Journal articles: 'Applied theory of oscillations'

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PHILLIPSON, PAUL E., and PETER SCHUSTER. "DYNAMICS OF RELAXATION OSCILLATIONS." International Journal of Bifurcation and Chaos 11, no. 05 (May 2001): 1471–82. http://dx.doi.org/10.1142/s0218127401002845.

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Relaxation oscillations are characteristic of periodic processes consisting of segments which differ greatly in time: a long-time span when the system is moving slowly and a relatively short time span when the system is moving rapidly. The period of oscillation, the sum of these contributions, is usually treated by singular perturbation theory which starts from the premise that the long span is asymptotically extended and the short span shrinks asymptotically to a single instant. Application of the theory involves the analysis of adjacent dynamical regions and multiple time scales. The relaxation oscillations of the Stoker–Haag piecewise-linear discontinuous equation and the van der Pol equation are investigated using a simpler analytical method requiring only the connection at a point of the two dynamical fast and slow regions. Compared to the results of singular perturbation theory, the quantitative results of the present method are more accurate in the Stoker–Haag case and marginally less accurate in the van der Pol case. The relative simplicity of the formulation suggests extension to three-dimensional systems where relaxation oscillations can become unstable leading to bistability, multiple periodicity and chaos.

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Zevin, A. A. "The theory of parametric oscillations." Journal of Applied Mathematics and Mechanics 78, no. 1 (2014): 30–38. http://dx.doi.org/10.1016/j.jappmathmech.2014.05.004.

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Gredelj, Sanel. "The Methodology of Distinguish Between Random and Chaotic Machine Tool Oscillations." IOP Conference Series: Materials Science and Engineering 1208, no. 1 (November 1, 2021): 012009. http://dx.doi.org/10.1088/1757-899x/1208/1/012009.

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Abstract Machine tool oscillations are irregular or aperiodic. Most often, these oscillations are chaotic but, in some cases, they can be quasi-periodic or random. The methodology for characterizing oscillations in the first of two steps uses the nonparametric hypothesis tests which the observed oscillations confirmed as irregular. The methodology for the final characterization of oscillations is based on chaos quantifiers. A time series defined as the measured values of oscillations in the time domain is the basis for calculating the quantifiers of chaos. There are four quantifiers of chaos: the Lyapunov exponent, Kolmogorov entropy, fractal dimension and correlation dimension. The correlation dimension and Kolmogorov entropy are important for distinguishing between random and chaotic oscillations. Other quantifiers of chaos are not used for this purpose. The methodology requires a multidisciplinary approach based on combining Nonlinear Dynamics and Probability Theory and Statistics. The methodology can be applied to many oscillating phenomena. Therefore, the paper mainly used the term oscillations, not vibrations, chatter, etc.

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PHILLIPSON, PAUL E., and PETER SCHUSTER. "AN ANALYTIC PICTURE OF NEURON OSCILLATIONS." International Journal of Bifurcation and Chaos 14, no. 05 (May 2004): 1539–48. http://dx.doi.org/10.1142/s0218127404010151.

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Current induced oscillations of a space clamped neuron action potential demonstrates a bifurcation scenario originally encapsulated by the four-dimensional Hodgkin–Huxley equations. These oscillations were subsequently described by the two-dimensional FitzHugh–Nagumo Equations in close agreement with the Hodgkin–Huxley theory. It is shown that the FitzHugh–Nagumo equations can to close approximation be reduced to a generalized van der Pol oscillator externally driven by the current. The current functions as an external constant force driving the action potential. As a consequence approximate analytic expressions are derived which predict the bifurcation scenario, the amplitudes of the oscillations and the oscillation periods in terms of the current and the physiological constants of the FitzHugh–Nagumo model. A second reduction permits explicit analytic solution and results in a spiking model which can be multiply coupled and extended to include the dynamics of phase locking, entrainment and chaos characteristic of time-dependent synaptic inputs.

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CHANG, YU, LILI ZHOU, and JINLIANG WANG. "HOPF BIFURCATION IN A CALCIUM OSCILLATION MODEL AND ITS CONTROL: FREQUENCY DOMAIN APPROACH." International Journal of Bifurcation and Chaos 23, no. 01 (January 2013): 1350012. http://dx.doi.org/10.1142/s0218127413500120.

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The Hopf bifurcation in a calcium oscillation model is theoretically analyzed by Hopf bifurcation theory in frequency domain. Approximation expressions for frequencies and amplitudes of periodic orbits arising from Hopf bifurcation are provided by using second-order harmonic balance method. In addition, a new method is proposed to control the amplitudes of the periodic orbits. Numerical simulations show the effectiveness of the method for suppressing periodic oscillations.

