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Journal articles on the topic 'Nonlinear oscillation'

Author: Grafiati

Published: 4 June 2021

Last updated: 18 February 2022

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1

Chon, Ki H., Ramakrishna Raghavan, Yu-Ming Chen, Donald J. Marsh, and Kay-Pong Yip. "Interactions of TGF-dependent and myogenic oscillations in tubular pressure." American Journal of Physiology-Renal Physiology 288, no. 2 (February 2005): F298—F307. http://dx.doi.org/10.1152/ajprenal.00164.2004.

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We have previously shown that there are two oscillating components in spontaneously fluctuating single-nephron blood flow obtained from Sprague-Dawley rats (Yip K-P, Holstein-Rathlou NH, and Marsh DJ. Am J Physiol Renal Physiol 264: F427–F434, 1993). The slow oscillation (20–30 mHz) is mediated by tubuloglomerular feedback (TGF), whereas the fast oscillation (100 mHz) is probably related to spontaneous myogenic activity. The fast oscillation is rarely detected in spontaneous tubular pressure because of its small magnitude and the fact that tubular compliance filters pressure waves. We detected myogenic oscillation superimposed on TGF-mediated oscillation when ambient tubular flow was interrupted. Two well-defined peaks are present in the mean power spectrum of stop-flow pressure (SFP) centering at 25 and 100 mHz ( n = 13), in addition to a small peak at 125–130 mHz. Bispectral analysis indicates that two of these oscillations (30 and 100 mHz) interact nonlinearly to produce the third oscillation at 125–130 mHz. The presence of nonlinear interactions between TGF and myogenic oscillations indicates that estimates of the relative contribution of each of these mechanisms in renal autoregulation need to account for this interaction. The magnitude of myogenic oscillations was considerably smaller in the SFP measured from spontaneously hypertensive rats (SHR, n = 13); consequently, nonlinear interactions were not observed with bispectral analysis. Reduced augmentation of myogenic oscillations in SFP of SHR might account for the failure in detecting nonlinear interactions in SHR.

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2

Zaitsev, V. V., and Ar V. Karlov. "NONLINEAR RESONANCE IN OSCILLATORY CIRCUIT WITH FRACTAL CAPACITY." Vestnik of Samara University. Natural Science Series 18, no. 6 (June 9, 2017): 136–42. http://dx.doi.org/10.18287/2541-7525-2012-18-6-136-142.

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A model of oscillation circuit containing a nonlinear fractal component of capacity is proposed. The diﬀerential equation of motion of fractional order for forced oscillations under the action of an external signal is obtained. An approximate analytical solution of the equation of motion is conducted by methods of equivalent linearization and slowly varying amplitudes. The amplitude-frequency and phase response of fractional oscillator with cubic nonlinearity are analyzed.

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3

ZHANG, JIANXIN, XIAODONG CHEN, and ANTHONY C. DAVIES. "LOOP GAIN AND ITS RELATION TO NONLINEAR BEHAVIOR AND CHAOS IN A TRANSFORMER-COUPLED OSCILLATOR." International Journal of Bifurcation and Chaos 14, no. 07 (July 2004): 2503–12. http://dx.doi.org/10.1142/s0218127404010783.

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A relationship between the loop gain and nonlinear behavior of a transformer-coupled oscillator is established in this paper. With increase of the loop gain, the system undergoes a series of changes in its dynamical behavior, i.e. no oscillation, near-sinusoidal, period-doubling, chaotic and squegging oscillations. It is expected that the approach of loop gain analysis can be universally applied to investigating the nonlinear behavior, especially chaos, in other oscillators.

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4

Bathellier, Brice, Alan Carleton, and Wulfram Gerstner. "Gamma Oscillations in a Nonlinear Regime: A Minimal Model Approach Using Heterogeneous Integrate-and-Fire Networks." Neural Computation 20, no. 12 (December 2008): 2973–3002. http://dx.doi.org/10.1162/neco.2008.11-07-636.

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Fast oscillations and in particular gamma-band oscillation (20–80 Hz) are commonly observed during brain function and are at the center of several neural processing theories. In many cases, mathematical analysis of fast oscillations in neural networks has been focused on the transition between irregular and oscillatory firing viewed as an instability of the asynchronous activity. But in fact, brain slice experiments as well as detailed simulations of biological neural networks have produced a large corpus of results concerning the properties of fully developed oscillations that are far from this transition point. We propose here a mathematical approach to deal with nonlinear oscillations in a network of heterogeneous or noisy integrate-and-fire neurons connected by strong inhibition. This approach involves limited mathematical complexity and gives a good sense of the oscillation mechanism, making it an interesting tool to understand fast rhythmic activity in simulated or biological neural networks. A surprising result of our approach is that under some conditions, a change of the strength of inhibition only weakly influences the period of the oscillation. This is in contrast to standard theoretical and experimental models of interneuron network gamma oscillations (ING), where frequency tightly depends on inhibition strength, but it is similar to observations made in some in vitro preparations in the hippocampus and the olfactory bulb and in some detailed network models. This result is explained by the phenomenon of suppression that is known to occur in strongly coupled oscillating inhibitory networks but had not yet been related to the behavior of oscillation frequency.

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5

Quek, Zhi Hao, Wei Khim Ng, Aik Hui Chan, and Choo Hiap Oh. "Nonlinear Dirac Neutrino Oscillations." EPJ Web of Conferences 240 (2020): 07010. http://dx.doi.org/10.1051/epjconf/202024007010.

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Neutrino oscillations are a possible way to probe beyond Standard Model physics. The propagation of Dirac neutrinos in a massive medium is governed by the Dirac equation modified with an effective Hamiltonian that de- pends on the number density of surrounding matter fields. At the same time, quantum nonlinearities may contribute to neutrino oscillations by further mod- ifying the Dirac equation. A possible nonlinearity is computationally studied using Mathematica at low energies. We find that the presence of a uniform, static background matter distribution may significantly alter the oscillation am- plitude and wavelength; the considered nonlinearity may further reduce both oscillation amplitude and wavelength. In addition, the presence of matter al- lows the effects of the nonlinearity to be more readily observed for the chosen background densities and neutrino energy.

