley & Sons, Ltd.     

The influence of coupling on chaotic maps modelling bursting cells

Bursting behavior is ubiquitous in physical and biological systems, specially in neural cells where it plays an important role in information processing. This activity refers to a complex oscillation characterized by a slow alternation between spiking behavior and quiescence. In this paper, the interesting phenomena which transpire when two cells are coupled together, is studied in terms of symbolic dynamics. More specifically, we characterize the topological entropy of a map used to examine the role of coupling on identical bursters. The strength of coupling leads to the introduction of a second topological invariant that allows us to distinguish isentropic dynamics. We illustrate the significant effect of the strength parameter on the topological invariants with several numerical results.     

The chaos and control of a food chain model supplying additional food to top-predator

The control and management of chaotic population is one of the main objectives for constructing mathematical model in ecology today. In this paper, we apply a technique of controlling chaotic predator–prey population dynamics by supplying additional food to top-predator. We formulate a three species predator–prey model supplying additional food to top-predator. Existence conditions and local stability criteria of equilibrium points are determined analytically. Persistence conditions for the system are derived. Global stability conditions of interior equilibrium point is calculated. Theoretical results are verified through numerical simulations. Phase diagram is presented for various quality and quantity of additional food. One parameter bifurcation analysis is done with respect to quality and quantity of additional food separately keeping one of them fixed. Using MATCONT package, we derive the bifurcation scenarios when both the parameters quality and quantity of additional food vary together. We predict the existence of Hopf point (H), limit point (LP) and branch point (BP) in the model for suitable supply of additional food. We have computed the regions of different dynamical behaviour in the quantity–quality parametric plane. From our study we conclude that chaotic population dynamics of predator prey system can be controlled to obtain regular population dynamics only by supplying additional food to top predator. This study is aimed to introduce a new non-chemical chaos control mechanism in a predator–prey system with the applications in fishery management and biological conservation of prey predator species.     

Three-species food web model with impulsive control strategy and chaos

A three-species ecological model with impulsive control strategy is developed using the theory and methods of ecology and ordinary differential equation. Conditions for extinction of the system are given based on the theory of impulsive equation and small amplitude perturbation. Using comparison involving multiple Lyapunov functions, the system is shown to be permanent. Further, the influence of the impulsive perturbation on the inherent oscillation are studied numerically and is found to depict rich dynamics, such as the period-doubling bifurcation, the period-halving bifurcation, a chaotic band, a narrow or wide periodic window, and chaotic crises. In addition, the largest Lyapunov exponent is computed. This computation demonstrates the chaotic dynamic behavior of the model. The qualitative nature of concerned strange attractors is also investigated through their computed Fourier spectra. The foregoing results have the potential to be useful for the study of the dynamic complexity of ecosystems.     

The chaos and optimal control of cancer model with complete unknown parameters

In this paper, we study the chaos and optimal control of cancer model with completely unknown parameters. The stability analysis of the biologically feasible steady-states of this model will be discussed. It is proved that the system appears to exhibit periodic and quasi-periodic limit cycles and chaotic attractors for some ranges of the system parameters. The necessary optimal controllers input for the asymptotic stability of some positive equilibrium states are derived. Numerical analysis and extensive numerical examples of the uncontrolled and controlled systems were carried out for various parameters values and different initial densities.     

Bifurcations and chaos in a generic management model

This paper illustrates how period-doubling bifurcations and chaotic behaviour can be internally generated in a typical management system.A company is assumed to allocate resources to its production and marketing departments in accordance with shifts in inventory and/or backlog. When order backlogs are small, additional resources are provided to the marketing department in order to recruit new customers. At the same time, resoures are removed from the production line to prevent a build-up of excessive inventories. In the face of large order backlogs, on the other hand, the company redirects resources from sales to production. Delays in adjusting production and sales create the potential for oscillatory behaviour. If reallocation of resources is strong enough, this behaviour is destabilized, and the system starts to perform self-sustained oscillations.To complete the model, we have included a feedback which represents customer's reaction to varying delivery delays. As the loss of customers in response to high delivery delays is increased, the simple limit cycle oscillation becomes unstable, and through a cascade of period-doubling bifurcations the systems develops into a chaotic state. A relatively detailed analysis of this bifurcation sequence is presented. A Poincaré section and return map are constructed for the chaotic case, and the largest Lyapunov exponent is evaluated. Finally, a parameter plane analysis of the transition to chaos is presented.     