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Han, Hongfang, and Qinsheng Bi. "Bursting Oscillations as well as the Mechanism in a Filippov System with Parametric and External Excitations." International Journal of Bifurcation and Chaos 30, no. 12 (September 30, 2020): 2050168. http://dx.doi.org/10.1142/s0218127420501680.

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The main purpose of this paper is to explore the bursting oscillations as well as the mechanism of a parametric and external excitation Filippov type system (PEEFS), in which different types of bursting oscillations such as fold/nonsmooth fold (NSF)/fold/NSF, fold/NSF/fold and fold/fold bursting oscillations can be observed. By employing the overlap of the transformed phase portrait and the equilibrium branches of the generalized autonomous system, the mechanisms of the bursting oscillations are investigated. Our results show that the fold bifurcation and the boundary equilibrium bifurcation (BEB) can cause the transitions between the quiescent states and repetitive spiking states. The oscillating frequencies of the spiking states can be approximated theoretically by their occurring mechanisms, which agree well with the numerical simulations. Furthermore, some nonsmooth evolutions are investigated by employing differential inclusions theory, which reveals that the positional relationship between the points of the trajectory interacting with the nonsmooth boundary and the related sliding boundary of the nonsmooth system may affect the nonsmooth evolutions.

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Li, Bingtuan. "Discrete oscillations." Journal of Difference Equations and Applications 2, no. 4 (January 1996): 389–99. http://dx.doi.org/10.1080/10236199608808073.

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MATARRESE, SABINO, and MASSIMO PIETRONI. "BARYONIC ACOUSTIC OSCILLATIONS VIA THE RENORMALIZATION GROUP." Modern Physics Letters A 23, no. 01 (January 10, 2008): 25–32. http://dx.doi.org/10.1142/s0217732308026182.

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Renormalization Group techniques, successfully employed in quantum field theory and statistical physics, are applied to study the dynamics of structure formation in the Universe. A semi-analytic approach to the computation of the nonlinear power-spectrum of dark matter density fluctuations is proposed. The method can be applied down to zero redshift and to length scales where perturbation theory fails. Our predictions accurately fit the results of numerical simulations in reproducing the "acoustic oscillations" features of the power spectrum, which will be accurately measured in future galaxy surveys and will provide a probe to distinguish among different dark energy models.

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Gilson, James G. "Oscillations of a polarizable vacuum." Journal of Applied Mathematics and Stochastic Analysis 4, no. 2 (January 1, 1991): 95–110. http://dx.doi.org/10.1155/s1048953391000060.

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A classical basis for one-dimensional Schrödinger quantum theory is constructed from simple vacuum polarization harmonic oscillators within standard stochastic theory. The model is constructed on a two-dimensional phase configuration surface with phase velocity vectors that have a speed of light zitterbewegung behaviour character. The system supplies a natural Hermitian scalar product describing probability density which is derived from angular momentum considerations. The generality of the model which is extensive is discussed.

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Chernyshev, Vladimir, Leonid Savin, and Olga Fominova. "Indirect Control of Oscillations: Elements of Theory." SPIIRAS Proceedings 18, no. 1 (December 1, 2018): 148–75. http://dx.doi.org/10.15622/sp.18.1.148-175.

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A brief review of the main research areas in the field of controlled vibration protection systems is given. It is shown that Vibration systems with indirect control processes of oscillations allow with a minimum expenditure of energy to ensure programmable switching parameters and structures, in which the dissipative restoring and inertial forces generated on the basis of active impact. Within synthesis of indirect control the chains of new auxiliary mathematical constructs for finding optimal synthesizing functions of the elastic-damping units parameters control are obtained. It enabled to separate a base model with intermittent damping and base model with impulse trap. As a result of the study, based on the harmonic balance method, the dynamic properties of the basic model with intermittent damping, calculation formulas are obtained for determining the parameters of the compensation effect and calculating the dynamic coefficient. It is established that, with an optimal sequence of damping switching, the resonant phenomena are eliminated, and the transient processes decay within one period of the kinematic perturbation. The basic model with a pulse trap imitates the limiting variant of intermittent damping and realizes the process of superimposing constraining bonds, the sequence and duration of which are new variables essentially increasing controllability. And for indirect pulse control, there exicts a certain minimum of power consumption independent of the achieved effect of vibration protection. A regulated increase in the duration of the application of the restraining coupling in the low-frequency region and a decrease in this duration in the high-frequency region provides a monotonically decreasing dependence on the dynamic coefficients over the entire frequency range. An example of a solution to the optimization problem of controlling the damping process for a basic model of a vibration isolation system is considered. It is established that intermittent damping is an indispensable feature of the optimality of the vibration isolation system: the damper switches on when the sign of the object's speed has changed and turns off when the object's displacement sign has changed.