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6

D.S, Dilip. "Oscillation and Non oscillation of Solutions of Generalized Nonlinear Difference Equation of Second Order." Journal of Computational Mathematica 3, no. 2 (December 30, 2019): 23–32. http://dx.doi.org/10.26524/cm51.

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7

Becker, E., W. J. Hiller, and T. A. Kowalewski. "Experimental and theoretical investigation of large-amplitude oscillations of liquid droplets." Journal of Fluid Mechanics 231 (October 1991): 189–210. http://dx.doi.org/10.1017/s0022112091003361.

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Finite-amplitude, axially symmetric oscillations of small (0.2 mm) liquid droplets in a gaseous environment are studied, both experimentally and theoretically. When the amplitude of natural oscillations of the fundamental mode exceeds approximately 10% of the droplet radius, typical nonlinear effects like the dependence of the oscillation frequency on the amplitude, the asymmetry of the oscillation amplitude, and the interaction between modes are observed. As the amplitude decreases due to viscous damping, the oscillation frequency and the amplitude decay factor reach their asymptotical values predicted by linear theory. The initial behaviour of the droplet is described quite satisfactorily by a proposed nonlinear inviscid theoretical model.

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8

Longuet-Higgins, Michael S. "Resonance in nonlinear bubble oscillations." Journal of Fluid Mechanics 224 (March 1991): 531–49. http://dx.doi.org/10.1017/s0022112091001866.

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In two recent papers (Longuet-Higgins 1989a,b) the author showed that the shape oscillations of bubbles can emit sound like a monopole source, at second order in the distortion parameter ε. In the second paper (LH2) it was predicted that the emission would be amplified when the second harmonic frequency 2σn of the shape oscillation approaches the frequency ω of the breathing mode. This ‘resonance’ would however be drastically limited by damping due to acoustic radiation and thermal diffusion. The predictions were confirmed by further numerical calculations in Longuet-Higgins (1990a).Ffowcs Williams & Guo (1991) have questioned the conclusions of LH2 on the grounds that near resonance there is a slow (secular) transfer of energy between the shape oscillation and the volumetric mode which tends to diminish the amplitude of the shape oscillation and hence falsify the perturbation analysis. They have also argued that the volumetric mode never grows sufficiently to produce sound of the stated order of magnitude. In the present paper we show that these assertions are unfounded. Ffowcs Williams & Guo considered only undamped oscillations. Here we show that when the appropriate damping is included in their analysis the secular transfer of energy becomes completely insignificant. The resulting pressure pulse (figure 5 below) is found to be essentially identical to that calculated in LH2, figure 3. Moreover, in the initial-value problem considered in LH2, the excitation of the volumetric mode takes place not by a secular energy transfer but by a resonance during the first few cycles of the shape oscillation. This accounts for the amplification near resonance found in Longuet-Higgins (1990a). Finally, it is pointed out that the initial energy of the shape oscillations is far greater than is required to produce the O(ε2) volume pulsations that were studied in LH2, and which were used for a comparison with field data. This acoustic radiation was not calculated by Ffowcs Williams & Guo.

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9

Shevtsov, Boris, and Olga Shevtsova. "Fluctuations and nonlinear oscillations in complex natural systems." E3S Web of Conferences 62 (2018): 02006. http://dx.doi.org/10.1051/e3sconf/20186202006.

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Resonance propagation of radiation in the ionosphere, solar activity, magnetic dynamos, lightning discharges, fracture processes, plastic deformations, seismicity, turbulence and hydrochemical variability are considered as examples of complex dynamical systems in which similar fluctuation and nonlinear oscillation regimes arise. Collective effects in the systems behavior and chaotic oscillations in individual subsystems, the ratio of random and deterministic, the analysis of variability factors and the change of dynamic regimes, the scaling relation between the elements of the system and the interaction of scales are discussed. It is shown that consolidation and branching in disruptions or thunderstorm activity is the transfer of disturbances to up and down of cascades as in turbulence, and the alpha-omega effects of the magnetic dynamo are the same cascade processes, but in the presence of an external magnetic field or rotation that removes the degeneracy in the system by directions. Particular attention is paid to natural generators and oscillation amplifiers, in which the Lorentz triplet plays the role of a universal model of a nonlinear oscillator.

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10

Markov, Nikolay, Viktor Dmitriev, Svetlana Maltseva, and Andrey Dmitriev. "Application of the Nonlinear Oscillations Theory to the Study of Non-equilibrium Financial Market." Financial Assets and Investing 7, no. 3 (November 30, 2016): 5–19. http://dx.doi.org/10.5817/fai2016-3-1.

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The research deals with the construction, implementation and analysis of the model of the non-equilibrium financial market using econophysical approach and the theory of nonlinear oscillations. We used the scaled variation of supply and demand prices and elasticity of these two variables as dynamic variables in the simulation of the non-equilibrium financial market. View of the dynamic variables data was determined based on the strength of econophysical prerequisites using the model of hydrodynamic type. As a result, we found that the non-equilibrium market can be described with a good degree of accuracy with oscillator models with nonlinear rigidity and a self-oscillating system with inertial self-excitation. The most important states of model of oscillation non-equilibrium model of the market were found, including the appearance of chaos and its mechanisms. We have made the calculations of the correlation dimension for the financial time series. The results show that all observed time series have a clearly defined chaotic dynamic nature.

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11

Shcherbak, Volodymyr, and Iryna Dmytryshyn. "Estimation of oscillation velocities of oscillator network." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 32 (December 28, 2018): 182–89. http://dx.doi.org/10.37069/1683-4720-2018-32-17.

