一类五次方振子系统的叉形分叉及振动共振研究

研究了一类具有分数阶导数阻尼的五次方振子系统中的叉形分叉及振动共振现象. 基于快慢变量分离法, 消去系统中的高频激励成分, 得到关于慢变量的等效系统, 根据等效系统中稳态平衡点的变化情况研究了系统的叉形分叉现象. 结果表明: 高频信号幅值的递增变化会引起亚临界叉形分叉, 高频信号频率和分数阶导数阻尼阶数的递增变化都会引起超临界叉形分叉; 振动共振和叉形分叉是关联的, 当叉形分叉发生时, 振动共振曲线会出现两个峰值, 否则只会出现一个峰值. 通过解析结果和数值模拟结果的对比, 验证了解析分析的正确性. 关键词： 亚临界叉形分叉 超临界叉形分叉 分数阶导数阻尼 振动共振     

Vibrational resonance in Duffing systems with fractional-order damping

The phenomenon of vibrational resonance (VR) is investigated in over- and under-damped Duffing systems with fractional-order damping. It is found that the factional-order damping can induce change in the number of the steady stable states and then lead to single- or double-resonance behavior. Compared with vibrational resonance in the ordinary systems, the following new results are found in the fractional-order systems. (1) In the overdamped system with double-well potential and ordinary damping, there is only one kind of single-resonance, whereas there are double-resonance and two kinds of single-resonance for the case of fractional-order damping. The necessary condition for these new resonance behaviors is the value of the fractional-order satisfies α?>?1. (2) In the overdamped system with single-well potential and ordinary damping, there is no resonance, whereas there is a single-resonance for the case of fractional-order damping. The necessary condition for the new result is α?>?1. (3) In the underdamped system with double-well potential and ordinary damping, there are double-resonance and one kind of single-resonance, whereas there are double-resonance and two kinds of single-resonance for the case of fractional-order damping. The necessary condition for the new single-resonance is α?相似文献   

Bifurcation and resonance in a fractional Mathieu-Duffing oscillator

The bifurcation and resonance phenomena are investigated in a fractional Mathieu-Duffing oscillator which contains a fast parametric excitation and a slow external excitation. We extend the method of direct partition of motions to evaluate the response for the parametrically excited system. Besides, we propose a numerical method to simulate different types of local bifurcation of the equilibria. For the nonlinear dynamical behaviors of the considered system, the linear stiffness coefficient is a key factor which influences the resonance phenomenon directly. Moreover, the fractional-order damping brings some new results that are different from the corresponding results in the ordinary Mathieu-Duffing oscillator. Especially, the resonance pattern, the resonance frequency and the resonance magnitude depend on the value of the fractional-order closely.     

双频信号驱动含分数阶内、外阻尼Duffing振子的振动共振

本文利用解析和数值的方法研究了由双频周期信号驱动含分数阶内、外阻尼的Duffing振子的振动共振现象,并讨论了分数阶阶数对上述现象的影响. 研究发现：双频周期信号同时驱动的分数阶Duffing振子响应幅值增益Q可随着高频周期激励幅值的改变达到最大值,即出现了和整数阶非线性动力系统相似的振动共振现象,而相应的分数阶导数项则分别为系统提供了内、外两种阻尼力从而导致了系统有效势函数的改变,进而引发了比整数阶动力系统更为丰富的振动共振现象. 关键词： 振动共振 Duffing振子 分数阶阻尼 分数阶系统     

含分数阶微分的线性单自由度振子的动力学分析

研究了一个含分数阶微分的线性单自由度振子, 通过平均法得到了系统的近似解析解. 在近似解中, 分数阶微分项的系数和阶次以等效线性阻尼和等效线性刚度的形式影响着系统的动力学特性, 这一点与现有文献中直接将分数阶微分项归类为阻尼进行处理的方法完全不同. 比较了近似解析解和数值解, 二者的符合精度很高, 证明了近似解析解的准确性. 分析了分数阶系数和分数阶阶次对系统响应特性的影响, 发现分数阶系数和分数阶阶次都既可以通过等效线性阻尼影响系统的共振振幅, 又可以通过等效线性刚度影响系统的共振频率.     

