具有分数阶导数阻尼的强迫振动共振现象

研究了一类具有分数阶导数阻尼的强迫振动共振现象.首先构造渐近解,然后利用Riemann-Liouville分数阶导数定义及性质,求出分数阶导数项的表达式.再利用多重尺度法,求出各共振的共振频率.这些共振既包括主共振也包括次共振.对于每个共振频率,引入解谐参数,消除长期项.利用数学软件画出共振振幅及初相位在不同的分数指数下的数值解的图形,发现分数指数对共振的影响,并对每个共振频率求出渐近解的一阶近似表达式.     

Oscillatory region and asymptotic solution of fractional van der Pol oscillator via residue harmonic balance technique

In this paper, a powerfully analytical technique is proposed for predicting and generating the steady state solution of the fractional differential system based on the method of harmonic balance. The zeroth-order approximation using just one Fourier term is applied to predict the parametric function for the boundary between oscillatory and non-oscillatory regions of the fractional van der Pol oscillator. The unbalanced residues due to Fourier truncation are considered iteratively by solving linear algebraic equations to improve the accuracy of the solutions successively. The highly accurate solutions to the angular frequency and limit cycle of fractional van der Pol oscillator are obtained and compared. The results reveal that the technique described in this paper is very effective and simple for obtaining asymptotic solution of nonlinear system having fractional order derivative.     

Principal resonance responses of SDOF systems with small fractional derivative damping under narrow-band random parametric excitation

The principal resonance responses of nonlinear single-degree-of-freedom (SDOF) systems with lightly fractional derivative damping of order α (0 < α < 1) subject to the narrow-band random parametric excitation are investigated. The method of multiple scales is developed to derive two first order stochastic differential equation of amplitude and phase, and then to examine the influences of fractional order and intensity of random excitation on the first-order and second-order moment. As an example, the stochastic Duffing oscillator with fractional derivative damping is considered. The effects of detuning frequency parameter, the intensity of random excitation and the fractional order derivative damping on stability are studied through the largest Lyapunov exponent. The corresponding theoretical results are well verified through direct numerical simulations. In addition, the phenomenon of stochastic jump is analyzed for parametric principal resonance responses via finite differential method. The stochastic jump phenomena indicates that the most probable motion is around the larger non-trivial branch of the amplitude response when the intensity of excitation is very small, and the probable motion of amplitude responses will move from the larger non-trivial branch to trivial branch with the increasing of the intensity of excitation. Such stochastic jump can be considered as bifurcation.     

Nonlinear fractional cone systems with the Caputo derivative

Nonlinear fractional cone systems involving the Caputo fractional derivative are considered. We establish sufficient conditions for the existence of at least one cone solution to such systems. Sufficient conditions for the unique existence of the cone solution to a nonlinear fractional cone system are given.     

Asymptotics for some semilinear hyperbolic equations with non-autonomous damping

Let V and H be Hilbert spaces such that $V?H?V^{\prime }$ with dense and continuous injections. Consider a linear continuous operator $A:V\to V^{\prime }$ which is assumed to be symmetric, monotone and semi-coercive. Given a function $f:V\to H$ and a map $\gamma \in W_{\mathit{loc}}^{1,1}({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ such that ${\lim }_{t\to +\infty }\gamma (t)=0$, our purpose is to study the asymptotic behavior of the following semilinear hyperbolic equation$\frac{d^{2}u}{d{t}^2}(t)+\gamma (t)\frac{du}{dt}(t)+Au(t)+f(u(t))=0,t?0.$ The nonlinearity f is assumed to be monotone and conservative. Condition ${\int }_0^{+\infty }\gamma (t)dt=+\infty$ guarantees that some suitable energy function tends toward its minimum. The main contribution of this paper is to provide a general result of convergence for the trajectories of (E): if the quantity $\gamma (t)$ behaves as $k/t^{\alpha }$, for some $\alpha \in ]0,1[$, $k>0$ and t large enough, then $u(t)$ weakly converges in V toward an equilibrium as $t\to +\infty$. Strong convergence in V holds true under compactness or symmetry conditions. We also give estimates for the speed of convergence of the energy under some ellipticity-like conditions. The abstract results are applied to particular semilinear evolution problems at the end of the paper.     