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The study of the collective behavior of multiscale dynamic processes is currently one of the most urgent problems of nonlinear dynamics. Such systems arise on modelling of many cyclical biological or physical processes. It is of fundamental importance for understanding the basic laws of synchronous dynamics of distributed active subsystems with oscillations, such as neural ensembles, biomechanical models of cardiac or locomotor activity, models of turbulent media, etc. Since the nonlinear oscillations that are observed in such systems have a stable limit cycle , which does not depend on the initial conditions, then a system of interconnected nonlinear oscillators is usually used as a model of multiscale processes. The equations of Lienar type are often used as the main dynamic model of each of these oscillators. In a number of practical control problems of such interconnected oscillators it is necessary to determine the oscillation velocities by known data. This problem is considered as observation problem for nonlinear dynamical system. A new method – a synthesis of invariant relations is used to design a nonlinear observer. The method allows us to represent unknowns as a function of known quantities. The scheme of the construction of invariant relations consists in the expansion of the original dynamical system by equations of some controlled subsystem (integrator). Control in the additional system is used for the synthesis of some relations that are invariant for the extended system and have the attraction property for all of its trajectories. Such relations are considered in observation problems as additional equations for unknown state vector of initial oscillators ensemble. To design the observer, first we introduce a observer for unique oscillator of Lienar type and prove its exponential convergence. This observer is then extended on several coupled Lienar type oscillators. The performance of the proposed method is investigated by numerical simulations.

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12

Olshanskii, Vasiliy, and Stanislav Olshanskii. "Modeling the motion of an oscillator with a soft elastic characteristic." Physico-mathematical modelling and informational technologies, no. 25 (May 25, 2017): 113–24. http://dx.doi.org/10.15407/fmmit2017.25.113.

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The free oscillations of a system with one degree of freedom are considered under the assumption that the elasticity of a spring is proportional to the cubic root of its deformation. Two forms of the analytical solution of the nonlinear differential equation of motion of the oscillator are obtained. In the first displacement of the oscillator in time is expressed in terms of incomplete elliptic integrals of the first and second kind. In the second form, the solution is expressed in terms of periodic Ateb-functions. The tables of the involved functions are made, which simplify the calculation. Formulas are also derived for calculating the oscillation periods when the oscillator is signaled or the initial deviation from the equilibrium position or the initial velocity (instantaneous pulse) in this position. The dependence of the oscillation period on the parameters of the oscillator and the initial conditions is established. Examples of calculations of oscillations are presented with the use of compiled tables of special functions and using the proposed approximations of the Ateb-functions. Comparison of numerical results obtained by different methods is made.

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13

Luo, X. N., Yong Zhou, and C. F. Li. "Oscillation of a nonlinear difference equation with several delays." Mathematica Bohemica 128, no. 3 (2003): 309–17. http://dx.doi.org/10.21136/mb.2003.134176.

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14

Thandapani, E. "Oscillation theorems for second order damped nonlinear difference equations." Czechoslovak Mathematical Journal 45, no. 2 (1995): 327–35. http://dx.doi.org/10.21136/cmj.1995.128516.

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15

Schmeidel, Ewa. "Oscillation of nonlinear three-dimensional difference systems with delays." Mathematica Bohemica 135, no. 2 (2010): 163–70. http://dx.doi.org/10.21136/mb.2010.140693.

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16

Olshanskiy, Vasyl, Maksym Slipchenko, Oleksandr Spolnik, and Oleksіі Tokarchuk. "POST-IMPACT VIBRATIONS OF A SQUARE NONLINEAR DISPATIVE OSCILLATOR." Vibrations in engineering and technology, no. 4(99) (December 18, 2020): 29–39. http://dx.doi.org/10.37128/2306-8744-2020-4-4.

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An oscillator damped by viscous linear resistance, due to the instantaneous increase in its mass after impact, can become a dissipative oscillatory system under the action of dry or positional friction. In the article describes the oscillations of a dissipative oscillator with an asymmetric quadratically nonlinear elastic characteristic and dry Coulomb friction, arising as a result of an inelastic vertical impact of a rigid body on it. In the article, the Cox model is used, which does not take into account local deformations of solid bodies subjected to impact. The paper establishes the dependences on the impact velocity and the values of other parameters at which the effect of asymmetry of the elastic characteristic of the system may appear or may not appear. The conditions are derived when the dynamic effect of asymmetry of the power characteristic is manifested in the system. It consists in the fact that the maximum displacement of the oscillator (oscillation range) in the direction of the shock pulse is less than the opposite extreme displacement (range) after the shock oscillations. The existence of such a critical value of the shock impulse is established, the excess of which leads to the loss of motion stability. The second integral of the oscillation equation describes the movement of the oscillator in time, expressed in terms of Jacobi elliptic functions. An approximate formula for their calculation is proposed. Formulas are also derived to determine the time to reach extreme deviations of the system from the equilibrium position. This time is expressed in terms of elliptic integrals of the first kind, which refer to the tabulated functions. Examples of calculations are considered, where, in addition to using the derived formulas, numerical computer integration of the original nonlinear differential equation of motion is carried out. A comparison of the results obtained for the displacement values of a quadratically nonlinear oscillator with dry friction expressed in terms of Jacobi elliptic functions and obtained by numerical integration is carried out. Good consistency of the calculation results in two ways confirmed the adequacy of the obtained analytical solutions of the nonlinear Cauchy problem.

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17

Dao, Nguyen Van. "Interaction between the elements characterizing the forced and parametric excitations." Vietnam Journal of Mechanics 20, no. 1 (March 30, 1998): 9–20. http://dx.doi.org/10.15625/0866-7136/10008.