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Yagasaki, Kazuyuki. "The Melnikov Theory for Subharmonics and Their Bifurcations in Forced Oscillations." SIAM Journal on Applied Mathematics 56, no. 6 (December 1996): 1720–65. http://dx.doi.org/10.1137/s0036139995281317.

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Thomson, G. R., and C. Constanda. "A matrix of fundamental solutions in the theory of plate oscillations." Applied Mathematics Letters 22, no. 5 (May 2009): 707–11. http://dx.doi.org/10.1016/j.aml.2008.04.016.

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DOWLING, A. P. "Nonlinear self-excited oscillations of a ducted flame." Journal of Fluid Mechanics 346 (September 10, 1997): 271–90. http://dx.doi.org/10.1017/s0022112097006484.

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Self-excited oscillations of a confined flame, burning in the wake of a bluff-body flame-holder, are considered. These oscillations occur due to interaction between unsteady combustion and acoustic waves. According to linear theory, flow disturbances grow exponentially with time. A theory for nonlinear oscillations is developed, exploiting the fact that the main nonlinearity is in the heat release rate, which essentially ‘saturates’. The amplitudes of the pressure fluctuations are sufficiently small that the acoustic waves remain linear. The time evolution of the oscillations is determined by numerical integration and inclusion of nonlinear effects is found to lead to limit cycles of finite amplitude. The predicted limit cycles are compared with results from experiments and from linear theory. The amplitudes and spectra of the limit-cycle oscillations are in reasonable agreement with experiment. Linear theory is found to predict the frequency and mode shape of the nonlinear oscillations remarkably well. Moreover, we find that, for this type of nonlinearity, describing function analysis enables a good estimate of the limit-cycle amplitude to be obtained from linear theory.Active control has been successfully applied to eliminate these oscillations. We demonstrate the same effect by adding a feedback control system to our nonlinear model. This theory is used to explain why any linear controller capable of stabilizing the linear flow disturbances is also able to stabilize finite-amplitude oscillations in the nonlinear limit cycles.

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Gajjar, Gopal R., and Shreevardhan Soman. "Power System Oscillation Modes Identifications: Guidelines for Applying TLS-ESPRIT Method." International Journal of Emerging Electric Power Systems 14, no. 1 (May 30, 2013): 57–66. http://dx.doi.org/10.1515/ijeeps-2013-0023.

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Abstract Fast measurements of power system quantities available through wide-area measurement systems enables direct observations for power system electromechanical oscillations. But the raw observations data need to be processed to obtain the quantitative measures required to make any inference regarding the power system state. A detailed discussion is presented for the theory behind the general problem of oscillatory mode indentification. This paper presents some results on oscillation mode identification applied to a wide-area frequency measurements system. Guidelines for selection of parametes for obtaining most reliable results from the applied method are provided. Finally, some results on real measurements are presented with our inference on them.

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SPINDEL, PH, and M. ZINQUE. "ASYMPTOTIC BEHAVIOR OF THE BIANCHI IX COSMOLOGICAL MODELS IN THE R2 THEORY OF GRAVITY." International Journal of Modern Physics D 02, no. 03 (September 1993): 279–94. http://dx.doi.org/10.1142/s0218271893000210.

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The asymptotic method of Belinskii, Khalatnikov and Lifshitz is applied to the study of the behavior near singularities of generic Bianchi IX cosmological models in the framework of the R2 theory of gravity. Three main kinds of asymptotic forms for the metric are obtained: a de Sitter geometry, a monotonic fall on a curvature singularity after a finite number of oscillations, an infinite sequence of regular oscillations. No chaos appears.

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Reese, D. R., G. M. Mirouh, F. Espinosa Lara, M. Rieutord, and B. Putigny. "Oscillations of 2D ESTER models." Astronomy & Astrophysics 645 (January 2021): A46. http://dx.doi.org/10.1051/0004-6361/201935538.

