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Chaos and chaos control in MEMS resonators under power law noise

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arXiv 2019年

作者： Wang, Yunfang Lei, Youming Department of Applied Mathematics Northwestern Polytechnical University Xi'an710072 China MIIT Key Laboratory of Dynamics and Control of Complex Systems Northwestern Polytechnical University Xi'an710072 China

The chaotic dynamics and its control under power law noise in Micro-electromechanical systems (MEMS) resonators with electrostatic excitation are probed. On the basis of the stochastic Melnikov method in the mean-square sense and the mean largest Lyapunov exponent, the threshold value of power law noise intensity for the onset of chaos is obtained analytically and numerically. We show that the threshold of noise intensity decreases with the increasing of the frequency exponent of power law noise in the parameter space. Numerical simulations, such as phase diagram and time history, are employed to verify the results acquired by the stochastic Melnikov method. Inspired by the analytical results, a time-delay feedback control algorithm is proposed for controlling the chaotic motion in the resonators. The effectiveness of this controller is certificated by the above numerical method. Copyright © 2019, The Authors. All rights reserved.

关键词： Stochastic systems

Rate-dependent bifurcation dodging in a thermoacoustic system driven by colored noise

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arXiv 2020年

作者： Zhang, Xiaoyu Xu, Yong Liu, Qi Kurths, Jürgen Grebogi, Celso School of Mathematics and Statistics Northwestern Polytechnical University Xi’an710072 China MIIT Key Laboratory of Dynamics and Control of Complex Systems Northwestern Polytechnical University Xian710072 China Potsdam Institute for Climate Impact Research Potsdam14412 Germany Institute for Complex Systems and Mathematical Biology School of Natural and Computing Sciences Kings College University of Aberdeen AberdeenAB24 3UE United Kingdom

Tipping in multistable systems occurs usually by varying the input slightly, resulting in the output switching to an often unsatisfactory state. This phenomenon is manifested in thermoacoustic systems. This thermoacoustic instability may lead to the disintegration of rocket engines, gas turbines and aeroengines, so it is necessary to design control measures for its suppression. It was speculated that such unwanted instability states may be dodged by changing quickly enough the bifurcation parameters. Thus, in this work, based on a fundamental mathematical model of thermoacoustic systems driven by colored noise, the corresponding Fokker-Planck-Kolmogorov equation of the amplitude is derived by using a stochastic averaging method. A transient dynamical behavior is identified through a probability density analysis. We find that the rate of change of parameters and the correlation time of the noise are helpful to dodge thermoacoustic instability, while a relatively large noise intensity is a disadvantageous factor. In addition, power-law relationships between the maximum amplitude and the noise parameters are explored, and the probability of successfully dodging a thermoacoustic instability is calculated. These results serve as a guide to the design of engines and to propose an effective control strategy, which is of great significance to aerospace-related fields. Copyright © 2020, The Authors. All rights reserved.

关键词： White noise

Averaging principles for non-autonomous two-time-scale stochastic reaction-diffusion equations with polynomial growth

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arXiv 2019年

作者： Wang, Ruifang Xu, Yong Pei, Bin Department of Applied Mathematics Northwestern Polytechnical University Xi'an710072 China MIIT Key Laboratory of Dynamics and Control of Complex Systems Northwestern Polytechnical University Xi'an710072 China School of Mathematical Sciences Fudan University Shanghai200433 China

In this paper, we develop the averaging principle for a class of two-time-scale stochastic reaction-diffusion equations driven by Wiener processes and Poisson random measures. We assume that all coefficients of the equation have polynomial growth, and the drift term of the equation is non-Lipschitz. Hence, the classical formulation of the averaging principle under the Lipschitz condition is no longer available. To prove the validity of the averaging principle, the existence and uniqueness of the mild solution are proved firstly. Then, the existence of time-dependent evolution family of measures associated with the fast equation is studied, by which the averaged coefficient is obtained. Finally, the validity of the averaging principle is *** Codes 70K70, 60H15, 34K33, 37B55, 60J75 Copyright © 2019, The Authors. All rights reserved.

