New chaos generators

Originating from a sinusoidal oscillator circuit that employs the current feedback op amp (CFOA) as the active building block, a new chaos generator that utilizes a discrete nonlinear device of asymmetrical nonlinearity is proposed. Two different positions of the nonlinearity within the circuit structure are investigated. A slight modification incorporating the addition of a single resistor to the generator is shown to facilitate its tuning and affect the dynamics of its chaotic behavior. Experimental laboratory results agree well with PSpice simulations, and both are included.     

Impulsive periodic and quasi-periodic orbits of coupled oscillators with essential stiffness nonlinearity

We study the impulsive responses of a grounded linear oscillator coupled to a light nonlinear attachment through an essentially nonlinear (nonlinearizable) stiffness. We analyze the periodic and quasi-periodic dynamics of the undamped system forced by a single impulse on the linear oscillator and being initially at rest, by considering separately low-, moderate- and high-energy impulsive motions. The motivation for studying the impulsive dynamics of this system centers on passive targeted energy transfer properties of the corresponding weakly damped one, that is, of the possibility of one-way, irreversible transfer of energy from the linear oscillator to the nonlinear attachment. A rather surprising aspect of this work is the complexity of the analysis required to study the impulsive dynamics of this system, due to its high degeneracy, as it undergoes a co-dimension three bifurcation.     

Reflection of short rectangular pulses in the ideal string attached to strongly nonlinear oscillator

The system under consideration is ideal elastic string attached to strongly nonlinear oscillator with cubic nonlinearity by two different ways – immediately and by weak linear spring. The reflection of short rectangular pulses from the oscillator is accompanied by excitation of vibrations. The type of mode excited determines the amount of energy transferred to the oscillator as well as the structure of the reflected wave.     

Anharmonic oscillator systems

The anharmonic oscillator is solved quickly, easily, and elegantly by Adomian's methods for solution of nonlinear stochastic differential equations emphasizing its applicability to nonlinear deterministic equations as well as stochastic equations. No difficulty is encountered in treating the case of the forced anharmonic oscillator or the stochastic case or of any nonlinear oscillating system such as the Duffing or Van der Pol oscillators, for example, with coefficients, as well as forcing functions, which are stochastic processes, since statistical separability is inherent in the Adomian method.     

Discontinuous dynamics of a non-linear, self-excited, friction-induced, periodically forced oscillator

The discontinuous dynamics of a non-linear, friction-induced, periodically forced oscillator is studied. The analytical conditions for motion switchability at the velocity boundary in such a nonlinear oscillator are developed to understand the motion switching mechanism. Using such analytical conditions of motion switching, numerical predictions of the switching scenarios varying with excitation frequency and amplitude are carried out, and the parameter maps for specific periodic motions are presented. Chaotic and periodic motions are illustrated through phase planes and switching sections for a better understanding of motion mechanism of the nonlinear friction oscillator. The periodic motions with switching in such a nonlinear, frictional oscillator cannot be obtained from the traditional analysis (e.g., perturbation and harmonic balance method).     

Exponential averaging for Hamiltonian evolution equations

We derive estimates on the magnitude of non-adiabatic interaction between a Hamiltonian partial differential equation and a high-frequency nonlinear oscillator. Assuming spatial analyticity of the initial conditions, we show that the dynamics can be transformed to the uncoupled dynamics of an infinite-dimensional Hamiltonian system and an anharmonic oscillator, up to coupling terms which are exponentially small in a certain power of the frequency of the oscillator. The result is derived from an abstract averaging theorem for infinite-dimensional analytic evolution equations in Gevrey spaces. Refining upon a similar result by Neishtadt for analytic ordinary differential equations, the temporal estimate crucially depends on the spatial regularity of the initial condition. The result shows to what extent the strong resonances between rapid forcing and highly oscillatory spatial modes can be suppressed by the choice of sufficiently smooth initial data. An application is provided by a system of nonlinear Schrödinger equations, coupled to a rapidly forcing single mode, representing small-scale oscillations. We provide an example showing that the estimates for partial differential equations we derive here are necessarily different from those in the context of ordinary differential equations.

