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.     

Bifurcation and chaos in an epidemic model with nonlinear incidence rates

This paper investigates a discrete-time epidemic model by qualitative analysis and numerical simulation. It is verified that there are phenomena of the transcritical bifurcation, flip bifurcation, Hopf bifurcation types and chaos. Also the largest Lyapunov exponents are numerically computed to confirm further the complexity of these dynamic behaviors. The obtained results show that discrete epidemic model can have rich dynamical behavior.     

Bifurcation and chaos in discrete-time predator–prey system

The discrete-time predator–prey system obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory. And numerical simulation results not only show the consistence with the theoretical analysis but also display the new and interesting dynamical behaviors, including period-3, 5, 6, 7, 8, 9, 10, 12, 18, 20, 22, 30, 39-orbits in different chaotic regions, attracting invariant circle, period-doubling bifurcation from period-10 leading to chaos, inverse period-doubling bifurcation from period-5 leading to chaos, interior crisis and boundary crisis, intermittency mechanic, onset of chaos suddenly and sudden disappearance of the chaotic dynamics, attracting chaotic set, and non-attracting chaotic set. In particular, we observe that when the prey is in chaotic dynamic, the predator can tend to extinction or to a stable equilibrium. The computations of Lyapunov exponents confirm the dynamical behaviors. The analysis and results in this paper are interesting in mathematics and biology.     

Bifurcation and chaos in a discrete genetic toggle switch system

A discrete genetic toggle switch system obtained by Euler method is first investigated. The conditions of existence for fold bifurcation and flip bifurcation are derived by using center manifold theorem and bifurcation theory. The numerical simulations, including bifurcation diagrams, phase portraits, and computation of Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behavior. We show the period 3 to 13 windows in different chaotic regions, period-doubling bifurcation or inverse period-doubling bifurcation from period-2 to 12 orbits leading to chaos, different kind of interior crisis and boundary crisis, intermittency behavior, chaotic set, chaotic non-attracting set, coexistence of period points with invariant cycles, and so on. The influence of the amplitude and frequency of excitable forcing on the system are also first considered by using numerical simulation. A different type of quasiperiodic orbits, jumping behaviors of quasiperiodic set from one set to another set, and the processes from quasiperiodic orbits to strange non-chaotic attractor are found.     

Bifurcation and chaos in a system simulating magnetic field configurations

In the present article, the behaviour of a nonlinear dynamical system has been analysed using the approach of bifurcation theory. The system is important due to the fact that it can simulate the magnetic field configurations in various situations. The nature of bifurcation has been explored in the parameter space with the help of continuation algorithm. The various limit and bifurcation points (BPs) are classified. In the second part, we have studied the temporal evolution of the system which also shows a chaotic behaviour. The system under consideration shows instability both with respect to parameter variation and evolution of time. Lastly, some mechanisms have been studied to control such chaotic scenario.     

Dynamics and chaos control in nonlinear electrostatic transducers

In this paper, we analyze the dynamics of a system consisting of two coupled nonlinearly Duffing oscillators, obtained from a nonlinear electrostatic device which is a prototype of emitters and receivers in communication engineering. Inverse or backward period doubling cascades and sudden transition to chaos are observed. A sliding mode controller is applied to control the electrostatic transducers system. The sliding surface used is one dimension higher than the traditional surface and guarantees its passage through the initial states of the controlled system. By means of the design of sliding mode dynamics characteristics, the controlled system performance is arbitrarily determined by assigning the switching gain of the sliding mode dynamics. Therefore, using the characteristic of this sliding mode we aim to design a controller that can meet the desired specification and use less control energy by comparing with the result in the current literature. The results show that the proposed controller can steer electrostatic transducers to the desired reference trajectory without chattering phenomenon and abrupt state change.     

Bifurcation and chaos analysis of spur gear pair with and without nonlinear suspension

This study performs a systematic analysis of the dynamic behavior of a single degree-of-freedom spur gear system with and without nonlinear suspension. The dynamic orbits of the system are observed using bifurcation diagrams plotted using the dimensionless damping coefficient and the dimensionless rotational speed ratio as control parameters. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, Lyapunov exponents and fractal dimension of the gear system. The numerical results reveal that the system exhibits a diverse range of periodic, sub-harmonic and chaotic behaviors. The results presented in this study provide an understanding of the operating conditions under which undesirable dynamic motion takes place in a spur gear system and therefore serve as a useful source of reference for engineers in designing and controlling such systems.     

