Fractional derivative and time delay damper characteristics in Duffing–van der Pol oscillators

In this paper, we investigate the damping characteristics of two Duffing–van der Pol oscillators having damping terms described by fractional derivative and time delay respectively. The residue harmonic balance method is presented to find periodic solutions. No small parameter is assumed. Highly accurate limited cycle frequency and amplitude are captured. The results agree well with the numerical solutions for a wide range of parameters. Based on the obtained solutions, the damping effects of these two oscillators are investigated. When the system parameters are identical, the steady state responses and their stability are qualitatively different. The initial approximations are obtained by solving a few harmonic balance equations. They are improved iteratively by solving linear equations of increasing dimension. The second-order solutions accurately exhibit the dynamical phenomena when taking the fractional derivative and time delay as bifurcation parameters respectively. When damping is described by time delay, the stable steady state response is more complex because time delay takes past history into account implicitly. Numerical examples taking time delay and fractional derivative are respectively given for feature extraction and convergence study.     

On limit cycle approximations in the van der Pol oscillator

This paper applies bifurcation analysis to the well-known van der Pol oscillator to obtain approximations of its periodic solutions in the nearly sinusoidal regime. A frequency domain method based on harmonic balance approximations is used for small values of the bifurcation parameter. Moreover, a comparison with some other frequency domain approaches is also given. Finally, a total harmonic distortion is computed using the information provided by the frequency domain approach.     

Delay induced Hopf bifurcation in a dual model of Internet congestion control algorithm

This paper focuses on the delay induced Hopf bifurcation in a dual model of Internet congestion control algorithms which can be modeled as a time-delay system described by a one-order delay differential equation (DDE). By choosing communication delay as the bifurcation parameter, we demonstrate that the system loses its stability and a Hopf bifurcation occurs when communication delay passes through a critical value. Moreover, the bifurcating periodic solution of the system is calculated by means of the perturbation method. Discussion of stability of the periodic solutions involves the computation of Floquet exponents by considering the corresponding Poincaré–Lindstedt series expansion. Finally, numerical simulations for verifying the theoretical analysis are provided.     

Spatiotemporal patterns induced by delay and cross-fractional diffusion in a predator-prey model describing intraguild predation

In this article, we study a reaction-diffusion predator-prey model that describes intraguild predation. We mainly consider the effects of time delay and cross-fractional diffusion on dynamical behavior. By using delay as the bifurcation parameter, we perform a detailed Hopf bifurcation analysis and derive the algorithm for determining the direction and stability of the bifurcating periodic solutions. We also demonstrate that proper cross-fractional diffusion can induce Turing pattern, and the smaller the order of fractional diffusion is, the more easily Turing pattern is able to occur.     

Forward residue harmonic balance for autonomous and non-autonomous systems with fractional derivative damping

Both the autonomous and non-autonomous systems with fractional derivative damping are investigated by the harmonic balance method in which the residue resulting from the truncated Fourier series is reduced iteratively. The first approximation using a few Fourier terms is obtained by solving a set of nonlinear algebraic equations. The unbalanced residues due to Fourier truncation are considered iteratively by solving linear algebraic equations to improve the accuracy and increase the number of Fourier terms of the solutions successively. Multiple solutions, representing the occurrences of jump phenomena, supercritical pitchfork bifurcation and symmetry breaking phenomena are predicted analytically. The interactions of the excitation frequency, the fractional order, amplitude, phase angle and the frequency amplitude response are examined. The forward residue harmonic balance method is presented to obtain the analytical approximations to the angular frequency and limit cycle for fractional order van der Pol oscillator. Numerical results reveal that the method is very effective for obtaining approximate solutions of nonlinear systems having fractional order derivatives.     

The use of partition polynomial series in Laplace inversion of composite functions with applications in fractional calculus

This paper presents an analytical method towards Laplace transform inversion of composite functions with the aid of Bell polynomial series. The presented results are used to derive the exact solution of fractional distributed order relaxation processes as well as time‐domain impulse response of fractional distributed order operators in new series forms. Evaluation of the obtained series expansions through computer simulations is also given. The results are then used to present novel series expansions for some special functions, including the one‐parameter Mittag‐Leffler function. It is shown that truncating these series expansions when combined with using potential partition polynomials provides efficient approximations for these functions. At the end, the results are shown to be also useful in studying asymptotical behavior of partial Bell polynomials. Numerical simulations as well as analytical examples are provided to verify the results of this paper.     

