A modified averaging scheme with application to the secondary Hopf bifurcation of a delayed van der Pol oscillator

In this paper, a modified averaging scheme is presented for a class of time-delayed vibration systems with slow variables. The new scheme is a combination of the averaging techniques proposed by Hale and by Lehman and Weibel, respectively. The averaged equation obtained from the modified scheme is simple enough but it retains the required information for the local nonlinear dynamics around an equilibrium. As an application of the present method, the delay value for which a secondary Hopf bifurcation occurs is successfully located for a delayed van der Pol oscillator.

Impulsive Control for T-S Fuzzy Model Based Chaotic Systems Consisting of Van der Pol Oscillators Coupled to Linear Oscillators

This paper proposes an impulsive control scheme for chaotic systems consisting of Van der Pol oscillators coupled to linear oscillators （VDPL） based on their Takagi-Sugeno （T-S） fuzzy models. A T-S fuzzy model is utilized to represent the chaotic VDPL system. By using comparison method, a general asymptotical stability criterion by means of linear matrix inequality （LMI） is derived for the T-S fuzzy model of VDPL system with impulsive effects. The simulation results demonstrate the effectiveness of the proposed scheme.

Phase synchronization and synchronization frequency of two-coupled van der Pol oscillators with delayed coupling

In this paper, phase synchronization and the frequency of two synchronized van der Pol oscillators with delay coupling are studied. The dynamics of such a system are obtained using the describing function method, and the necessary conditions for phase synchronization are also achieved. Finding the vicinity of the synchronization frequency is the major advantage of the describing function method over other traditional methods. The equations obtained based on this method justify the phenomenon of the synchronization of coupled oscillators on a frequency either higher, between, or lower than the highest, in between, or lowest natural frequency of the aggregate oscillators. Several numerical examples simulate the different cases versus the various synchronization frequency delays.

A mathematical model of circadian rhythms synchronization using fractional differential equations system of coupled van der Pol oscillators

This paper presents an alternative representation of a system of differential equations qualitatively showing the behavior of the biological rhythm of a crayfish during their transition from juvenile to adult stages. The model focuses on the interaction of four cellular oscillators coupled by diffusion of a hormone, a parameter μ is used to simu- late the quality of communication among the oscillators, in biological terms, it mea- sures developmental maturity of the crayfish. Since some quorum-sensing mechanism is assumed to be responsible for the synchronization of the biological oscillators, it is nat- ural to investigate the possibility that the underlying diffusion process is not standard, i.e. it may be a so-called anomalous diffusion. In this case, it is well understood that diffusion equations with fractional derivatives describe these processes in a more realis- tic way. The alternative formulation of these equations contains fractional operators of Liouville-Caputo and Caputo-Fabrizio type. The numerical simulations of the equations reflect synchronization of ultradian rhythms leading to a circadian rhythm. The classical behavior is recovered when the order of the fractional derivative is V = 1. We discuss possible biological implications.

An equivalent linearization method for nonlinear Van der Pol oscillator subjected to random vibration using orthogonal functions

This paper deals with the idea of the orthogonal functions in the equivalent linearization of the nonlinear systems. Block Pulse （BP） function gives effective tools to approximate complex problems. The aim of this work is on using properties of the BP function as an orthogonal function in process of linearization. The BP functions have been used to propose an equivalent linearization method in the time domain to determine the unknown linearization coefficients. The accuracy of the proposed method compared with the other equivalent linearization approaches, including the regulation linearization and the dual criterion linearization methods. This study exploited the nonlinear Van der Pol oscillator system under stationary random excitation to demonstrate the feasibility of the proposed method. The validity of the analytical method is verified by applying different values of nonlinearity and intensity of excitation. Besides, by comparing the mean-square responses and frequency response functions of the linearized systems for a wide range of nonlinearity depicted the present method is in agreement with other methods.

An efficient approach to solving fractional Van der Pol-Duffing jerk oscillator

The motive behind the current work is to perform the solution of the Van der Pol–Duffing jerk oscillator,involving fractional-order by the simplest method.An effective procedure has been introduced for executing the fractional-order by utilizing a new method without the perturbative approach.The approach depends on converting the fractional nonlinear oscillator to a linear oscillator with an integer order.A detailed solving process is given for the obtained oscillator with the traditional system.

Bifurcation and dynamic response analysis of rotating blade excited by upstream vortices

A reduced model is proposed and analyzed for the simulation of vortexinduced vibrations （VIVs） for turbine blades. A rotating blade is modelled as a uniform cantilever beam, while a van der Pol oscillator is used to represent the time-varying characteristics of the vortex shedding, which interacts with the equations of motion for the blade to simulate the fluid-structure interaction. The action for the structural motion on the fluid is considered as a linear inertia coupling. The nonlinear characteristics for the dynamic responses are investigated with the multiple scale method, and the modulation equations are derived. The transition set consisting of the bifurcation set and the hystere- sis set is constructed by the singularity theory and the effects of the system parameters, such as the van der Pol damping. The coupling parameter on the equilibrium solutions is analyzed. The frequency-response curves are obtained, and the stabilities are determined by the Routh-Hurwitz criterion. The phenomena including the saddle-node and Hopf bifurcations are found to occur under certain parameter values. A direct numerical method is used to analyze the dynamic characteristics for the original system and verify the va- lidity of the multiple scale method. The results indicate that the new coupled model is useful in explaining the rich dynamic response characteristics such as possible bifurcation phenomena in the VIVs.

Dynamics and synchronization of a complex-valued star network

Complex networks have been extensively investigated in recent years.However,the dynamics,especially chaos and bifurcation,of the complex-valued complex network are rarely studied.In this paper,a star network of coupled complex-valued van der Pol oscillators is proposed to reveal the mechanism of star coupling.By the aid of bifurcation diagram,Lyapunov exponent spectrum and phase portrait in this study,chaos,hyper-chaos,and multi-existing chaotic attractors are observed from the star network,although there are only periodic states in a complex-valued van der Pol oscillator.Complexity versus coupling strength and nonlinear coefficient shows that the bigger the network size,the larger the parameter range within the chaotic(hyper-chaotic)region.It is revealed that the chaotic bifurcation path is highly robust against the size variation of the star network,and it always evolves to chaos directly from period-1 and quasi-periodic states,respectively.Moreover,the coexistence of chaotic phase synchronization and complete synchronization among the peripherals is also found from the star network,which is a symmetrybreaking phenomenon.