     

Solving the nonlinear equations for one-dimensional nano-sized model including Rydberg and Varshni potentials and Casimir force using the decomposition method

The Adomian decomposition method has been applied to solve the nonlinear equations from the one-dimensional model for a nano-sized-oscillator. The model includes Rydberg and Varshni potentials as well as Casimir force with fractional damping. New approximate solution of the equations of motion for anharmonic vibrations of a nano-sized oscillator with the elastic force deriving from the Rydberg and Varshni potentials has been obtained.     

附加非线性振子的双稳态电磁式振动能量捕获器动力学特性研究

随着微机电科技的进步,利用环境振动进行系统自供电已经成为目前非线性动力学研究的热点.将质量-弹簧-阻尼系统与双稳态振动能量捕获系统相结合,提出了附加非线性振子的双稳态电磁式振动能量捕获器,建立系统的力学模型及控制方程.通过数值仿真研究了简谐激励下质量比和调频比发生变化时附加非线性振子的双稳态电磁式振动能量捕获器的动力学响应.通过与附加线性振子双稳态系统的对比,获得了上述参数对附加非线性振子的双稳态电磁式振动能量捕获器发生大幅运动的影响规律,显示出附加非线性振子的双稳态电磁式振动能量捕获器的优越性,并获得了附加非线性振子的双稳态电磁式振动能量捕获器发生连续大幅混沌运动的最优参数配合.上述研究结果为双稳态电磁式振动能量捕获系统的相关研究提供了理论基础.     

Nonlinear vibration of a quarter-car model excited by the road surface profile

We examine the Melnikov criterion for a transition to chaos in case of a single–degree–of–freedom nonlinear oscillator with the Duffing potential with a nonlinear hard stiffness and a kinematic excitation term caused by the road profile. Using the new effective Hamiltonian we have examined appearance of homoclinic orbits in a quarter car model. Cross–sections of stable and unstable manifolds defined the condition of transition to chaos through a homoclinic bifurcation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some thermodynamic features of ideal systems with nonlinear interaction

The thermodynamic properties of two two-dimensional systems are investigated comparatively. One system corresponds to a bound state of a two-dimensional harmonic oscillator, and the other corresponds to a nonlinear anharmonic oscillator. The spectrum for the harmonic oscillator is presented; for the nonlinear oscillator, the WKB spectrum and the wave functions are first calculated. The Ω-potential behavior is essentially different at high temperatures. This difference may possibly be observed by investigating the temperature dependence of the heat capacity in planar anisotropic media. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 116, No. 3, pp. 431–441, September, 1998.     

耦合Van der Pol-Duffing振子的强共振分叉解

本文用多尺度方法研究了一非线性耦合Ｖａｒ ｄｅｒ Ｐｏｌ－Ｄｕｆｆｉｎｇ振子在强共振情形下的分叉解,研究表明,当分叉参数取不同值时,此系统将出现单个振子的周期运动、两个振子的锁频分叉运动和拟周期分叉运动,同时,本文也给出一些数值结果,以验证理论的正确性。     

Suppression of hysteresis in a forced van der Pol–Duffing oscillator

This paper examines the suppression of hysteresis in a forced nonlinear self-sustained oscillator near the fundamental resonance. The suppression is studied by applying a rapid forcing on the oscillator. Analytical treatment based on perturbation analysis is performed to capture the entrainment zone, the quasiperiodic modulation domain and the hysteresis area as well. The analysis leads to a strategy for the suppression of hysteresis occurring between 1:1 frequency-locked motion and quasiperiodic response. This hysteresis suppression causes the disappearance of nonlinear effects leading to a smooth transition between the quasiperiodic and the frequency-locked responses. Specifically, it appears that a rapid forcing introduces additional apparent nonlinear stiffness which can effectively suppress hysteresis in a certain range of the rapid excitation frequency. This work was motivated by the important issue of controlling and eliminating hysteresis often undesirable in mechanical systems, in general, and in application to microscale devices, especially.     

Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach

In this work, the Hamiltonian approach is applied to obtain the natural frequency of the Duffing oscillator, the nonlinear oscillator with discontinuity and the quintic nonlinear oscillator. The Hamiltonian approach is then extended to the second and third orders to find more precise results. The accuracy of the results obtained is examined through time histories and error analyses for different values for the initial conditions. Excellent agreement of the approximate frequencies and the exact solution is demonstrated. It is shown that this method is powerful and accurate for solving nonlinear conservative oscillatory systems.     