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In nonlinear systems, the first order of smallness terms of nonresonance forced and parametric excitations have no effect on the oscillation in the first an approximation. However, they do interact one with another in the second approximation.Using the asymptotic method of nonlinear mechanics [1] we obtain the equations for the amplitudes and phases of oscillation. The amplitude curves are drawn digital computer. The stationary oscillations and their stability are of special interest.

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18

Vrabel, Robert, Marcel Abas, Michal Kopcek, and Michal Kebisek. "Active Control of Oscillation Patterns in the Presence of Multiarmed Pitchfork Structure of the Critical Manifold of Singularly Perturbed System." Mathematical Problems in Engineering 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/650354.

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We analyze the possibility of control of oscillation patterns for nonlinear dynamical systems without the excitation of oscillatory inputs. We propose a general method for the partition of the space of initial states to the areas allowing active control of the stable steady-state oscillations. Furthermore, we show that the frequency of oscillations can be controlled by an appropriately positioned parameter in the mathematical model. This paper extends the knowledge of the nature of the oscillations with emphasis on its consequences for active control. The results of the analysis are numerically verified and provide the feedback for further design of oscillator circuits.

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19

Ganiev, R. F., D. A. Zhebynev, and A. M. Feldman. "EXCITATION OF NONLINEAR PRESSURE OSCILLATIONS IN LOW-PRESSURE FLUID FLOW USING A HIGH-PRESSURE HYDRODYNAMIC GENERATOR." Spravochnik. Inzhenernyi zhurnal, no. 280 (July 2020): 7–13. http://dx.doi.org/10.14489/hb.2020.07.pp.007-013.

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The results of the study of the excitation process of nonlinear oscillations of finite amplitude pressure in a low-pressure (treated) fluid flow using a hydrodynamic oscillator of flow type, the working fluid of which has no direct contact with the fluid of the treated flow, are presented. It is shown that the oscillations from the hydrodynamic generator can be transmitted to the fluid flow through the interface (matching device) in the form of a disk with a certain acoustic resistance. It was found that resonant oscillations can be disturbed in the low-pressure flow processing chamber. The conditions of excitation of resonant oscillations in the processing chamber with a flowing liquid are found. Numerical values of the oscillation span and resonance frequencies are given.

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20

Ganiev, R. F., D. A. Zhebynev, and A. M. Feldman. "EXCITATION OF NONLINEAR PRESSURE OSCILLATIONS IN LOW-PRESSURE FLUID FLOW USING A HIGH-PRESSURE HYDRODYNAMIC GENERATOR." Spravochnik. Inzhenernyi zhurnal, no. 280 (July 2020): 7–13. http://dx.doi.org/10.14489/hb.2020.07.pp.007-013.

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The results of the study of the excitation process of nonlinear oscillations of finite amplitude pressure in a low-pressure (treated) fluid flow using a hydrodynamic oscillator of flow type, the working fluid of which has no direct contact with the fluid of the treated flow, are presented. It is shown that the oscillations from the hydrodynamic generator can be transmitted to the fluid flow through the interface (matching device) in the form of a disk with a certain acoustic resistance. It was found that resonant oscillations can be disturbed in the low-pressure flow processing chamber. The conditions of excitation of resonant oscillations in the processing chamber with a flowing liquid are found. Numerical values of the oscillation span and resonance frequencies are given.

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21

AZUMA, HISAO, and SHOICHI YOSHIHARA. "Three-dimensional large-amplitude drop oscillations: experiments and theoretical analysis." Journal of Fluid Mechanics 393 (August 25, 1999): 309–32. http://dx.doi.org/10.1017/s0022112099005728.

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Three-dimensional large-amplitude oscillations of a mercury drop were obtained by electrical excitation in low gravity using a drop tower. Multi-lobed (from three to six lobes) and polyhedral (including tetrahedral, hexahedral, octahedral and dodecahedral) oscillations were obtained as well as axisymmetric oscillation patterns. The relationship between the oscillation patterns and their frequencies was obtained, and it was found that polyhedral oscillations are due to the nonlinear interaction of waves.A mathematical model of three-dimensional forced oscillations of a liquid drop is proposed and compared with experimental results. The equations of drop motion are derived by applying the variation principle to the Lagrangian of the drop motion, assuming moderate deformation. The model takes the form of a nonlinear Mathieu equation, which expresses the relationships between deformation amplitude and the driving force's magnitude and frequency.

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22

Pushpalatha, G., and S. A. Vijaya Lakshmi. "Oscillation of Even Order Nonlinear Neutral Differential Equations of E." International Journal of Trend in Scientific Research and Development Volume-2, Issue-3 (April 30, 2018): 632–37. http://dx.doi.org/10.31142/ijtsrd11060.

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23

Grace, S. R., and Bikkar S. Lalli. "Oscillation theorems for certain nonlinear differential equations with deviating arguments." Czechoslovak Mathematical Journal 36, no. 2 (1986): 268–74. http://dx.doi.org/10.21136/cmj.1986.102090.

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24

Thandapani, E., and S. Pandian. "Oscillation theorems for certain second order perturbed nonlinear difference equations." Czechoslovak Mathematical Journal 45, no. 4 (1995): 757–66. http://dx.doi.org/10.21136/cmj.1995.128557.

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25

Hamedani, G. G. "Some remarks on a recent third order nonlinear oscillation result." Časopis pro pěstování matematiky 110, no. 3 (1985): 237–40. http://dx.doi.org/10.21136/cpm.1985.118230.

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26

Švaňa, Peter. "Oscillation criteria for forced nonlinear elliptic equations of arbitrary order." Časopis pro pěstování matematiky 113, no. 2 (1988): 169–78. http://dx.doi.org/10.21136/cpm.1988.118340.