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Context. Recent numerical and theoretical considerations have shown that low-degree acoustic modes in rapidly rotating stars follow an asymptotic formula. In parallel, recent studies have revealed the presence of regular pulsation frequency patterns in rapidly rotating δ Scuti stars that seem to match theoretical expectations. Aims. In this context, a key question is whether strong gradients or discontinuities can adversely affect the asymptotic frequency pattern to the point of hindering its identification. Other important questions are how rotational splittings are affected by the 2D rotation profiles expected from baroclinic effects and whether it is possible to probe the rotation profile using these splittings. Methods. In order to address these questions, we numerically calculate stellar pulsation modes in continuous and discontinuous rapidly rotating models produced by the 2D Evolution STEllaire en Rotation (ESTER) code. This code self-consistently calculates the rotation profile based on baroclinic effects and uses a spectral multi-domain approach, thus making it possible to introduce discontinuities at the domain interfaces without loss of numerical accuracy. The pulsation calculations are carried out using an adiabatic version of the Two-dimensional Oscillation Program (TOP) code. The variational principle is then used to confirm the high numerical accuracy of the pulsation frequencies and to derive an integral formula for the generalised rotational splittings. Acoustic glitch theory, combined with ray dynamics, is applied to the discontinuous models in order to interpret their pulsation spectra. Results. Our results show that the generalised rotational splittings are very well approximated by the integral formula, except for modes involved in avoided crossings. This potentially allows the application of inverse theory for probing the rotation profile. We also show that glitch theory applied along the island mode orbit can correctly predict the periodicity of the glitch frequency pattern produced by the discontinuity or Γ1 dip related to the He II ionisation zone in some of the models. Furthermore, the asymptotic frequency pattern remains sufficiently well preserved to potentially allow its detection in observed stars.

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Milik, Alexandra, Peter Szmolyan, Helwig Löffelmann, and Eduard Gröller. "Geometry of Mixed-Mode Oscillations in the 3-D Autocatalator." International Journal of Bifurcation and Chaos 08, no. 03 (March 1998): 505–19. http://dx.doi.org/10.1142/s0218127498000322.

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We present a geometric explanation of a basic mechanism generating mixed-mode oscillations in a prototypical simple model of a chemical oscillator. Our approach is based on geometric singular perturbation theory and canard solutions. We explain how the small oscillations are generated near a special point, which is classified as a folded saddle-node for the reduced problem. The canard solution passing through this point separates small oscillations from large relaxation type oscillations. This allows to define a one-dimensional return map in a natural way. This bimodal map is capable of explaining the observed bifurcation sequence convincingly.

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Aebischer, H. A., and Yu S. Sayasov. "Drift waves and magnetic field oscillations in cylindrical plasmas." Journal of Plasma Physics 40, no. 2 (October 1988): 319–36. http://dx.doi.org/10.1017/s0022377800013301.

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A general investigation of linear drift-wave phenomena in cylindrically bounded plasmas, immersed in a magnetic field without shear and curvature, is performed within the two-fluid hydrodynamical approximation, taking into account electron-temperature oscillations and inhomogeneous radial distributions of the undisturbed electron density and temperature. For plasmas in which the electron temperature strongly exceeds the ion temperature the problem is reduced to an ordinary complex second-order differential equation describing the radial distribution of the oscillating electric potential. It is shown that the presence of electron-temperature oscillations (which must always exist in order to satisfy electron-energy conservation) and of radial gradients in the undisturbed electron temperature (which must always exist owing to cooling of the plasma at the boundary) leads to an important modification of the theory of drift waves in cylindrical plasmas (with regard to their stability and the radial distribution of the oscillating quantities) compared with previous papers in which these phenomena were disregarded. A numerical program for solving the corresponding complex-eigenvalue problem has been derived that allows a realistic calculation of all the quantities pertaining to drift-wave phenomena. It has been applied, in particular, to the calculation of the radial distribution of the oscillating coherent magnetic fields accompanying the coherent drift waves. The numerical results prove to be in good agreement with experiments performed with a helium plasma.

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Bollen, J. A. M. "Synchronization theory for forced oscillations in second-order systems." Journal of Optimization Theory and Applications 45, no. 4 (April 1985): 545–76. http://dx.doi.org/10.1007/bf00939134.

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Shul’ga, O. M. "Theory of oscillations of discrete-continuous systems with periodic properties." Journal of Mathematical Sciences 101, no. 1 (August 2000): 2818–20. http://dx.doi.org/10.1007/bf02918837.

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Bulgakov, V., I. Holovach, and M. Berezovyy. "Longitudinal oscillations of the sugar beet root crop body at vibrational digging up from soil." Research in Agricultural Engineering 51, No. 3 (February 7, 2012): 99–104. http://dx.doi.org/10.17221/4910-rae.