Generating the periodic solutions for forcing van der Pol oscillators by the Iteration Perturbation method

In this paper, the iteration perturbation method proposed by He [J.H. He, Non-perturbative methods for strongly nonlinear problems, Dissertation. de-Verlag im Internet GmbH, 2006; J.H. He, Limit cycle and bifurcation of nonlinear problems, Chaos Solitons Fractals 26 (2005) 827–833] is used to generate periodic solutions of van der Pol oscillator with a forcing term, forcing oscillator with quadratic type damping and van der Pol oscillator with excitation term. The comparison of the obtained results verifies its convenience and effectiveness.     

Frequency–energy plots of steady-state solutions for forced and damped systems,and vibration isolation by nonlinear mode localization

We study the structure of the periodic steady-state solutions of forced and damped strongly nonlinear coupled oscillators in the frequency–energy domain by constructing forced and damped frequency – energy plots (FEPs). Specifically, we analyze the steady periodic responses of a two degree-of-freedom system consisting of a grounded forced linear damped oscillator weakly coupled to a strongly nonlinear attachment under condition of 1:1 resonance. By performing complexification/averaging analysis we develop analytical approximations for strongly nonlinear steady-state responses. As an application, we examine vibration isolation of a harmonically forced linear oscillator by transferring and confining the steady-state vibration energy to the weakly coupled strongly nonlinear attachment, thereby drastically reducing its steady-state response. By comparing the nonlinear steady-state response of the linear oscillator to its corresponding frequency response function in the absence of a nonlinear attachment we demonstrate the efficacy of drastic vibration reduction through steady-state nonlinear targeted energy transfer. Hence, our study has practical implications for the effective passive vibration isolation of forced oscillators.     

Temporal and spectral responses of a softening Duffing oscillator undergoing route-to-chaos

Because nonlinear responses are oftentimes transient and consist of complex amplitude and frequency modulations, linearization would inevitably obscure the temporal transition attributable to the nonlinear terms, thus also making all inherent nonlinear effects inconspicuous. It is shown that linearization of a softening Duffing oscillator underestimates the variation of the frequency response, thereby concealing the underlying evolution from bifurcation to chaos. In addition, Fourier analysis falls short of capturing the time evolution of the route-to-chaos and also misinterprets the corresponding response with fictitious frequencies. Instantaneous frequency along with the empirical mode decomposition is adopted to unravel the multi-components that underlie the bifurcation-to-chaos transition, while retaining the physical features of each component. Through considering time and frequency responses simultaneously, a better understanding of the particular Duffing oscillator is achieved.     

On the motion of an oscillator with a periodically time-varying mass

The stability of the motion of an oscillator with a periodically time-varying mass is under consideration. The key idea is that an adequate change of variables leads to a newtonian equation, where classical stability techniques can be applied: Floquet theory for the linear oscillator, KAM method in the nonlinear case. To illustrate this general idea, first we have generalized the results of [W.T. van Horssen, A.K. Abramian, Hartono, On the free vibrations of an oscillator with a periodically time-varying mass, J. Sound Vibration 298 (2006) 1166–1172] to the forced case; second, for a weakly forced Duffing’s oscillator with variable mass, the stability in the nonlinear sense is proved by showing that the first twist coefficient is not zero.     

Iterative reproducing kernel method for nonlinear oscillator with discontinuity

In this paper, iterative reproducing kernel method is applied to obtain the analytical approximate solution of a nonlinear oscillator with discontinuities. The solution obtained by using the method takes the form of a convergent series with easily computable components. An illustrative example is given to demonstrate the effectiveness of the present method. The results obtained using the scheme presented here show that the numerical scheme is very effective and convenient for the nonlinear oscillator with discontinuities.     

具调和振子的非线性Schrodinger方程

考虑具调和振子的非线性Schrodinger方程的Cauchy问题，采用Galerkin方法证明了整体强解的存在性，使用能量估计方法证明了整体强解的唯一性。