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27

Šutová, Zuzana, and Róbert Vrábeľ. "Active Control Of Oscillation Patterns In Nonlinear Dynamical Systems And Their Mathematical Modelling." Research Papers Faculty of Materials Science and Technology Slovak University of Technology 22, no. 341 (December 1, 2014): 29–33. http://dx.doi.org/10.2478/rput-2014-0004.

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Abstract The article deals with the active control of oscillation patterns in nonlinear dynamical systems and its possible use. The purpose of the research is to prove the possibility of oscillations frequency control based on a change of value of singular perturbation parameter placed into a mathematical model of a nonlinear dynamical system at the highest derivative. This parameter is in singular perturbation theory often called small parameter, as ε → 0+. Oscillation frequency change caused by a different value of the parameter is verified by modelling the system in MATLAB.

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28

Goupil, M. J. "Some Insights into Stellar Structure from Nonlinear Pulsations." International Astronomical Union Colloquium 134 (1993): 231–33. http://dx.doi.org/10.1017/s0252921100014263.

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Efficient tools of investigation of stellar pulsation are the integral relations which link oscillation frequencies to the static structure of stellar models, as provided by the linear theory of pulsation (for a review, see Saio, this conference).Similarly, oscillation amplitudes and phases, which arise from nonlinear processes, can be related to the stellar structure by means of amplitude equation formalisms (for a review, see Buchler, this conference).For the simple case of a monoperiodic oscillation, involving only one unstable marginal mode, such a formalism shows that the (limit cycle) radius variations, at time t and mass level m, can be approximated, up to second order of approximation, (Buchler and Goupil, 1984; Buchler and Kovàca, 1986) by:where A, R, Ω, ĸ, £r(m) respectively are the amplitude, stellar radius, linear nonadiabatic frequency, growth rate, radius eigenfunction. Second order nonlinearities generated first harmonic oscillations and change in equilibrium radius about which the star oscillates, as represented by the last two terms in (la) respectively. Analogous expressions are obtained for velocity and light variations, that can be compared with observations.

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Itoh, Makoto, and Leon Chua. "Chaotic Oscillation via Edge of Chaos Criteria." International Journal of Bifurcation and Chaos 27, no. 11 (October 2017): 1730035. http://dx.doi.org/10.1142/s021812741730035x.

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In this paper, we show that nonlinear dynamical systems which satisfy the edge of chaos criteria can bifurcate from a stable equilibrium point regime to a chaotic regime by periodic forcing. That is, the edge of chaos criteria can be exploited to engineer a phase transition from ordered to chaotic behavior. The frequency of the periodic forcing can be derived from this criteria. In order to generate a periodic or a chaotic oscillation, we have to tune the amplitude of the periodic forcing. For example, we engineer chaotic oscillations in the generalized Duffing oscillator, the FitzHugh–Nagumo model, the Hodgkin–Huxley model, and the Morris–Lecar model. Although forced oscillators can exhibit chaotic oscillations even if the edge of chaos criteria is not satisfied, our computer simulations show that forced oscillators satisfying the edge of chaos criteria can exhibit highly complex chaotic behaviors, such as folding loci, strong spiral dynamics, or tight compressing dynamics. In order to view these behaviors, we used high-dimensional Poincaré maps and coordinate transformations. We also show that interesting nonlinear dynamical systems can be synthesized by applying the edge of chaos criteria. They are globally stable without forcing, that is, all trajectories converge to an asymptotically-stable equilibrium point. However, if we apply a forcing signal, then the dynamical systems can oscillate chaotically. Furthermore, the average power delivered from the forced signal is not dissipated by chaotic oscillations, but on the contrary, energy can be generated via chaotic oscillations, powered by locally-active circuit elements inside the one-port circuit [Formula: see text] connected across a current source.

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Fu, Sheng-Chen, and Long-Yi Tsai. "Oscillation in nonlinear difference equations." Computers & Mathematics with Applications 36, no. 10-12 (November 1998): 193–201. http://dx.doi.org/10.1016/s0898-1221(98)80020-5.

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Bitsadze, R. "Nonlinear Characteristic Problem for a Nonlinear Oscillation Equation." Journal of Mathematical Sciences 206, no. 4 (March 13, 2015): 341–47. http://dx.doi.org/10.1007/s10958-015-2315-7.

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Pejovic, Branko, Pavel Kovac, Vladimir Pucovsky, and Aleksandar Todic. "Theoretical Study of Tool Holder Self-excited Oscillation in Turning Processes Using a Nonlinear Model." International Journal of Acoustics and Vibration 23, No 3, September 2018 (September 2018): 307–13. http://dx.doi.org/10.20855/ijav.2018.23.31058.

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In this paper, by observing a system which is composed from a workpiece and a tool holder, a dynamic and a mathematical nonlinear model is acquired. These models can be used as a theoretical foundation for research of self-excited oscillations, which are the object of research in this paper. All relevant oscillator factors are taken into consideration including a frictional force between flank surface and a machined new surface, which is dependent on relative system speed. For obtaining more reliable results, the characteristic friction function is expanded into the Taylor series with an arbitrary number of members regarding required accuracy. The main nonlinear differential equation of the system is solved by the method of slowly varying coefficients, which is elaborated on in detail here. One assumption is made, which states that the system has a weak nonlinearity, and respectively small damping factor. After obtaining the law of motion with relation to a larger number of influential factors, the amplitude of self-excited oscillation is determined in two different ways. Previously, this is conducted for two characteristic phases—for stationary and nonstationary modes. At the end of the paper, an analytic determination and occurrence condition of self-excited oscillations is established. This is an important factor for practical use. This is also the stability condition. The starting point for this determination was a type of experimental friction function. Derived relationships allow detailed quantitative analysis of certain parameters’ influence, determination of stability, and give a reliable description of the process, which is not the case with the existing linear model. After the theoretical analysis of obtained results, a possibility for application of the suggested method in the machine tool area is presented. The derived general model based on the method of slowly varying coefficients can be directly applied in all cases where nonlinearity is not too large, which is usually the case in the field of machine tools. The greater damping factor causes a smaller amplitude of self-excited steady oscillation. Characteristics of the selfexcited oscillations in the described model mostly depend on the character of the friction force. Angular frequency in mentioned nonlinear oscillations depends on the amplitude and initial conditions of movement, which is not the case in free oscillations.