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The longitudinal vibrations theory of a continuous elastic body with one fixed extremity is designed. The Ostrogradskii–Hamilton principle of a stationary operation is applied. By Ritz’s method the Ritz’s equation of frequencies for viewed oscillatory process is obtained. In particular analytical expressions for definition of the first and second fundamental frequencies of body oscillations and forced oscillations amplitude of its any cross-section are obtained.

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Ghayesh, Mergen H., Ali Farajpour, and Hamed Farokhi. "Asymmetric Oscillations of AFG Microscale Nonuniform Deformable Timoshenko Beams." Vibration 2, no. 2 (June 14, 2019): 201–21. http://dx.doi.org/10.3390/vibration2020013.

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A nonlinear vibration analysis is conducted on the mechanical behavior of axially functionally graded (AFG) microscale Timoshenko nonuniform beams. Asymmetry is due to both the nonuniform material mixture and geometric nonuniformity. Using the Timoshenko beam theory, the continuous models for translation/rotation are developed via an energy balance. Size-dependence is incorporated via the modified couple stress theory and the rotation via the Timoshenko beam theory. Galerkin’s method of discretization is applied and numerical simulations are conducted for a size-dependent vibration of the AFG microscale beam. Effects of material gradient index and axial change in the cross-sectional area on the force and frequency diagrams are investigated.

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Doctors, Lawrence J., Alexander H. Day, and David Clelland. "Unsteady Effects During Resistance Tests on a Ship Model in a Towing Tank." Journal of Ship Research 52, no. 04 (December 1, 2008): 263–73. http://dx.doi.org/10.5957/jsr.2008.52.4.263.

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It is known that there are oscillations in the wave resistance during the constant-velocity phase of a towing-tank resistance test on a ship model. In this work, the unsteady thin-ship resistance theory has been applied to this case. The results have been compared with experiment data obtained using a towing carriage the velocity history of which can be programmed. It is demonstrated here that generally excellent correlation exists between the theory and the experiments. In particular, one can predict the influence of Froude number, rate of acceleration, and type of smoothing of the acceleration on the characteristics of the oscillations. These characteristics include the amplitude, rate of decay, frequency, and phasing of the oscillations in the curve of wave resistance versus time.

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CHENG, SUI SUN, SHENG-LI XIE, and BING-GEN ZHANG. "QUALITATIVE THEORY OF PARTIAL DIFFERENCE EQUATIONS (III): FORCED OSCILLATIONS OF PARABOLIC TYPE PARTIAL DIFFERENCE EQUATIONS." Tamkang Journal of Mathematics 26, no. 2 (June 1, 1995): 177–92. http://dx.doi.org/10.5556/j.tkjm.26.1995.4395.

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Parabolic type partial difference equations with a forcing term is stud- ied in this paper. By means of three averaging techniques, the problem of oscillation of these equations is reduced to that of recurrence relations in one variable. Avariety of oscillation criteria is given for these recurrence relations which in turn yield oscillation criteria for the partial difference equations.

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da Silva Fernandes, Sandro. "Application of the Hori Method in the Theory of Nonlinear Oscillations." Mathematical Problems in Engineering 2012 (2012): 1–32. http://dx.doi.org/10.1155/2012/239357.

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Some remarks on the application of the Hori method in the theory of nonlinear oscillations are presented. Two simplified algorithms for determining the generating function and the new system of differential equations are derived from a general algorithm proposed by Sessin. The vector functions which define the generating function and the new system of differential equations are not uniquely determined, since the algorithms involve arbitrary functions of the constants of integration of the general solution of the new undisturbed system. Different choices of these arbitrary functions can be made in order to simplify the new system of differential equations and define appropriate near-identity transformations. These simplified algorithms are applied in determining second-order asymptotic solutions of two well-known equations in the theory of nonlinear oscillations: van der Pol equation and Duffing equation.

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Wiercigroch, Marian, Marcin Kapitaniak, Vahid Vaziri, and Krishnan Nandakumar. "Complex dynamics of drill-strings: Theory and experiments." MATEC Web of Conferences 211 (2018): 01002. http://dx.doi.org/10.1051/matecconf/201821101002.

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We investigate complex drill-string dynamics in a downhole drilling where strong nonlinear interactions between various types of vibration take place. First, we present a low dimensional model of the downhole drilling where a drill-bit cutting a rock formation has a strong coupling between torsional and axial oscillations. The model can be used to study drilling stability as an example results are given. Then we introduce a new experimental rig developed by the Centre for Applied Dynamics Research at the University of Aberdeen, capable of reproducing all major types of drill-string vibration. One of the most important features of this versatile experimental rig is the fact that commercial drill-bits, employed in the drilling industry, and real rock-samples are used. The rig operate in different configurations, which enables the experimental study of various phenomena, such as stick-slip oscillations, whirling and drill-bit bounce. It also allows to determine mechanical characteristics of the drill-bits, which are used to calibrate mathematical models.

