In recent papers by the authors (S.~Motonaga and K.~Yagasaki, Obstructions to integrability of nearly integrable dynamical systems near regular level sets, submitted for publication, and K.~Yagasaki, Nonintegrability of nearly integrable dynamical systems near resonant periodic orbits, submitted for publication), two different techniques which allow us to prove the real-analytic or complex-meromorphic nonintegrability of forced nonlinear oscillators having the form of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems were provided. Here the concept of nonintegrability in the Bogoyavlenskij sense is adopted and the first integrals and commutative vector fields are also required to depend real-analytically or complex-meromorphically on the small parameter. In this paper we review the theories and continue to demonstrate their usefulness. In particular, we consider the periodically forced damped pendulum and prove its nonintegrability in the above meaning.
We prove that general three- or four-dimensional systems %of differential equations are real-analytically nonintegrable near degenerate equilibria in the Bogoyavlenskij sense under additional weak conditions when the Jacobian matrices have a zero and pair of purely imaginary eigenvalues or two incommensurate pairs of purely imaginary eigenvalues at the equilibria. For this purpose, we reduce their integrability to that of the corresponding Poincare-Dulac normal forms and further to that of simple planar systems, and use a novel approach for proving the analytic nonintegrability of planar syste…
In a recent paper by the author (K. Yagasaki, Nonintegrability of the restricted three-body problem, submitted for publication), a technique was developed for determining whether nearly integrable systems are not meromorphically Bogoyavlenskij-integrable such that the first integrals and commutative vector fields also depend meromorphically on the small parameter. Here we continue to demonstrate the technique for some classes of dynamical systems. In particular, we consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and discuss a relationship of the technique w…
Stability pockets of a periodically forced oscillator in a model for seasonality
在季节性模型中的周期性强迫振荡器在参数平面中显示稳定性口袋及其链。振荡器的频率和由0和1之间的值指示的季节是两个参数。本研究旨在作为Schmal等人在2015年的数值研究的理论补充。在他们的术语中，稳定性口袋或Arnol'd洋葱。我们构造了受迫振子的Poincaré映射，并证明了Arnol'd舌被一个有许多折叠的映射带入稳定口袋。在van der Pol \& Strutt于1928年发表的一篇文章中已经观察到了稳定性口袋，后来Broer \& Levi在1995年解释了这一点。
Periodically-forced finite networks of heterogeneous coupled oscillators: a low-dimensional approach
我们研究了一个由500个耦合的改进型van der Pol振荡器组成的网络。与每个振荡器相关联的参数的值从正态分布中得出，从而给出异构网络。对于足够强的耦合，所有振荡器都具有相同的周期，并且我们考虑网络在这种状态下的周期性强迫。通过利用振荡器的状态和它的参数的值之间的快速发展的相关性，我们得到一个近似的低维描述的系统在多项式混沌展开的前几个系数。然后，可以对这个低维系统进行标准的分叉分析，并且由此获得的结果很好地预测了随机参数的任何实现集的高维系统的行为。还讨论了该方法开始失败的情况。
Frequency control of singularly perturbed forced Duffing's oscillator
本文研究了Duffing型受迫奇摄动微分方程的动力学问题。我们解释了解的大频率非线性振动的出现。它表明，频率可以控制的一个小参数在最高的导数。本文推广了B.S.吴伟平、孙伟平及C.W. Lim，解析近似双阱Duffing振子在大振幅振荡中的应用，Journal of Sound and Vibration，Volume 307，Issues 3-5，（2007），pp. 953-960.提出了一种基于坐标动态变化的非线性振动分析新方法。
The paper revisits a well-known model of forced vibro-impact oscillator with Amonton-Coulomb friction. In vast majority of the existing studies, this model included also viscous friction, and its global dynamics in the state space is governed by periodic, quasiperiodic or chaotic attractors. We demonstrate that removal of the viscous friction leads to qualitative modification of the global dynamics. Namely, the state space is divided into the regions with "regular" attraction to the aforementioned special solutions, and the regions with profoundly Hamiltonian dynamics. The latter regions conta…
We study the jump phenomenon present in the forced asymmetric Duffing oscillator using the known steady-state asymptotic solution. The major result is the computation of the jump manifold, which encodes global information about all possible jumps.
We perform a detailed study of the dynamics of a nonlinear, one-dimensional oscillator driven by a periodic force under hysteretic damping, whose linear version was originally proposed and analyzed by Bishop in [1]. We first add a small quadratic stiffness term in the constitutive equation and construct the periodic solution of the problem by a systematic perturbation method, neglecting transient terms as $t\rightarrow \infty$. We then repeat the analysis replacing the quadratic by a cubic term, which does not allow the solutions to escape to infinity. In both cases, we examine the dependence …
One-shot learning for the complex dynamical behaviors of weakly nonlinear forced oscillators
State Key Lab of Disaster Reduction in Civil Engineering（灾害防治ivil工程国家重点实验室）；Department of Civil and Environmental Engineering（土木与环境工程系）；Guangxi Laboratory of Whole Life Safety for Land-Sea Corridor Engineering（广西陆海通道工程全生命周期安全实验室）
The ability of a circadian system to entrain to the 24-hour light-dark cycle is one of its most important properties. A new tool, called the entrainment map, was recently introduced to study this process for a single oscillator. Here we generalize the map to study the effects of light-dark forcing in a hierarchical system consisting of a central circadian oscillator that drives a peripheral circadian oscillator. We develop techniques to reduced the higher dimensional phase space of the coupled system to derive a generalized 2-D entrainment map. Determining the nature of various fixed points, t…
Distributed delays modeled by 'weak generic kernels' are introduced in the well-known coupled Landau-Stuart system, as well as a chaotic van der Pol-Rayleigh system with parametric forcing. The systems are close via the 'linear chain trick'. Linear stability analysis of the systems and conditions for Hopf bifurcation which initiates oscillations are investigated, including deriving the normal form at bifurcation, and deducing the stability of the resulting limit cycle attractor. The value of the delay parameter $a=a_{Hopf}$ at Hopf bifurcation picks out the onset of Amplitude Death(AD) in all …
We study the existence of real-analytic first integrals and real-analytic integrability for perturbations of integrable systems in the sense of Bogoyavlenskij including non-Hamiltonian ones. We especially assume that there exists a family of periodic orbits on a regular level set of the first integrals having a connected and compact component and give sufficient conditions for nonexistence of the same number of real-analytic first integrals in the perturbed systems as the unperturbed ones and for their real-analytic nonintegrability near the level set such that the first integrals and commutat…