关键词： Stochastic systems

Transport and escape in a deformable channel driven by fractional Gaussian noise

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Physical Review E 2019年第2期100卷 022114-022114页

作者： Ruoxing Mei Yong Xu Jürgen Kurths Department of Applied Mathematics Northwestern Polytechnical University Xi'an 710072 China. Department of Applied Mathematics Northwestern Polytechnical University Xi'an 710072 China and MIIT Key Laboratory of Dynamics and Control of Complex Systems Northwestern Polytechnical University Xi'an 710072 China. Potsdam Institute for Climate Impact Research Potsdam 14412 Germany.

Fractional Gaussian noise (FGN) with the Hurst exponent H is an important tool to model various phenomena in biophysical systems, like subdiffusion in a single protein molecule. Considering that there also exists a confined structure which can be modeled as a channel in these systems, transport and escape driven by FGN in a deformable channel are investigated in this paper. By calculating the mean velocity, and the mean first passage time (MFPT) for crossing the nearest bottleneck and the probability distribution of the final position, effects of FGN and channel structure on the system dynamics are illustrated. Our results indicate that FGN has a complex influence mechanism under different combinations of H and the noise intensity. For a persistence case (H>0.5), the mean velocity decreases but MFPT increases with the increase of the noise intensity and H. While for an antipersistence case (H<0.5), when H is small, the relationships among the mean velocity, MFPT and the noise intensity are exactly the opposite to persistence cases. When H has a large value, the mean velocity tends to first decrease and then increase. Moreover, effects of the bottleneck and channel asymmetry are investigated. It is shown that a small H and a large channel width can lead to a large mean velocity and fast crossing. Besides, a channel asymmetry can affect the system dynamics by inducing asymmetric structure and adjusting the width of bottleneck. However, the effect of the bottleneck is the main factor. Therefore, a combination of channel with wide bottleneck and FGN in an antipersistence regime is the optimal choice to promote the transport and escape. These results provide a basis for the explanation of molecular activity in living organisms and the design of particle mixture separators.

关键词： Noise Particle phenomena Transport phenomena

Exploring the impact of core expansion on the vortex distribution in superconducting–normal-metal hybrid nanostructures

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Physical Review B 2019年第5期100卷 054519-054519页

作者： Ritika Panghotra Matias Timmermans Cun Xue Bart Raes Victor Moshchalkov Joris Van de Vondel Laboratory of Solid State Physics and Magnetism Department of Physics and Astronomy KU Leuven Celestijnenlaan 200D B–3001 Leuven Belgium Catalan Institute of Nanoscience and Nanotechnology (ICN2) CSIC and BIST Campus UAB Bellaterra 08193 Barcelona Spain School of Mechanics Civil Engineering and Architecture and MIIT Key Laboratory of Dynamics and Control of Complex Systems Northwestern Polytechnical University Xi'an 710072 People's Republic of China

The superconducting condensate in a bilayer normal-metal–superconducting film has a nonuniform dependence throughout the layer. For sufficiently thin normal metallic films, the superconducting correlations induced by the contacting superconducting layer result in the formation of a minigap and the characteristic length scales governing the response of the condensate will be different. In this work we use scanning tunneling spectroscopy to visualize the vortex states as a function of the applied magnetic field in a Au-MoGe nanostructure. By comparing the obtained zero-bias conductance maps with time-dependent Ginzburg-Landau simulations, we directly confirm that the observed vortex distributions can only be explained by taking into account the impact of the normal metallic layer. We illustrate this impact on the vortex state for two lithographically fabricated mesoscopic bilayer structures containing an identical antidot array but having different lateral sizes.