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Wang, Jiangbin, Ling Liu, Chongxin Liu, and Jian Liu. "Fixed-Time Synergetic Control for a Seven-Dimensional Chaotic Power System Model." International Journal of Bifurcation and Chaos 29, no. 10 (September 2019): 1950130. http://dx.doi.org/10.1142/s021812741950130x.

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Differing from the existing literature that only focus on controlling some simple chaotic power system models, this paper aims to control chaotic oscillations in complex seven-dimensional power system model. First, based on fixed-time stability theory, a novel fixed-time synergetic controller is proposed to make its macro variable enter into an invariant manifold within a fixed-time upper bound by a constant, depending only on control parameters that can be changed by the designer and calculated theoretically. The presented controller can eliminate chattering and achieve exact convergence of the macro variable. Then, the proposed control method is applied to suppress chaos in the seven-dimensional power system model. Based on the important idea that chaotic oscillation in a power system is caused by its excess energy, a model of energy storage device controller is employed in the controller design process to absorb active power from the entire controlled system. Finally, several simulation examples are given to confirm the effectiveness, the superiority and the robustness of the proposed control scheme. Compared with the existing literature, a relatively general method of suppressing chaotic oscillations in power systems is developed.

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LINET, B., and P. TEYSSANDIER. "QUANTUM PHASE SHIFT AND NEUTRINO OSCILLATIONS IN A STATIONARY, WEAK GRAVITATIONAL FIELD." Modern Physics Letters A 26, no. 23 (July 30, 2011): 1737–51. http://dx.doi.org/10.1142/s0217732311036115.

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A new method based on Synge's world function is developed for determining within the WKB approximation the gravitationally induced quantum phase shift of a particle propagating in a stationary spacetime. This method avoids any calculation of geodesics. A detailed treatment is given for relativistic particles within the weak field, linear approximation of any metric theory. The method is applied to the calculation of the oscillation terms governing the interference of neutrinos considered as the superposition of two eigenstates having different masses. It is shown that the neutrino oscillations are not sensitive to the gravitomagnetic components of the metric as long as the spin contributions can be ignored. Explicit calculations are performed when the source of the field is a spherical, homogeneous body. A comparison is made with previous results obtained in Schwarzschild spacetime.

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Piña-Fuentes, Dan, Martijn Beudel, Simon Little, Jonathan van Zijl, Jan Willem Elting, D. L. Marinus Oterdoom, Martje E. van Egmond, J. Marc C. van Dijk, and Marina A. J. Tijssen. "Toward adaptive deep brain stimulation for dystonia." Neurosurgical Focus 45, no. 2 (August 2018): E3. http://dx.doi.org/10.3171/2018.5.focus18155.

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The presence of abnormal neural oscillations within the cortico-basal ganglia-thalamo-cortical (CBGTC) network has emerged as one of the current principal theories to explain the pathophysiology of movement disorders. In theory, these oscillations can be used as biomarkers and thereby serve as a feedback signal to control the delivery of deep brain stimulation (DBS). This new form of DBS, dependent on different characteristics of pathological oscillations, is called adaptive DBS (aDBS), and it has already been applied in patients with Parkinson’s disease. In this review, the authors summarize the scientific research to date on pathological oscillations in dystonia and address potential biomarkers that might be used as a feedback signal for controlling aDBS in patients with dystonia.

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Bulgakov, V., V. Adamchuk, I. Holovach, and D. Orszaghova. "The theory of longitudinal vibrations of a conical elastic body in an elastic medium." Agricultural Science and Practice 3, no. 1 (April 15, 2016): 27–35. http://dx.doi.org/10.15407/agrisp3.01.027.

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Aim. To elaborate the theory of longitudinal vibrations of a solid elastic body with one fi xed end in the elastic medium. The example of such a body may be found in a sugar beet root in soil, the latter being elastic medium. Methods. The principle of stationary action of Ostrogradsky-Hamilton and the Ritz method were applied in the work. Results. The Ritz method was applied to obtain the Ritz frequency equation for the oscillating process under investigation. The analytic expressions were defi ned to determine the fi rst and second eigenfrequencies of vibration and the amplitude of constrained vibrations of any of its cross-sections. The values of the fi rst and second eigenfrequencies of the elastic body under investigation with specifi c geometric and physical pa- rameters were found. The dependency diagrams for the fi rst and second eigenfrequencies on the coeffi cient of elastic contraction of soil as the elastic medium, and the dependency diagrams for the amplitude of constrained oscillations of the mentioned body on the coeffi cient c of elastic deformation of soil and the distance of the cross-section of the body from the conditional point of fi xation were drawn. The dependency diagrams for the amplitude of constrained oscillations of the elastic body on the change in the amplitude and the frequency of perturbing force were obtained. Conclusions. The impossibility of resonance occurrence was substantiated as the frequency of the perturbing force cannot equal the frequency of eigenvibrations of the elastic body due to technological and technical reasons. It was proven that the breaking of the elastic body is impossible with lon- gitudinal deformations due to the shortness of the amplitude of longitudinal vibrations of the mentioned body.