Bifurcation and synchronization of synaptically coupled FHN models with time delay

This paper presents an investigation of dynamics of the coupled nonidentical FHN models with synaptic connection, which can exhibit rich bifurcation behavior with variation of the coupling strength. With the time delay being introduced, the coupled neurons may display a transition from the original chaotic motions to periodic ones, which is accompanied by complex bifurcation scenario. At the same time, synchronization of the coupled neurons is studied in terms of their mean frequencies. We also find that the small time delay can induce new period windows with the coupling strength increasing. Moreover, it is found that synchronization of the coupled neurons can be achieved in some parameter ranges and related to their bifurcation transition. Bifurcation diagrams are obtained numerically or analytically from the mathematical model and the parameter regions of different behavior are clarified.     

Hybrid control on bifurcation for a delayed fractional gene regulatory network

A novel delayed fractional-order gene regulatory network model is proposed in this paper. Firstly, the sum of delays is chosen as the bifurcation parameter, the conditions of the existence for the Hopf bifurcation are obtained by analyzing the associated characteristic equation. Then, it is the first time that a hybrid controller is designed to control the Hopf bifurcation for the proposed network. It is demonstrated that both fractional order and feedback gain can effectively control the dynamics of such network, the stability domain can be extended under the small fractional order or feedback gain. The controller is enough to delay the onset of the Hopf bifurcation and achieve some desirable dynamical behaviors by changing the appropriate order or feedback gain. Finally, numerical simulations are employed to justify the validity of our theoretical analysis.     

On problems of Aizerman and Kalman

The problems of the stability of nonlinear control systems posed by Aizerman and Kalman have stimulated the development of methods for detecting hidden periodic oscillations in multidimensional dynamical systems. In the 1950s, Pliss developed an analytical method for detecting periodic oscillations in third-order systems satisfying the generalized Routh-Hurwitz conditions. It has turned out that this generalized method of Pliss can be regarded as a special version of the describing function method in the critical case. Being combined with computational procedures based on applied bifurcation theory, this method makes it possible to obtain new classes of systems for which the conjectures of Aizerman and Kalman are false. The known approaches to constructing counterexamples to Aizerman’s and Kalman’s conjectures proposed by Fitts, Barabanov, and Llibre are reviewed. A new effective analytical-numerical method for constructing such counterexamples is presented. The method is based on combining the classical theory of small parameter, bifurcation theory, and the method of harmonic linearization. It is applied to numerically construct a series of counterexamples to Aizerman’s and Kalman’s conjectures.     

Nonlinear dynamic analysis of fractional order rub-impact rotor system

Nonlinear dynamic characteristics of rub-impact rotor system with fractional order damping are investigated. The model of rub-impact comprises a radial elastic force and a tangential Coulomb friction force. The fractional order damped rotor system with rubbing malfunction is established. The four order Runge–Kutta method and ten order CFE-Euler method are introduced to simulate the fractional order rub-impact rotor system equations. The effects of the rotating speed ratio, derivative order of damping and mass eccentricity on the system dynamics are investigated using rotor trajectory diagrams, bifurcation diagrams and Poincare map. Various complicated dynamic behaviors and types of routes to chaos are found, including period doubling bifurcation, sudden transition and quasi-periodic from periodic motion to chaos. The analysis results show that the fractional order rub-impact rotor system exhibits rich dynamic behaviors, and that the significant effect of fractional order will contribute to comprehensive understanding of nonlinear dynamics of rub-impact rotor.     

Stability and Hopf bifurcation in a delayed competition system

In this paper, Hopf bifurcation for two-species Lotka–Volterra competition systems with delay dependence is investigated. By choosing the delay as a bifurcation parameter, we prove that the system is stable over a range of the delay and beyond that it is unstable in the limit cycle form, i.e., there are periodic solutions bifurcating out from the positive equilibrium. Our results show that a stable competition system can be destabilized by the introduction of a maturation delay parameter. Further, an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by using the theory of normal forms and center manifolds, and numerical simulations supporting the theoretical analysis are also given.     

Hopf-pitchfork bifurcation and periodic phenomena in nonlinear financial system with delay

In this paper, we identify the critical point for a Hopf-pitchfork bifurcation in a nonlinear financial system with delay, and derive the normal form up to third order with their unfolding in original system parameters near the bifurcation point by normal form method and center manifold theory. Furthermore, we analyze its local dynamical behaviors, and show the coexistence of a pair of stable periodic solutions. We also show that there coexist a pair of stable small-amplitude periodic solutions and a pair of stable large-amplitude periodic solutions for different initial values. Finally, we give the bifurcation diagram with numerical illustration, showing that the pair of stable small-amplitude periodic solutions can also exist in a large region of unfolding parameters, and the financial system with delay can exhibit chaos via period-doubling bifurcations as the unfolding parameter values are far away from the critical point of the Hopf-pitchfork bifurcation.     