关键词： Coherence length Methods in superconductivity Proximity effect Vortices in superconductors

First-passage-time distribution in a moving parabolic potential with spatial roughness

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Physical Review E 2019年第5期99卷 052203-052203页

作者： Yongge Li Yong Xu Jürgen Kurths Center for Mathematical Sciences & School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Department of Applied Mathematics Northwestern Polytechnical University Xi'an 710072 China MIIT Key Laboratory of Dynamics and Control of Complex Systems Northwestern Polytechnical University Xi'an 710072 China Potsdam Institute for Climate Impact Research Potsdam 14412 Germany Human and Animal Physiology Department Saratov State University Saratov 410000 Russia

In this paper, we investigate the first-passage-time distribution (FPTD) within a time-dependent parabolic potential in the presence of roughness with two methods: the Kramers theory and a nonsingular integral equation. By spatially averaging, the rough potential is equivalent to the combination of an effective smooth potential and an effective diffusion coefficient. Based on the Kramers theory, we first obtain Kramers approximations (KAs) of FPTD for both smooth and rough potentials. As expected, KA is valid only for high barriers and small external forces, and generally applicable for high barriers in rough potentials. To overcome the shortcoming of KA, a probability asymptotic approximation (PAA) based on an integral equation is proposed, which uses the transient probability density function (PDF) of the natural boundary conditions instead of the absorbing boundary conditions. We find that PAA fits very well even for large external forces. This enables us to analytically solve the FPTD for large external forces and low barriers as a strong extension to KA. In addition, we show that in the presence of a rough potential, the PAA of FPTD is in good agreement with numerical simulations for low barrier potentials. The PAA makes it possible to investigate the first-passage problem with ultrafast varying potentials and short exiting time. Thus, KA and PAA are complementary in determining the FPTD both for various barriers and external forces. Finally, the mean first-passage time (MFPT) is studied, which illustrates that the PAA of MFPT is effective almost in the whole range of external forces, while the KA of MFPT is valid only for small external forces.

关键词： Chaos & nonlinear dynamics First passage problems Fokker–Planck equation

Metamorphosis of discontinuity lines and rectification of magnetic flux avalanches in the presence of noncentrosymmetric pinning forces

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Physical Review B 2021年第22期103卷 224514-224514页

作者： M. Motta L. Burger Lu Jiang J. D. González Acosta Ž. L. Jelić F. Colauto W. A. Ortiz T. H. Johansen M. V. Milošević C. Cirillo C. Attanasio Cun Xue A. V. Silhanek B. Vanderheyden Departamento de Física Universidade Federal de São Carlos 13565-905 São Carlos São Paulo Brazil Montefiore Research Unit Department of Electrical Engineering and Computer Science Université de Liège 4000 Sart Tilman Belgium School of Aeronautics Northwestern Polytechnical University Xi'an 710072 China Facultad de Ingeniería Universidad del Magdalena Santa Marta Colombia Departement Fysica Universiteit Antwerpen Groenenborgerlaan 171 B-2020 Antwerpen Belgium Nokia Bell Labs Antwerp Belgium Department of Physics University of Oslo POB 1048 Blindern 0316 Oslo Norway CNR-SPIN c/o Università degli Studi di Salerno I-84084 Fisciano (Sa) Italy Dipartimento di Fisica “E.R. Caianiello” Università degli Studi di Salerno I-84084 Fisciano (Sa) Italy School of Mechanics Civil Engineering and Architecture and MIIT Key Laboratory of Dynamics and Control of Complex Systems Northwestern Polytechnical University Xi'an 710072 China Experimental Physics of Nanostructured Materials Q-MAT CESAM Université de Liège B-4000 Sart Tilman Belgium