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Henry, Molly J., and Björn Herrmann. "Low-Frequency Neural Oscillations Support Dynamic Attending in Temporal Context." Timing & Time Perception 2, no. 1 (2014): 62–86. http://dx.doi.org/10.1163/22134468-00002011.

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Behaviorally relevant environmental stimuli are often characterized by some degree of temporal regularity. Dynamic attending theory provides a framework for explaining how perception of stimulus events is affected by the temporal context within which they occur. However, the precise neural implementation of dynamic attending remains unclear. Here, we provide a suggestion for a potential neural implementation of dynamic attending by appealing to low-frequency neural oscillations. The current review will familiarize the reader with the basic theoretical tenets of dynamic attending theory, and review empirical work supporting predictions derived from the theory. The potential neural implementation of dynamic attending theory with respect to low-frequency neural oscillations will be outlined, covering stimulus processing in regular and irregular contexts. Finally, we will provide some more speculative connections between dynamic attending and neural oscillations, and suggest further avenues for future research.

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Headley, Velmer. "Elliptic Oscillation Theory." Zeitschrift für Analysis und ihre Anwendungen 9, no. 5 (1990): 429–32. http://dx.doi.org/10.4171/zaa/413.

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CHENG, SUI SUN, BING GEN ZHANG, and SHENG-LI XIE. "QUALITATIVE THEORY OF PARTIAL DIFFERENCE EQUATIONS (IV): FORCED OSCILLATIONS OF HYPERBOLIC TYPE NONLINEAR PARTIAL DIFFERENCE EQUATIONS." Tamkang Journal of Mathematics 26, no. 4 (December 1, 1995): 337–60. http://dx.doi.org/10.5556/j.tkjm.26.1995.4414.

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Nonlinear hyperbolic type partial difference equations with a forcing term are studied in this paper. By means of two averaging techniques, the problems of oscillation of characteristic initial value problem and of initial boundary value problem are reduced to that of forced and/ or unforced recurrence relations in one variable. A variety of oscillation criteria is given for these relations which in turn yield oscillation criteria for the partial difference equations.

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Gao, Chunyan, Haihong Liu, Zengrong Liu, Yuan Zhang, and Fang Yan. "Oscillatory behavior of p53-Mdm2 system driven by transcriptional and translational time delays." International Journal of Biomathematics 13, no. 05 (May 28, 2020): 2050034. http://dx.doi.org/10.1142/s1793524520500345.

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Biological experiments clarify that p53-Mdm2 module is the core of tumor network and p53 oscillation plays an important role in determining the tumor cell fate. In this paper, we investigate the effect of time delay on the oscillatory behavior induced by Hopf bifurcation in p53-Mdm2 system. First, the stability of the unique positive equilibrium point and the existence of Hopf bifurcation are investigated by using the time delay as the bifurcation parameter and by applying the bifurcation theory. Second, the explicit criteria determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are developed based on the normal form theory and the center manifold theorem. In addition, the combination of numerical simulation results and theoretical calculation results indicates that time delays in p53-Mdm2 system are critical for p53 oscillations. The results may help us to better understand the biological functions of p53 pathway and provide clues for treatment of cancer.

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Al-Hussein, Abdul-Basset A., Fadhil Rahma Tahir, and Karthikeyan Rajagopal. "Chaotic Power System Stabilization Based on Novel Incommensurate Fractional-Order Linear Augmentation Controller." Complexity 2021 (September 29, 2021): 1–13. http://dx.doi.org/10.1155/2021/3334609.