含分数阶微分的线性单自由度振子的动力学分析(Ⅱ)

研究了含两类分数阶微分项的线性单自由度振子, 通过平均法得到了系统的近似解析解. 在近似解中, 两类分数阶微分项的系数和阶次均以等效线性阻尼和等效线性刚度的形式影响着系统的动力学特性, 这一点和现有文献中大多数直接将分数阶微分项归类为阻尼进行处理是完全不同的. 对近似解析解和数值解进行了比较, 二者符合精度很高, 证明了该结果的准确性. 然后分析了两类分数阶微分项的系数和阶次对系统响应特性的影响, 发现两类分数阶微分项的系数和阶次都既可以影响系统的共振振幅, 又可以影响系统的共振频率. 最后研究了第二类分数阶微分项对共振频率的影响, 指出了在振动控制工程中如何通过选取合适的第二类分数阶微分项的系数达到满意的控制效果.     

分数阶时滞反馈对Duffing振子动力学特性的影响

研究了含分数阶时滞耦合反馈的Duffing自治系统, 通过平均法得到了系统周期解的一阶近似解析形式, 定义了以反馈系数、分数阶阶次、时滞参数表示的等效刚度和等效阻尼系数, 发现分数阶时滞耦合反馈同时具有速度时滞反馈和位移时滞反馈的作用. 比较了三种参数条件下近似解析解与数值积分的结果, 二者的吻合精度都很高, 证明了近似解析解的正确性和准确性. 分析了反馈系数、分数阶阶次和非线性刚度系数等参数对系统分岔点、周期解稳定性、周期解的存在范围、零解的稳定性以及稳定性切换次数等系统动力学特性的影响.     

单参数Lorenz混沌系统的电路设计与实现

为了研究混沌系统的性质及其应用,采用分立元件设计并实现了单参数Lorenz混沌系统,系统参数与电路元件参数一一对应.通过调节电路中的可变电阻,观察到了该单参数系统的极限环、叉式分岔、倍周期分岔和混沌等动力学现象,以及该系统由倍周期分岔进入混沌的过程.研究了分数阶单参数Lorenz系统存在混沌的必要条件,找出了分数阶单参数Lorenz系统出现混沌的最低阶数以及最低阶数随系统参数变化的一般规律.电路仿真与电路实现研究表明,单参数Lorenz系统具有物理可实现性、丰富的动力学特性以及理论分析与实验结果的一致性.     

Double Loops and Pitchfork Symmetry Breaking Bifurcations of Optical Solitons in Nonlinear Fractional Schrödinger Equation with Competing Cubic-Quintic Nonlinearities

Symmetry breaking bifurcations of solitons are investigated in framework of a nonlinear fractional Schrödinger equation (NLFSE) with competing cubic-quintic nonlinearity. Some prototypical characteristics of the symmetry breaking, featured by transformations of symmetric and antisymmetric soliton families into asymmetric ones, are found. Stable asymmetric solitons emerge from unstable symmetric and antisymmetric ones by way of two different symmetry breaking scenarios. A twisting branch, featured with double loops bifurcation, bifurcates off from the base branch of symmetric soliton solutions and crosses it, then merges into the base branch driven by the competitive nonlinear effect. A supercritical pitchfork bifurcation is bifurcated from the branch of antisymmetric soliton solutions and gives rise to a supercritical pitchfork bifurcation. Stability of the soliton families is explored by linear stability analysis. With the increase of the Lévy index, stability region induced by the twisting loops bifurcation is expanded. However, stability region of the pitchfork bifurcation is shrunk on the parameter plane of the Lévy index and the soliton power.     

Stability of a linear oscillator with damping force of the fractional-order derivative

It begins with the study of damping representation of a linear vibration system of single degree of freedom (SDOF),from the view point of fractional calculus. By using the idea of stability switch,it shows that the linear term involving the fractional-order derivative of an order between 0 and 2 always acts as a damping force,so that the unique equilibrium is asymp-totically stable. Further,based on the idea of stability switch again,the paper proposes a scheme for determining the stable gain region of a li...     