Chaos control on autonomous and non-autonomous systems with various types of genetic algorithm-optimized weak perturbations

Recently, we proposed a chaos control strategy with weak Fourier signals optimized by using a genetic algorithm (GA) and demonstrated its merits in controlling Lorenz and Rössler systems (Physical Review E, 2004). In this continuation work, performance of various types of signals, namely periodic continuous, periodic discrete, and constant bias (non-periodic), applied to an autonomous (Rössler) system and a non-autonomous (Murali–Lakshmanan–Chua, MLC) system are investigated. An index of relative robustness is proposed for measuring the noise-resisting ability of the control signals. The results reveal that the constant signal has the strongest noise-resisting ability, the periodic pulse signal has the weakest, and the Fourier signal falls in between. Phase modulation generally shortens the transient time period and is additionally beneficial to non-autonomous systems in minimizing significantly the signal power. By searching with the present GA-optimization, it is demonstrated that the minimum-power signal for controlling the non-autonomous (MLC) system is the signal with a frequency exactly the same as that of the system forcing but with phase modulation. The effectiveness of the GA-optimized signals of extremely low power employed in alternatively switching control of non-autonomous systems is also demonstrated.     

Convergence for non-autonomous semidynamical systems with impulses

The present paper deals with impulsive non-autonomous systems with convergence. We show that the structure of the Levinson center of a compact dissipative system is preserved under homomorphism in impulsive convergent systems. Also, we present some criteria of convergence using Lyapunov functions.     

Exact decay rates for weakly coupled systems with indirect damping by fractional memory effects

This paper deals with the asymptotic behavior of a weakly coupled system of two equations in which one of them has a dissipative mechanism given by a memory term. This term depends on the fractional operator with exponent $\theta \in [0,1]$. We show that strong solutions of the system decay polynomially with a rate that depends on both the exponent θ and wave propagation speeds. Optimal decay rates are found and the results show a surprising aspect: More regular damping does not necessarily imply a faster decay.     

Pullback attractors for non-autonomous porous elastic system with nonlinear damping and sources terms

In this paper, we study the asymptotic behavior of a non-autonomous porous elastic systems with nonlinear damping and sources terms. By employing nonlinear semigroups and the theory of monotone operators, we establish existence and uniqueness of weak and strong solutions. We also prove the existence of minimal pullback attractors with respect to a universe of tempered sets defined by the sources terms. Finally, we prove the upper-semicontinuity of pullback attractors with respect to non-autonomous perturbations.     

Attractor for the non-autonomous long wave-short wave resonance interaction equation with damping

In this paper, the long wave-short wave resonance interaction equation with a nonlinear term in bounded domain was studied. When $\beta\geq\frac{3}{2}$, we obtained the existence and uniqueness of the weak solution of system (1.1)-(1.4) by Gal\"{e}rkin''s method, and further proved the existence of the compact uniform attractor for damped driven by the non-autonomous long wave-short wave resonance interaction equation.     

Permanence for non-autonomous food chain systems with delay

In this paper we first establish a new general criterion for the permanence of Kolmogorov-type systems of nonautonomous functional differential equations. Then, as applications of this criterion we study the permanence of a class of n-species general nonautonomous food chain systems with delay and new sufficient condition are established.     

Nonlinear transversely vibrating beams by the improved energy balance method and the global residue harmonic balance method

Mathematical modeling of many engineering systems such as beam structures often leads to nonlinear ordinary or partial differential equations. Nonlinear vibration analysis of the beam structures is very important in mechanical and industrial applications. This paper presents the high order frequency-amplitude relationship for nonlinear transversely vibrating beams with odd and even nonlinearities using the improved energy balance method and the global residue harmonic balance method. The accuracy of the energy balance method is improved based on combining features of collocation method and Galerkin–Petrov method, and an improved harmonic balance method is proposed which is called the global residue harmonic balance method. Unlike other harmonic balance methods, all the former global residual errors are introduced in the present approximation to improve the accuracy. Finally, preciseness of the present analytic procedures is evaluated in contrast with numerical calculations methods, giving excellent results.     