Considering a noncentrosymmetric pinning texture composed of a square array of triangular holes, the magnetic flux penetration and expulsion are investigated experimentally and theoretically. A direct visualization of the magnetic landscape obtained using a magneto-optical technique on a Nb film is complemented by a multiscale numerical modeling. This combined approach allows the magnetic flux dynamics to be identified from the single flux quantum limit up to the macroscopic electromagnetic response. Within the theoretical framework provided by time-dependent Ginzburg-Landau simulations, an estimation of the in-plane current anisotropy is obtained and its dependence with the radius of the curvature of hole vertices is addressed. These simulations show that current crowding plays an important role in channeling the flux motion, favoring hole-to-hole flux hopping rather than promoting interstitial flux displacement in between the holes. The resulting anisotropy of the critical current density gives rise to a distinct pattern of discontinuity lines for increasing and decreasing applied magnetic fields, in sharp contrast to the invariable patterns reported for centrosymmetric pinning potentials. This observation is partially accounted for by the rectification effect, as demonstrated by finite-element modeling. At low temperatures, where magnetic field penetration is dominated by thermomagnetic instabilities, highly directional magnetic flux avalanches with a fingerlike shape are observed to propagate along the easy axis of the pinning potential. This morphology is reproduced by numerical simulations. Our findings demonstrate that anisotropic pinning landscapes and, in particular, ratchet potentials produce subtle modifications to the critical state field profile that are reflected in the distribution of discontinuity lines.

关键词： Flux pinning Magneto-optical effect Monolayer films Superconductors Finite-element method Landau-Ginzburg theory

Solving Fokker-Planck equation using deep learning

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arXiv 2019年

作者： Xu, Yong Zhang, Hao Li, Yongge Zhou, Kuang Liu, Qi Kurths, Jürgen Department of Applied Mathematics Northwestern Polytechnical University Xi’an710072 China MIIT Key Laboratory of Dynamics and Control of Complex Systems Northwestern Polytechnical University Xi’an710072 China Department of Engineering Mechanics Northwestern Polytechnical University Xi’an710072 China Center for Mathematical Sciences School of Mathematics and Statistics HuazhongUniversity of Science and Technology Wuhan430074 China Potsdam Institute for Climate Impact Research Potsdam14412 Germany Department of Physics Humboldt University Berlin Berlin12489 Germany

The probability density function of stochastic differential equations is governed by the Fokker-Planck (FP) equation. A novel machine learning method is developed to solve the general FP equations based on deep neural networks. The proposed algorithm does not require any interpolation and coordinate transformation, which is different from the traditional numercial methods. The main novelty of this paper is that penalty factors are introduced to overcome the local optimization for the deep learning approach, and the corresponding setting rules are given. Meanwhile, we consider a normalization condition as a supervision condition to effectively avoid that the trial solution is zero. Several numerical examples are presented to illustrate performances of the proposed algorithm, including one- and two-dimensional systems. All the results suggest that the deep learning is quite feasible and effective to calculate the FP equation. Further, influences of the number of hidden layers, the penalty factors, and the optimization algorithm are discussed in detail. These results indicate that the performances of the machine learning technique can be improved through constructing the neural networks appropriately. Copyright © 2019, The Authors. All rights reserved.

关键词： Deep neural networks

Convergence of martingale solutions to the hybrid slow-fast system

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arXiv 2018年

作者： Xu, Yong Yang, Xiaoyu Pei, Bin Bai, Yuzhen School of Mathematics and Statistics Northwestern Polytechnical University Xi’an710072 China MIIT Key Laboratory of Dynamics and Control of Complex Systems Northwestern Polytechnical University Xi’an710072 China School of Mathematical Sciences Qufu Normal University Qufu273165 China

This paper is devoted to studying the weak convergence for a slow-fast system with jumps modulated by Markovian switching regimes with the martingale method. However, due to the coexistence of fast component and Markovian switching regimes, the martingale method and perturbed test functions can not be applied directly. In this situation, a combination of perturbed test functions and the time discretization is applied efficiently. And the choice of appropriate perturbed test functions, which are related to the averaged coefficients, plays a decisive role. Our results also cover the case of slow-fast system without Markovian switching regimes. Finally, some examples are presented, and numerical simulations are carried out to observe a good *** Codes 70K70, 60J25, 60G51 Copyright © 2018, The Authors. All rights reserved.

关键词： Markov processes