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Yeh, Y. S., T. H. Chang, C. T. Fan, C. L. Hung, J. N. Jhou, J. M. Huang, J. L. Shiao, Z. Q. Wu, and C. C. Chiu. "Nonlinear oscillation behavior of a driven gyrotron backward-wave oscillator." Physics of Plasmas 17, no. 11 (November 2010): 113112. http://dx.doi.org/10.1063/1.3520616.

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Grace, S. R. "Oscillation theorems of comparison type for neutral nonlinear functional differential equations." Czechoslovak Mathematical Journal 45, no. 4 (1995): 609–26. http://dx.doi.org/10.21136/cmj.1995.128562.

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35

Mathai, Varghese, Laura A. W. M. Loeffen, Timothy T. K. Chan, and Sander Wildeman. "Dynamics of heavy and buoyant underwater pendulums." Journal of Fluid Mechanics 862 (January 16, 2019): 348–63. http://dx.doi.org/10.1017/jfm.2018.867.

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The humble pendulum is often invoked as the archetype of a simple, gravity driven, oscillator. Under ideal circumstances, the oscillation frequency of the pendulum is independent of its mass and swing amplitude. However, in most real-world situations, the dynamics of pendulums is not quite so simple, particularly with additional interactions between the pendulum and a surrounding fluid. Here we extend the realm of pendulum studies to include large amplitude oscillations of heavy and buoyant pendulums in a fluid. We performed experiments with massive and hollow cylindrical pendulums in water, and constructed a simple model that takes the buoyancy, added mass, fluid (nonlinear) drag and bearing friction into account. To first order, the model predicts the oscillation frequencies, peak decelerations and damping rate well. An interesting effect of the nonlinear drag captured well by the model is that, for heavy pendulums, the damping time shows a non-monotonic dependence on pendulum mass, reaching a minimum when the pendulum mass density is nearly twice that of the fluid. Small deviations from the model’s predictions are seen, particularly in the second and subsequent maxima of oscillations. Using time-resolved particle image velocimetry (TR-PIV), we reveal that these deviations likely arise due to the disturbed flow created by the pendulum at earlier times. The mean wake velocity obtained from PIV is used to model an extra drag term due to incoming wake flow. The revised model significantly improves the predictions for the second and subsequent oscillations.

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Savelyev, S. V., and L. A. Morozova. "HIGH POWER CHAOTIC OSCILLATOR." Electronic engineering Series 2 Semiconductor devices 259, no. 4 (2020): 31–36. http://dx.doi.org/10.36845/2073-8250-2020-259-4-31-36.

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The article presents a novel design of a microwave chaotic oscillator. The oscillator contains an inertial converter for the nonlinear amplifier output signal that modulates the supply voltage of the transistor. When the inertia of the converter becomes less than 0.06, the oscillator demonstrates chaotic behavior. We present an experimental model of a chaotic oscillator based on a high power transistor 2T982A-2. The inertial converter circuit contains a diode that perform half-wave conversion of a part of the output signal and a RC circuit with a time constant equal to 0.05 of the duration of an oscillation at the central frequency of the oscillator. The output signal of the inertial converter has been applied to the emitter power supply circuit of the transistor. Modulation of the supply voltage caused the output signal of the oscillator to become a sequence of non-repeating oscillation trains with a random duration and initial phase. The frequency band of the generated chaos with an uneven power spectrum of 4 dB was in a range from 3.1 to 3.3 GHz with an integrated power of 1.2 W. The averaged spectral density of noise oscillations was 6 10-3 W / MHz. Efficiency of the oscillator was 15%.

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Ishida, Y., T. Ikeda, and T. Yamamoto. "Nonlinear Forced Oscillations Caused by Quartic Nonlinearity in a Rotating Shaft System." Journal of Vibration and Acoustics 112, no. 3 (July 1, 1990): 288–97. http://dx.doi.org/10.1115/1.2930507.

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This paper deals with nonlinear forced oscillations in a rotating shaft system which are caused by quartic nonlinearity in a restoring force. These oscillations are theoretically analyzed by paying attention to the nonlinear components represented by the polar coordinates. It is clarified which kind of nonlinear component has an influence on each oscillation. In experiments it was shown that, when the shaft was supported by double-row angular contact ball bearings, the restoring force had nonlinear spring characteristics involving quartic nonlinearity in addition to quadratic and cubic ones. Experimental results were compared with the theoretical results regarding the probability of occurrence and the shapes of the resonance curves.

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Ishida, Y., K. Yasuda, and S. Murakami. "Nonstationary Oscillation of a Rotating Shaft With Nonlinear Spring Characteristics During Acceleration Through a Major Critical Speed (A Discussion by the Asymptotic Method and the Complex-FFT Method)." Journal of Vibration and Acoustics 119, no. 1 (January 1, 1997): 31–36. http://dx.doi.org/10.1115/1.2889684.

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Nonstationary oscillations during acceleration through a major critical speed of a rotating shaft with nonlinear spring characteristics are discussed. First, the first approximate solutions of steady-state and nonstationary oscillations are obtained by the asymptotic method. Second, the amplitude variation curves of each oscillation component are obtained by the complex-FFT method. It is clarified that the first approximation of the asymptotic method has comparatively large quantitative error in the case of nonstationary solutions. In addition, the influences of each nonlinear component in polar coordinate expression on nonstationary oscillations are investigated.