Dynamics of fractional-order sinusoidally forced simplified Lorenz system and its synchronization

In this paper, dynamical behaviors of the fractional-order sinusoidally forced simplified Lorenz are investigated by employing the time-domain solution algorithm of fractional-order calculus. The system parameters and the fractional derivative orders q are treated as bifurcation parameters. The range of the bifurcation parameters in which the system generates chaos is determined by bifurcation, phase portrait, and Poincaré section, and different bifurcation motions are visualized by virtue of a systematic numerical analysis. We find that the lowest order of this system to yield chaos is 3.903. Based on fractional-order stability theory, synchronization is achieved by using nonlinear feedback control method. Simulation results show the scheme is effective and a chaotic secure communication scheme is present based on this synchronization.     

Bursting dynamics in parametrically driven memristive Jerk system

Compared with general nonlinear systems, multi-time scale system has complex bursting dynamics and has received widespread attention. A memristor-based Jerk system with parametric excitation is proposed in this study. As the selected excitation frequency is far less than the natural frequency, implying the existence of an order gap between the excitation frequency and the natural one, the system can be considered as a classic fast-slow system with two timescales. In our system, when the slow-varying parameters periodically pass through the critical pitchfork bifurcation point periodically, a distinct time delay behavior can be observed. Complex bursting oscillations induced by the delayed pitchfork are revealed with different excitation amplitudes. By virtue of the fast-slow analysis method, the corresponding generation mechanisms are discussed by the transformed phase portraits, the time series, and the phase portraits. As the delay time interval induced by the pitchfork bifurcation is dependant not only on the excitation amplitude, but also on the excitation frequency, some excitation frequency related bursting patterns are also considered in our study. Finally, numerical simulations are provided to verify the validity of the study.     

相位噪声诱发神经放电的单次或两次相干共振

神经元电活动可以从静息通过Hopf分岔到放电,放电频率有固定周期;也可以从静息通过鞍-结分岔到放电,放电频率接近零.在具有周期性的相位噪声作用下的Hopf分岔和鞍-结分岔点附近,都会产生相干共振.噪声的周期小于Hopf分岔点附近的放电的周期时,相位噪声可以引起神经系统产生一次相干共振,位于系统内在的固有频率附近;噪声的周期大于系统的固有周期时,相位噪声可以引起双共振,对应低噪声强度的共振产生在噪声频率附近,对应高噪声强度的共振产生在系统的固有频率附近;并对双共振的产生原因进行了解释.在鞍-结分岔点附近,无论噪声的周期是大是小,都只会引起一次共振,研究结果不仅揭示了相位噪声作用下平衡点分岔点相干共振的动力学特性和对应于两类分岔的两类神经兴奋性的差别,还对近期的相位噪声诱发产生单或双共振的不同研究结果给出了解释.     

窄带随机噪声作用下单自由度非线性干摩擦系统的响应

研究了单自由度非线性干摩擦系统在窄带随机噪声参数激励下的主共振响应问题.用Krylov-Bogoliubov平均法得到了关于慢变量的随机微分方程.在没有随机扰动情形,得到了系统响应幅值满足的代数方程.在有随机扰动情形,用线性化方法和矩方法给出了系统响应稳态矩计算的近似计算公式.讨论了系统阻尼项、非线性项、随机扰动项和干摩擦项等参数对于系统响应的影响.理论计算和数值模拟表明,当非线性强度增大时系统的响应显著变小,系统分岔点滞后;随着激励频率的增大系统响应变大,而当激励频率小于一定的值时,系统响应为零;增加干 关键词： 单自由度非线性干摩擦系统 主共振响应 Krylov-Bogoliubov平均法     

The dynamics of a memristor-based Rulkov neuron with fractional-order difference

The exploration of the memristor model in the discrete domain is a fascinating hotspot. The electromagnetic induction on neurons has also begun to be simulated by some discrete memristors. However, most of the current investigations are based on the integer-order discrete memristor, and there are relatively few studies on the form of fractional order. In this paper, a new fractional-order discrete memristor model with prominent nonlinearity is constructed based on the Caputo fractional-order difference operator. Furthermore, the dynamical behaviors of the Rulkov neuron under electromagnetic radiation are simulated by introducing the proposed discrete memristor. The integer-order and fractional-order peculiarities of the system are analyzed through the bifurcation graph, the Lyapunov exponential spectrum, and the iterative graph. The results demonstrate that the fractional-order system has more abundant dynamics than the integer one, such as hyper-chaos, multi-stable and transient chaos. In addition, the complexity of the system in the fractional form is evaluated by the means of the spectral entropy complexity algorithm and consequences show that it is affected by the order of the fractional system. The feature of fractional difference lays the foundation for further research and application of the discrete memristor and the neuron map in the future.     