Lebesgue regularity for differential difference equations with fractional damping

We provide necessary and sufficient conditions for the existence and uniqueness of solutions belonging to the vector‐valued space of sequences for equations that can be modeled in the form where X is a Banach space, A is a closed linear operator with domain D(A) defined on X, and G is a nonlinear function. The operator Δ^γ denotes the fractional difference operator of order γ>0 in the sense of Grünwald‐Letnikov. Our class of models includes the discrete time Klein‐Gordon, telegraph, and Basset equations, among other differential difference equations of interest. We prove a simple criterion that shows the existence of solutions assuming that f is small and that G is a nonlinear term.     

Uniform attractor for non-autonomous plate equation with a localized damping and a critical nonlinearity

In this paper, we consider dynamical behavior of non-autonomous plate-type evolutionary equations with critical nonlinearity. We prove the existence of a uniform attractor in the space .     

Periodic wave solutions of coupled integrable dispersionless equations by residue harmonic balance

We introduce the residue harmonic balance method to generate periodic solutions for nonlinear evolution equations. A PDE is firstly transformed into an associated ODE by a wave transformation. The higher-order approximations to the angular frequency and periodic solution of the ODE are obtained analytically. To improve the accuracy of approximate solutions, the unbalanced residues appearing in harmonic balance procedure are iteratively considered by introducing an order parameter to keep track of the various orders of approximations and by solving linear equations. Finally, the periodic solutions of PDEs result. The proposed method has the advantage that the periodic solutions are represented by Fourier functions rather than the sophisticated implicit functions as appearing in most methods.     

Well-posedness of equations with fractional derivative

We study the well-posedness of the equations with fractional derivative D^αu（t） = Au（t） ＋ f（t）,0≤ t ≤ 2π, where A is a closed operator in a Banach space X, α 〉 0 and D^α is the fractional derivative in the sense of Weyl. Using known results on LP-multipliers, we give necessary and/or sufficient conditions for the LP-well-posedness of this problem. The conditions we give involve the resolvent of A and the Rademacher boundedness. Corresponding results on the well-posedness of this problem in periodic Besov spaces, periodic Triebel-Lizorkin spaces and periodic Hardy spaces are also obtained.     

On a final value problem for fractional reaction-diffusion equation with Riemann-Liouville fractional derivative

In this paper, we study a backward problem for a fractional diffusion equation with nonlinear source in a bounded domain. By applying the properties of Mittag-Leffler functions and Banach fixed point theorem, we establish some results above the existence, uniqueness, and regularity of the mild solutions of the proposed problem in some suitable space. Moreover, we also show the ill-posedness of our problem in the sense of Hadamard. The regularized solution is given, and the convergence rate between the regularized solution and the exact solution is also obtained.     

The stochastic obstacle problem for the harmonic oscillator with damping

We study the Kolmogorov equation associated with a second order stochastic variational inequality related to the harmonic oscillator.     

Mathematical analysis of autonomous and non-autonomous age-structured reaction-diffusion-advection population model

In this paper, we study an age-structured reaction-diffusion-advection population model. First, we use a non-densely defined operator to the linear age-structured reaction-diffusion-advection population model in a patchy environment. By spectral analysis, we obtain the asynchronous exponential growth of the population model. Then we consider nonlinear death rate and birth rate, which all depend on the function related to the generalized total population, and we prove the existence of a steady state of the system. Finally, we study the age-structured reaction-diffusion-advection population model in non-autonomous situations. We give the comparison principle and prove the eventual compactness of semiflow by using integrated semigroup. We also prove the existence of compact attractors under the periodic situation.