Antiphase synchronization of chaos by noncontinuous coupling: two impacting oscillators

We show that two identical chaotic oscillators can evolve in antiphase synchronization regime when noncontinuous coupling between them is introduced. As an example, we consider dynamics of two mechanical oscillators coupled by impacts.     

Stochastic chaos in a Duffing oscillator and its control

Stochastic chaos discussed here means a kind of chaotic responses in a Duffing oscillator with bounded random parameters under harmonic excitations. A system with random parameters is usually called a stochastic system. The modifier ‘stochastic’ here implies dependent on some random parameter. As the system itself is stochastic, so is the response, even under harmonic excitations alone. In this paper stochastic chaos and its control are verified by the top Lyapunov exponent of the system. A non-feedback control strategy is adopted here by adding an adjustable noisy phase to the harmonic excitation, so that the control can be realized by adjusting the noise level. It is found that by this control strategy stochastic chaos can be tamed down to the small neighborhood of a periodic trajectory or an equilibrium state. In the analysis the stochastic Duffing oscillator is first transformed into an equivalent deterministic nonlinear system by the Gegenbauer polynomial approximation, so that the problem of controlling stochastic chaos can be reduced into the problem of controlling deterministic chaos in the equivalent system. Then the top Lyapunov exponent of the equivalent system is obtained by Wolf’s method to examine the chaotic behavior of the response. Numerical simulations show that the random phase control strategy is an effective way to control stochastic chaos.     

Complex dynamics and chaos control of duopoly Bertrand model in Chinese air-conditioning market

A dynamic duopoly Bertrand model with quadratic cost function which is closer to reality and different from previous researches is discussed. The model is applied into air-conditioning market where the boundary equilibrium point is locally stable. Numerical simulations illustrate that the stability of Nash equilibrium strongly depends on the speed of adjustment of bounded rational player. The adjustment speeds and the degree of substitutability may undermine the stability of the equilibrium and cause a market structure to behave chaotically. The Lyapunov dimension of the chaos attractor is 1.9585 under some conditions. The stabilization of the chaotic behavior can be obtained by reducing the degree of substitutability. The results have an important theoretical and practical significance to Chinese air-conditioning market.     

Feedback control of bursting and multistability in chaotic systems

With a modulated CO₂ laser as a standard model of periodically driven multistable systems, we experimentally demonstrate that a small-amplitude optoelectronic feedback perturbation can efficiently transform a bursting chaotic system to a nonchaotic one. Numerical simulations are in excellent agreement with the experimental results. The control has also been equally effective in the case of a driven FitzHugh-Nagumo model of Neuroscience.     

Mathematics analysis and chaos in an ecological model with an impulsive control strategy

In this paper, on the basis of the theories and methods of ecology and ordinary differential equation, an ecological model with an impulsive control strategy is established. By using the theories of impulsive equation, small amplitude perturbation skills and comparison technique, we get the condition which guarantees the global asymptotical stability of the lowest-level prey and mid-level predator eradication periodic solution. It is proved that the system is permanent. Further, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which shows rich dynamics, such as period-doubling bifurcation, period-halving bifurcation, chaotic band, narrow or wide periodic window, chaotic crises,etc. Moreover, the computation of the largest Lyapunov exponent demonstrates the chaotic dynamic behavior of the model. At the same time, we investigate the qualitative nature of strange attractor by using Fourier spectra. All these results may be useful for study of the dynamic complexity of ecosystems.     