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Luo, D. "A simple nonlinear model of low frequency (interseasonal) oscillations in the tropical atmosphere." Nonlinear Processes in Geophysics 3, no. 1 (March 31, 1996): 29–40. http://dx.doi.org/10.5194/npg-3-29-1996.

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Abstract. In this paper, for a prescribed normalized vertical convective heating profile, nonlinear Kelvin wave equations with wave-CISK heating over equatorial region is reduced to a sixth-order nonlinear ordinary differential equation by using the Galerkin spectral method in the case of considering nonlinear interaction between first and second baroclinic modes. Some numerical calculations are made with the fourth- order Rung-Kutta scheme. It is found that in a narrow range of the heating intensity parameter b, 30-60-day oscillation can occur through linear coupling between first and second baroclinic Kelvin wave-CISK modes for zonal wave-number one when the convective heating is confined to the lower and middle tropospheres. While for zonal wavenumber two, 30-60-day oscillation can be observed in a narrow range of b only when the convective heating is confined to the lower troposphere. However, in a wider range of this heating intensity parameter, 30-60-day oscillation can occur through nonlinear interaction between the first and second baroclinic Kelvin wave-CISK modes with zonal wavenumber one for three vertical convective heating profiles having a maximum in the upper, middle and lower tropospheres, and the total streamfield of the nonlinear first and second baroclinic Kelvin wave-CISK modes possesses a phase reversal between the upper- and lower-tropospheric wind fields. While for zonal wavenumber two, no 30-60-day oscillations can be found. Therefore, it appears that nonlinear interaction between vertical Kelvin wave-CISK modes favours the occurrence of 30-60-day oscillations, particularly, the importance of the vertical distribution of convective heating is re-emphasised.

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Becker, E., W. J. Hiller, and T. A. Kowalewski. "Nonlinear dynamics of viscous droplets." Journal of Fluid Mechanics 258 (January 10, 1994): 191–216. http://dx.doi.org/10.1017/s0022112094003290.

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Nonlinear viscous droplet oscillations are analysed by solving the Navier-Stokes equation for an incompressible fluid. The method is based on mode expansions with modified solutions of the corresponding linear problem. A system of ordinary differential equations, including all nonlinear and viscous terms, is obtained by an extended application of the variational principle of Gauss to the underlying hydrodynamic equations. Results presented are in a very good agreement with experimental data up to oscillation amplitudes of 80% of the unperturbed droplet radius. Large-amplitude oscillations are also in a good agreement with the predictions of Lundgren & Mansour (boundary integral method) and Basaran (Galerkin-finite element method). The results show that viscosity has a large effect on mode coupling phenomena and that, in contradiction to the linear approach, the resonant mode interactions remain for asymptotically diminishing amplitudes of the fundamental mode.

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Tokuda, Isao T., Akihiko Okamoto, Ritsuko Matsumura, Toru Takumi, and Makoto Akashi. "Potential contribution of tandem circadian enhancers to nonlinear oscillations in clock gene expression." Molecular Biology of the Cell 28, no. 17 (August 15, 2017): 2333–42. http://dx.doi.org/10.1091/mbc.e17-02-0129.

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Limit-cycle oscillations require the presence of nonlinear processes. Although mathematical studies have long suggested that multiple nonlinear processes are required for autonomous circadian oscillation in clock gene expression, the underlying mechanism remains controversial. Here we show experimentally that cell-autonomous circadian transcription of a mammalian clock gene requires a functionally interdependent tandem E-box motif; the lack of either of the two E-boxes results in arrhythmic transcription. Although previous studies indicated the role of the tandem motifs in increasing circadian amplitude, enhancing amplitude does not explain the mechanism for limit-cycle oscillations in transcription. In this study, mathematical analysis suggests that the interdependent behavior of enhancer elements including not only E-boxes but also ROR response elements might contribute to limit-cycle oscillations by increasing transcriptional nonlinearity. As expected, introduction of the interdependence of circadian enhancer elements into mathematical models resulted in autonomous transcriptional oscillation with low Hill coefficients. Together these findings suggest that interdependent tandem enhancer motifs on multiple clock genes might cooperatively enhance nonlinearity in the whole circadian feedback system, which would lead to limit-cycle oscillations in clock gene expression.

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Ringwood, John V., and Simon C. Malpas. "Slow oscillations in blood pressure via a nonlinear feedback model." American Journal of Physiology-Regulatory, Integrative and Comparative Physiology 280, no. 4 (April 1, 2001): R1105—R1115. http://dx.doi.org/10.1152/ajpregu.2001.280.4.r1105.

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Blood pressure is well established to contain a potential oscillation between 0.1 and 0.4 Hz, which is proposed to reflect resonant feedback in the baroreflex loop. A linear feedback model, comprising delay and lag terms for the vasculature, and a linear proportional derivative controller have been proposed to account for the 0.4-Hz oscillation in blood pressure in rats. However, although this model can produce oscillations at the required frequency, some strict relationships between the controller and vasculature parameters must be true for the oscillations to be stable. We developed a nonlinear model, containing an amplitude-limiting nonlinearity that allows for similar oscillations under a very mild set of assumptions. Models constructed from arterial pressure and sympathetic nerve activity recordings obtained from conscious rabbits under resting conditions suggest that the nonlinearity in the feedback loop is not contained within the vasculature, but rather is confined to the central nervous system. The advantage of the model is that it provides for sustained stable oscillations under a wide variety of situations even where gain at various points along the feedback loop may be altered, a situation that is not possible with a linear feedback model. Our model shows how variations in some of the nonlinearity characteristics can account for growth or decay in the oscillations and situations where the oscillations can disappear altogether. Such variations are shown to accord well with observed experimental data. Additionally, using a nonlinear feedback model, it is straightforward to show that the variation in frequency of the oscillations in blood pressure in rats (0.4 Hz), rabbits (0.3 Hz), and humans (0.1 Hz) is primarily due to scaling effects of conduction times between species.