Bifurcation and chaos in friction-induced vibration

Friction-induced vibration is a phenomenon that has received extensive study by the dynamics community. This is because of the important industrial relevance and the ever-evolving development of new friction models. In this paper, we report the result of bifurcation study of a single-degree-of-freedom mechanical oscillator sliding over a surface. The friction model we use is that developed by Canudas de Wit et al., a model that is receiving increasing acceptance from the mechanics community. Using this model, we find a stable limit cycle at intermediate sliding speed for a single-degree-of-freedom mechanical oscillator. Moreover, the mechanical oscillator can exhibit chaotic motions. For certain parameters, numerical simulation suggests the existence of a Silnikov homoclinic orbit. This is not expected in a single-degree-of-freedom system. The occurrence of chaos becomes possible because the friction model contains one internal variable. This demonstrates a unique characteristic of the friction model. Unlike most friction models, the present model is capable of simultaneously modeling self-excitation and predicting stick–slip at very low sliding speed as well.     

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.     

Bifurcation and chaos near sliding homoclinics

We study the chaotic behaviour of a time dependent perturbation of a discontinuous differential equation whose unperturbed part has a sliding homoclinic orbit that is a solution homoclinic to a hyperbolic fixed point with a part belonging to a discontinuity surface. We assume the time dependent perturbation satisfies a kind of recurrence condition which is satisfied by almost periodic perturbations. Following a functional analytic approach we construct a Melnikov-like function M(α) in such a way that if M(α) has a simple zero at some point, then the system has solutions that behave chaotically. Applications of this result to quasi-periodic systems are also given.     

Control of chaos in a piecewise smooth nonlinear system

This paper shows the stabilization of the unstable periodic orbit of any given piecewise smooth system with linear and/or nonlinear characteristics. By utilizing the periodicity of the switching action, we construct the Poincaré mapping including all information of the original system. This mapping offers a first step toward extending a novel technique for controlling chaos based on the appropriate state feedback in piecewise smooth nonlinear systems. We also apply this approach to Rayleigh type oscillator described by the piecewise smooth nonlinear systems.     

Bifurcation analysis and control of chaos for a hybrid ratio-dependent three species food chain

In this paper, a hybrid ratio-dependent three species food chain model with time delay is studied by using the theory of functional differential equation and Hopf bifurcation, the condition on which positive equilibrium exists and the quality of Hopf bifurcation are given. Chaotic solutions are observed and are controlled by delay parameter. Finally, we indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable state or a stable periodic orbit.     

Bifurcation modes in a nonlinear optical system with distributed field rotation

We consider a model of a nonlinear optical system with distributed field rotation described by a functional-differential diffusion equation. An existence theorem is proved for periodical spatially nonhomogeneous traveling-wave solutions, which are generated from a spatially homogeneous stationary solution by an Andronov-Hopf (cycle-generating) bifurcation. A series expansion of the solution in powers of a small parameter is obtained and a stability condition is given. Simulation results are used to discuss the properties of the model. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 89–99.     

Global chaos synchronization of new chaotic systems via nonlinear control

Nonlinear control is an effective method for making two identical chaotic systems or two different chaotic systems be synchronized. However, this method assumes that the Lyapunov function of error dynamic (e) of synchronization is always formed as V (e) = 1/2e^Te. In this paper, modification based on Lyapunov stability theory to design a controller is proposed in order to overcome this limitation. The method has been applied successfully to make two identical new systems and two different chaotic systems (new system and Lorenz system) globally asymptotically synchronized. Since the Lyapunov exponents are not required for the calculation, this method is effective and convenient to synchronize two identical systems and two different chaotic systems. Numerical simulations are also given to validate the proposed synchronization approach.     

Nonlinear dynamic analysis for bevel-gear system under nonlinear suspension-bifurcation and chaos

This study aims to analyze the dynamic behavior of bevel-geared rotor system supported on a thrust bearing and journal bearings under nonlinear suspension. The dynamic orbits of the system are observed using bifurcation diagrams plotted with both the dimensionless unbalance coefficient and the dimensionless rotational speed ratio as control parameters. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincaré maps, Lyapunov exponents, and fractal dimensions of the gear-bearing system. The numerical results reveal that the system exhibits a diverse range of periodic, sub-harmonic, and chaotic behaviors. The results presented in this study provide an understanding of the operating conditions under which undesirable dynamic motion takes place in a gear-bearing system and therefore serves as a useful source of reference for engineers in designing and controlling such systems.     

Stability,estimation of state,and control in one nonlinear system

A controlled system of nonlinear ordinary differential equations is considered. In this system, we distinguish the time-varying linear part, the part that linearly depends on the control, and the nonlinearity, which vanishes on the steady-state solution and satisfies the Lipschitz condition. In addition, there are some constraints imposed on the control.