Robust control of chaos in Chua’s circuit based on internal model principle

The paper treats the question of robust control of chaos in Chua’s circuit based on the internal model principle. The Chua’s diode has polynomial non-linearity and it is assumed that the parameters of the circuit are not known. A robust control law for the asymptotic regulation of the output (node voltage) along constant and sinusoidal reference trajectories is derived. For the derivation of the control law, the non-linear regulator equations are solved to obtain a manifold in the state space on which the output error is zero and an internal model of the k-fold exosystem (k = 3 here) is constructed. Then a feedback control law using the optimal control theory or pole placement technique for the stabilization of the augmented system including the Chua’s circuit and the internal model is derived. In the closed-loop system, robust output node voltage trajectory tracking of sinusoidal and constant reference trajectories are accomplished and in the steady state, the remaining state variables converge to periodic and constant trajectories, respectively. Simulation results are presented which show that in the closed-loop system, asymptotic trajectory control, disturbance rejection and suppression of chaotic motion in spite of uncertainties in the system are accomplished.     

Dynamic behavior and chaos control in a complex Riccati-type map

This paper is devoted to analyze the dynamic behavior of a Riccatitype map with complex variables and complex parameters. Fixed points and their asymptotic stability are studied. Lyapunov exponent is computed to indicate chaos. Bifurcation and chaos are discussed. Chaotic behavior of the map has been controlled by OGY feedback control method.     

Dynamics and chaos control of gyrostat satellite

We consider the chaotic motion of the free gyrostat consisting of a platform with a triaxial inertia ellipsoid and a rotor with a small asymmetry with respect to the axis of rotation. Dimensionless equations of motion of the system with perturbations caused by small asymmetries of the rotor are written in Andoyer-Deprit variables. These perturbations lead to separatrix chaos. For gyrostats with different ratios of moments of inertia heteroclinic and homoclinic trajectories are written in closed-form. These trajectories are used for constructing modified Melnikov function, which is used for determine the control that eliminates separatrix chaos. Melnikov function and phase space trajectory are built to show the effectiveness of the control.     

Melnikov analysis of chaos in a general epidemiological model

The purpose of this paper is to study a SIR model of epidemic dynamics with a periodically modulated nonlinear incidence rate. We must go, for the first time, through a series of coordinate transformations to bring the equations into amenable to Melnikov analysis. This analysis establishes mathematically the existence of chaotic motion of the models by Melnikov's method. The numerical simulations are made for the conclusions in this paper.     

Nonadiabatic couplings and chaos in a dynamical potential well model

The mechanism of nonadiabatic couplings between quantum states of a potential well model with finite heights and a dynamical width coordinate is investigated in detail. The system is described in a mixed quantum-classical approach in which the oscillations of the classical width coordinate induce transitions between the quantum states of a particle trapped inside the well. The dynamics of the system is considered in detail for transitions between two quantum states and resulting coupled Bloch-oscillator equations. Poincaré sections showing a mixed phase space with chaotic and regular behaviour are found by a numerical investigation. In particular, chaos results for high energies of the well width oscillations when the mixing between the adiabatic reference states is strong. The inclusion of relaxation is considered and shown that in this case the regimes of chaotic and regular dynamics are not separated as in the relaxation free case. In particular, for some initial conditions chaos can become a transient phenomena placed in a time window between regular oscillations of the system.     

Dynamical analysis and chaos control in a heterogeneous Kopel duopoly game

A dynamic of a nonlinear Kopel duopoly game with heterogeneous players is presented. By assuming two heterogeneous players where one player use naive expectation whereas the other employs a technique of adaptive. The stability conditions of equilibrium points are analyzed. Numerical simulations are used to show bifurcation diagrams, phase portraits and sensitive dependence on initial conditions. The chaotic behavior of the game has been controlled by using feedback control method.