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Royer, Baptiste, Shruti Puri, and Alexandre Blais. "Qubit parity measurement by parametric driving in circuit QED." Science Advances 4, no. 11 (November 2018): eaau1695. http://dx.doi.org/10.1126/sciadv.aau1695.

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Multiqubit parity measurements are essential to quantum error correction. Current realizations of these measurements often rely on ancilla qubits, a method that is sensitive to faulty two-qubit gates and that requires notable experimental overhead. We propose a hardware-efficient multiqubit parity measurement exploiting the bifurcation dynamics of a parametrically driven nonlinear oscillator. This approach takes advantage of the resonator’s parametric oscillation threshold, which depends on the joint parity of dispersively coupled qubits, leading to high-amplitude oscillations for one parity subspace and no oscillation for the other. We present analytical and numerical results for two- and four-qubit parity measurements, with high-fidelity readout preserving the parity eigenpaces. Moreover, we discuss a possible realization that can be readily implemented with the current circuit quantum electrodynamics (QED) experimental toolbox. These results could lead to substantial simplifications in the experimental implementation of quantum error correction and notably of the surface code.

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Šutová, Zuzana, Róbert Vrábeľ, Bohuslava Juhásová, and Martin Juhás. "Control of Oscillations in Second-Order Differential Equation." Research Papers Faculty of Materials Science and Technology Slovak University of Technology 22, no. 35 (December 1, 2014): 57–62. http://dx.doi.org/10.2478/rput-2014-0035.

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Abstract The article deals with the control of oscillations in a specific type of second-order differential equations. The purpose of the research is to prove the possibility of oscillation frequency control based on a change in the value of a singular perturbation parameter placed into a mathematical model of a nonlinear dynamical system at the highest derivative. The oscillation frequency change caused by a different value of the parameter is verified by numerically modelling the system.

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Althobati, Saad, Jehad Alzabut, and Omar Bazighifan. "Non-Linear Neutral Differential Equations with Damping: Oscillation of Solutions." Symmetry 13, no. 2 (February 7, 2021): 285. http://dx.doi.org/10.3390/sym13020285.

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The oscillation of non-linear neutral equations contributes to many applications, such as torsional oscillations, which have been observed during earthquakes. These oscillations are generally caused by the asymmetry of the structures. The objective of this work is to establish new oscillation criteria for a class of nonlinear even-order differential equations with damping. We employ different approach based on using Riccati technique to reduce the main equation into a second order equation and then comparing with a second order equation whose oscillatory behavior is known. The new conditions complement several results in the literature. Furthermore, examining the validity of the proposed criteria has been demonstrated via particular examples.

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BOGACHEK, E. N., I. V. KRIVE, I. O. KULIK, and A. S. ROZHAVSKY. "THE AHARONOV-CASHER AND BERRY’S PHASE EFFECTS IN SOLIDS." Modern Physics Letters B 05, no. 23 (October 10, 1991): 1607–11. http://dx.doi.org/10.1142/s0217984991001921.

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We consider the manifestations of charge-induced topological phase shift (Aharonov-Casher effect) in condensed matter physics. There will be an oscillating response to high voltage of the magnetic moment (persistent current) and conductivity, as well as a phase shift of the Aharonov-Bohm oscillation to a smaller voltage, for the normal metal ring threaded by a charged fiber. These oscillations shift in phase if the magnetic field vector rotates along the ring, as a consequence of the geometrical (Berry’s) phase associated with the electron spin.

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Liao, Shi-Jun, and A. T. Chwang. "Application of Homotopy Analysis Method in Nonlinear Oscillations." Journal of Applied Mechanics 65, no. 4 (December 1, 1998): 914–22. http://dx.doi.org/10.1115/1.2791935.

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In this paper, we apply a new analytical technique for nonlinear problems, namely the Homotopy Analysis Method (Liao 1992a), to give two-period formulas for oscillations of conservative single-degree-of-freedom systems with odd nonlinearity. These two formulas are uniformly valid for any possible amplitudes of oscillation. Four examples are given to illustrate the validity of the two formulas. This paper also demonstrates the general validity and the great potential of the Homotopy Analysis Method.

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Marinca, Vasile, Remus Daniel Ene, and Eugen Ioan Ghita. "On the Open Problem: An Optimal Analytical Solution for Duffing Oscillator." Applied Mechanics and Materials 801 (October 2015): 43–47. http://dx.doi.org/10.4028/www.scientific.net/amm.801.43.

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Based on the Optimal Homotopy Perturbation Method (OHPM), an analytical approximate solution to Duffing’s nonlinear oscillator problem is obtained. Using renormalization group method, Kirkinis [1] obtained an asymptotic solution to Duffing’s nonlinear oscillation problem. Kirkinis then asked if his asymptotic solution is optimal. In this paper, a negative answer to this open problem is given by means of OHPM.

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Zhao, Xu. "Nonlinear time transformation method for strong nonlinear oscillation systems." Acta Mechanica Sinica 8, no. 3 (August 1992): 279–88. http://dx.doi.org/10.1007/bf02489252.

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50

OSMAN, F., R. BEECH, and H. HORA. "Solutions of the nonlinear paraxial equation due to laser plasma–interactions." Laser and Particle Beams 22, no. 1 (March 2004): 69–74. http://dx.doi.org/10.1017/s0263034604221139.

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This article presents a numerical and theoretical study of the generation and propagation of oscillation in the semiclassical limit ħ → 0 of the nonlinear paraxial equation. In a general setting of both dimension and nonlinearity, the essential differences between the “defocusing” and “focusing” cases are observed. Numerical comparisons of the oscillations are made between the linear (“free”) and the cubic (defocusing and focusing) cases in one dimension. The integrability of the one-dimensional cubic nonlinear paraxial equation is exploited to give a complete global characterization of the weak limits of the oscillations in the defocusing case.

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