Resonance in a noise-driven excitable neuron model

The effects of variations in bifurcation parameter on coherence resonance in the noisy FitzHugh–Nagumo (FHN) neuron model are studied. We find that the coherence resonance effect depends monotonically on the firing bifurcation parameter, and this result is interpreted analytically. Then an external control method is presented to modulate coherence resonance in the excitable neuron model, and our scheme is based on the result that a weak periodic perturbation can be used to change the critical firing onset value. This method can be used to either enhance the effect of coherence resonance or delay the occurrence of coherence resonance, and the application of the method to another neuron model is also discussed.     

Suppressing chaos in a simplest autonomous memristor-based circuit of fractional order by periodic impulses

In this paper, the chaotic behavior of a simplest autonomous memristor-based circuit of fractional order is suppressed by periodic impulses applied to one or several state variables. The circuit consists of two passive linear elements, a capacitor and an inductor, as well as a nonlinear memristive element. It is shown that by applying a sequence of adequate (identical or different) periodic impulses to one or several variables, the chaotic behavior can be suppressed. Impulse values and control timing are determined numerically, based on the bifurcation diagram with impulses as bifurcation parameters. Empirically, the probability to have a reasonably wide range of impulses to suppress chaos is quite large, ensuring that chaos suppression can be implemented, as demonstrated by several examples presented.     

Analytical solution of time‐fractional Navier–Stokes equation in polar coordinate by homotopy perturbation method

In this letter, we implement a relatively new analytical technique, the homotopy perturbation method (HPM), for solving linear partial differential equations of fractional order arising in fluid mechanics. The fractional derivatives are described in Caputo derivatives. This method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations applied in engineering mathematics. The corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations. Some numerical examples are presented to illustrate the efficiency and reliability of HPM. He's HPM, which does not need small parameter is implemented for solving the differential equations. In this method, a homotopy is introduced to be constructed for the equation. The initial approximations can be freely chosen with possible unknown constants that can be determined by imposing the boundary and initial conditions. It is predicted that HPM can be found widely applicable in engineering. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Bifurcation and Stability Theory of Periodic Solutions For Integrodifferential Equations

This paper is concerned with a nonlinear integrodifferential equation (the delay logistic equation) governing the growth dynamics of a single species N(t) for time t > 0. This equation contains a positive parameter λ. Suppose that there exists a positive equilibrium solution N = c which is stable for all small values of λ. Assume also that this solution loses stability as λ is increased past a critical value λ*. This will correspond to a simple pure imaginary conjugate pair of roots of a characteristic equation associated with the linearized stability of N = c at λ = λ*. Then we will construct a unique bifurcating time periodic solution of the equation as a Taylor series in a parameter ε. Furthermore this solution exists either for supercritical values of the parameter (λ > λ*) or for subcritical values (λ < λ*). The stability behavior of this small periodic solution can be characterized according to whether the bifurcation is supercritical or subcritical-supercritical solutions are stable, but subcritical solutions are unstable. Therefore these results are analogous to Hoprs bifurcation theorem for autonomous systems of differential equations.     

弹性约束浅拱的非线性动力响应与分岔分析

浅拱采用竖向、转动方向弹性约束时,自振频率和模态与理想的铰支/固结边界存在差异,不同约束刚度将改变外激励下的非线性响应及各种分岔产生的参数域．由浅拱基本假定建立无量纲动力学方程, 采用在频率和模态中考虑约束刚度大小的方法,通过Galerkin全离散和多尺度摄动分析导出极坐标、直角坐标形式的平均方程, 其中方程系数与约束刚度一一对应．用数值方法分析了周期激励下竖向弹性约束系统最低两阶模态之间1∶2内共振时的动力行为, 所得结果与有限元的对比以及平均方程系数的收敛性证明了所采用方法是可行的．随着激励幅值、频率的变化存在若干分岔点,分岔发生时的参数分布与约束刚度值有关,在由分岔点连接的不稳定区或共振区附近,存在一系列稳态解、周期解、准周期解和混沌解窗口,且随参数的变化可观测到倍周期分岔．     

The effect of time delay on the dynamics of an SEIR model with nonlinear incidence

In this paper, a SEIR epidemic model with nonlinear incidence rate and time delay is investigated in three cases. The local stability of an endemic equilibrium and a disease-free equilibrium are discussed using stability theory of delay differential equations. The conditions that guarantee the asymptotic stability of corresponding steady-states are investigated. The results show that the introduction of a time delay in the transmission term can destabilize the system and periodic solutions can arise through Hopf bifurcation when using the time delay as a bifurcation parameter. Applying the normal form theory and center manifold argument, the explicit formulas determining the properties of the bifurcating periodic solution are derived. In addition, the effect of the inhibitory effect on the properties of the bifurcating periodic solutions is studied. Numerical simulations are provided in order to illustrate the theoretical results and to gain further insight into the behaviors of delayed systems.