Complex dynamical behavior and chaos control in fractional-order Lorenz-like systems

In this paper, the complex dynamical behavior of a fractional-order Lorenz-like system with two quadratic terms is investigated. The existence and uniqueness of solutions for this system are proved, and the stabilities of the equilibrium points are analyzed as one of the system parameters changes. The pitchfork bifurcation is discussed for the first time, and the necessary conditions for the commensurate and incommensurate fractional-order systems to remain in chaos are derived. The largest Lyapunov exponents and phase portraits are given to check the existence of chaos. Finally, the sliding mode control law is provided to make the states of the Lorenz-like system asymptotically stable. Numerical simulation results show that the presented approach can effectively guide chaotic trajectories to the unstable equilibrium points.     

非线性动力系统分岔点邻域内随机共振的特性

研究了叉形分岔系统和FitzHugh Nagumo(FHN)细胞模型两种非线性动力系统分岔点邻域内 随机共振的特性.研究结果表明：这两种系统在分岔发生时具有由一个吸引子变为两个吸引 子或者由两个吸引子变为一个吸引子共同的分岔特性，即在分岔点的邻域内, 系统在分岔点 的两侧有分岔前吸引子和分岔后吸引子存在，在噪声的作用下，系统的运动除了像传统随机 共振的机理那样在分岔点一侧共存的吸引子之间跃迁，还在分岔点两侧三个吸引子（分岔前 一个吸引子和分岔后两个吸引子）之间跃迁，并且这种跃迁单独诱发了随机共振 ；在两种 跃迁都发生的情况下, 在其分岔点的邻域内，由第二种跃迁诱发的随机共振在引起第一种跃 迁噪声的强度很大的范围内变化仍可维持, 而第一种跃迁诱发的随机共振在引起第二种跃迁 噪声的强度很小的范围内变化即迅速消失. 关键词： 随机共振 吸引子 分岔点 跃迁     

分数阶Newton-Leipnik系统的动力学分析

依据分数阶线性系统的稳定性理论,研究了具有双重混沌吸引子的Newton-Leipnik系统取不同分数阶时的动力学行为.研究表明该系统具有逆向Hopf分岔过程,即随着阶数的下降,分数阶Newton-Leipnik系统由双重混沌吸引子突变为单吸引子,其动力学行为将由混沌态历经短暂的周期态后收敛于稳定的平衡点.     

Vibrational resonance in fractional-order overdamped multistable systems

We investigated the vibrational resonance in fractional-order overdamped multistable systems theoretically and numerically. For a given fractional order p, our results show that the response amplitude exhibits a series of peaks as the frequency or the amplitude of the high frequency input varies. However, when the low-frequency input increases, the response amplitude exhibits unimodal resonance for 1?<?p?<?2. Additionally, for different values of p, whether the response amplitude changes monotonically depends on the degree of spatial potential asymmetry. The mechanism by which p affects the resonance behavior is analyzed. Our results indicate that the value of p affects the resonance behavior by either altering the conditions under which the response amplitude attains its extreme values, or by altering the oscillating frequency of the response amplitude with varying systematic parameters.     

带有分数阶阻尼的压电能量采集系统相干共振

研究了含分数阶阻尼的双稳态能量采集系统的相干共振. 建立了带有分数阶阻尼的轴向受压梁压电能量采集系统动力学模型. 对于分数阶方程, 采用Euler-Maruyama-Leipnik方法进行求解, 计算了不同阻尼阶数下的能量采集系统的信噪比、响应均值、跃迁数目等统计物理量. 结果表明: 此压电能量采集系统在随机激励下可以实现相干共振, 阻尼阶数对相干共振的临界噪声强度和相干共振幅值有很大影响. 关键词： 分数阶阻尼 随机激励 能量采集系统 相干共振