The variational theory of adiabatic motions,and its applications to linear and non-linear oscillators

This paper presents a variational derivation of the adiabatic periodic motion theorems and related time-integral-of-energy results, including the virial theorem, and some of their applications to linear and non-linear oscillators. Specifically, (i) first, the Maupertuis-Euler-Langrange (MEL) action principle is formulated for the most general (scleronomic and holonomic) system; in the derivation the time-dependent system parameters are treated just like additional generalized co-ordinates and subjected to similar variations; (ii) next, combination of MEL's principle with the first law of thermodynamics yields the adiabatic theorem; subsequent specializations of it lead to additional energetic equations; (iii) the theory is then applied to the one d.o.f. linear and non-linear oscillator; the effects of linear friction and of a harmonic external force are also discussed; useful relations for the adiabatically varying system parameters are thus obtained.     

Chirped-pulse oscillators: theory and experiment

Theory of chirped-pulse oscillators operating in the positive dispersion regime is presented. It is found that the chirped pulses can be described analytically as solitary pulse solutions of the nonlinear cubic-quintic complex Ginzburg–Landau equation. Due to the closed form of the solution, basic characteristics of the regime under consideration are easily traceable. Numerical simulations validate the analytical technique and the chirped-pulse stability. Experiments with 10 MHz Ti:Sa oscillator providing up to 150 nJ chirped pulses, which are compressible down to 30 fs, are in agreement with the theory. PACS 42.65.Re; 42.65.Tg; 42.55.Rz     

On the amplitude decay of strongly non-linear damped oscillators

The problem considered is that of the amplitude behavior of the strongly non-linear damped oscillator ü + 2 ?u? + u + αf(u) = 0 where ? is small, u and f(u) are O(1), α may be arbitrarily large, and f(u) is an odd function. A simple approximation to the amplitude decay is obtained by first linearizing the static terms and then using the WKB approximation. Of particular interest is the influence of the non-linearity on the amplitude decay. Several examples which illustrate this influence are provided.     

A simple theory of temperature-dependent energy levels: Application to He II

The statistical mechanics of interacting Bose quasi particles is considered for the case where energy levels are temperature-dependent and known from experiment. The results are applied to the thermodynamic properties of He II using neutron scattering data.     

耦合相振子系统同步的序参量理论

节律行为,即系统行为呈现随时间的周期变化,在我们的周围随处可见.不同节律之间可以通过相互影响、相互作用产生自组织,其中同步是最典型、最直接的有序行为,它也是非线性波、斑图、集群行为等的物理内在机制.不同的节律可以用具有不同频率的振子(极限环)来刻画,它们之间的同步可以用耦合极限环系统的动力学来加以研究.微观动力学表明,随着耦合强度增强,振子同步伴随着动力学状态空间降维到一个低维子空间,该空间由序参量来描述.序参量的涌现及其所描述的宏观动力学行为可借助于协同学与流形理论等降维思想来进行.本文从统计物理学的角度讨论了耦合振子系统序参量涌现的几种降维方案,并对它们进行了对比分析.序参量理论可有效应用于耦合振子系统的同步自组织与相变现象的分析,通过进一步研究序参量的动力学及其分岔行为,可以对复杂系统的涌现动力学有更为深刻的理解.     

Perturbation theory of odd anharmonic oscillators

We study the perturbation theory forH=p ²+x ²+x ²ⁿ⁺¹,n=1, 2, .... It is proved that when Im0,H has discrete spectrum. Any eigenvalue is uniquely determined by the (divergent) Rayleigh-Schrödinger perturbation expansion, and admits an analytic continuation to Im=0 where it can be interpreted as a resonance of the problem.Partially supported by G.N.F.M., C.N.R.     

Time dependent canonical perturbation theory II: Application to the Henon-Heiles system

In this communication, we report the results of the application of time-dependent perturbation theory to the Henon-Heiles system. We show that the predictions of the perturbation theory hold good for short times, and try to explain the increase of error in the predicted results with the increase in energy.     

Analytic approximations of statistical quantities and response of noisy oscillators

Nonlinear oscillators have been utilized in many contexts because they encompass a large class of phenomena. For a reduced phase oscillator model with weak noise forcing that is necessarily multiplicative, we provide analytic formulas for the stationary statistical quantities of the random period. This is an important quantity which we term ‘response’ (i.e., the spike times, instantaneous frequency in neuroscience, the cycle time in chemical reactions, etc.) that is often analytically intractable in noisy oscillator systems. The analytic formulas are accurate in the weak noise limit so that one does not have to numerically solve a time-varying Fokker-Planck equation. The steady-state and dynamic responses are also analyzed with deterministic forcing. A second order analytic formula is derived for the steady-state response, whereas the dynamic response with time-varying forcing is more complicated. We focus on the specific case where the forcing is sinusoidal and accurately capture the frequency response with an analytic approximation that is obtained with a rescaling of the equation. By utilizing various techniques in the weak noise regime, this work leads to a better understanding of how the random period of nonlinear oscillators are affected by multiplicative noise and external forcing. Comparisons of the asymptotic formulas with a full oscillator system confirm the qualitative accurateness of the theory.     

Analysis of non-linear autonomous conservative oscillators by a time transformation method

The problem considered is that of the free vibration of the class of strongly non-linear oscillators ü + mu + αf(u) = 0, where m = 1, 0 or ?1, f(u) is a non-linear function of the displacement u(t), and α is not small. Through a continuous stretching of the time t a new time T is found in whose domain the response is simple harmonic. This time transformation method allows the response u(t) and period τ to be calculated to any desired degree of accuracy; however, the primary objective is to develop reasonably simple, accurate approximations for the oscillators considered.     

A study of non-linear autoregressive processes: Differential theory

Summary It is shown how the general analytical expression of an autoregression, whose stationary solution has an arbitrary given form, can be obtained. The central-limit theorem is used to state a correspondence between autoregression and relevant diffusion equation which not only permits to give analytical form to the stationary distribution of a given autoregression, but also to obtain appropriate series expansions of its fundamental solution and the exact relaxation constants. The interest of the analysis in the context of Monte Carlo simulations of relaxation and steady-state processes is discussed. The procedure is illustrated by two examples of interest in the field of ionized gases.     

Synchronization of chaotic and nonchaotic oscillators: Application to bipolar disorder

In this Letter, we use a synchronization scheme on two bipolar disorder models consisting of a strong nonlinear system with multiplicative excitation and a nonlinear oscillator without parametric harmonic forcing. The stability condition following our control function is analytically demonstrated using the Lyapunov theory and Routh-Hurwitz criteria, we then have the condition for the existence of a feedback gain matrix. A convenient demonstration of the accuracy of the method is complemented by the numerical simulations from which we illustrate the synchronized dynamics between the two non-identical bipolar disorder patients.     

A study of non-linear autoregressive processes: Integral theory

Summary Autoregressions with constant parameters are special cases of a general iterative process which, in this paper, is given explicit form. No restriction is imposed to the process of variation of the random variable which is permitted to suffer both continuous variations and/or discrete jumps. As a consequence, the general autoregression assumes the same form of the iterative solution of an appropriate integral (master) equation. The procedure which leads to this conclusion is simple but not so direct and requires a special expedient that in this paper is discussed in detail.