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The nonlinear dynamics of an incommensurate fractional-order single-machine infinite-bus (SMIB) power system benchmark model are explored and studied by means of modern nonlinear analysis theories, such as bifurcation, chaos, power spectral density (PSD), and bicoherence methods. The effect of incommensurate order derivatives on power system dynamics is presented. The study reveals that the power system undergoes interesting dynamics such as periodic motion, chaotic oscillations, and multistability whenever the system parameter values fall into particular ranges. A new fractional-order linear augmentation-based control scheme is applied to damp out the power system’s chaotic oscillation, change the stability of the coexisting states, and drive the system from multistability to monostability. The stability of the proposed control system is derived using Lyapunov theory. Simulation results confirmed the effectiveness and robustness of the proposed control scheme in damping power system oscillations and achieving good overall performance. The results in this paper will give a better understanding of the nonlinear dynamic behaviors of the incommensurate fractional-order SMIB power system.

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Chen, Shu Ping, and Wei Zhang. "Further Reduction of Normal Forms for High Dimensional Nonlinear Systems and Application to a Composite Laminated Piezoelectric Plate." Applied Mechanics and Materials 291-294 (February 2013): 2662–65. http://dx.doi.org/10.4028/www.scientific.net/amm.291-294.2662.

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Normal form theory is robust and useful for direct bifurcation and stability analysis of nonlinear differential equations in real engineering problems. This paper develops a new computation method for obtaining a significant refinement of the normal forms for high dimensional nonlinear systems. In the theoretical model for the nonlinear oscillation of a composite laminated piezoelectric plate, the computation method is applied to compute the coefficients of the normal forms for the case of one double zero and a pair of pure imaginary eigenvalues. The algorithm is implemented in Maple V and the normal forms of the averaged equations and their coefficients for nonlinear oscillations of the composite laminated piezoelectric plate under combined parametric and transverse excitations are calculated.

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Chapman, P. B. "Perturbations of nonlinear autonomous oscillators." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 35, no. 4 (April 1994): 445–68. http://dx.doi.org/10.1017/s0334270000009541.

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AbstractA general theory is given for autonomous perturbations of non-linear autonomous second order oscillators. It is found using a multiple scales method. A central part of it requires computation of Fourier coefficients for representation of the underlying oscillations, and these coefficients are found as convergent expansions in a suitable parameter.

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van der Beek, C. G. A. "Normal forms and periodic solutions in the theory of non-linear oscillations. Existence and asymptotic theory." International Journal of Non-Linear Mechanics 24, no. 4 (January 1989): 263–79. http://dx.doi.org/10.1016/0020-7462(89)90045-0.

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Chen, Menghan, Jinchen Ji, Haihong Liu, and Fang Yan. "Periodic Oscillations in the Quorum-Sensing System with Time Delay." International Journal of Bifurcation and Chaos 30, no. 09 (July 2020): 2050127. http://dx.doi.org/10.1142/s0218127420501278.

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The main aim of this paper is to study the oscillatory behaviors of gene expression networks in quorum-sensing system with time delay. The stability of the unique positive equilibrium and the existence of Hopf bifurcation are investigated by choosing the time delay as the bifurcation parameter and by applying the bifurcation theory. The explicit criteria determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are developed based on the normal form theory and the center manifold theorem. Numerical simulations demonstrate good agreements with the theoretical results. Results of this paper indicate that the time delay plays a crucial role in the regulation of the dynamic behaviors of quorum-sensing system.

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Kuznetsov, N. V. "Theory of Hidden Oscillations and Stability of Control Systems." Journal of Computer and Systems Sciences International 59, no. 5 (September 2020): 647–68. http://dx.doi.org/10.1134/s1064230720050093.

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41

Kernytskyy, Ivan, Eugeniusz Koda, Bohdan Diveyev, Orest Horbay, Lyubomyr Sopilnyk, Ruslan Humenuyk, Yaroslav Sholudko, and Piotr Osinski. "Identification of Magnetorheological Layer Properties by Using Refined Plate Theory." Symmetry 13, no. 9 (August 31, 2021): 1601. http://dx.doi.org/10.3390/sym13091601.

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In this paper, the dynamic characteristics of sandwich plates with external rigid layers and an upper layer with magnetorheological properties (MR) are investigated. An analysis of the effect of the magnetic field on frequency and loss factor is presented. Vibration can be controlled by a magnetic-rheological viscoelastomer (MRVE), when used in sandwich plates. During vibration, MRVE exhibits an inhomogeneous complex module, which is controlled by an applied magnetic field and depends on the oscillation frequency. Using the dynamic equilibrium conditions, physical and kinematic relationships, and the partial differential equations for the conjugate transverse and longitudinal oscillations of a sandwich plate, are derived. This paper presents a new method for stress analysis that provides accurate stress distributions for multilayer plates subject to cylindrical bending. It uses an adaptive method that does not make strict assumptions about the plate model. Based on the depicted theoretical model, the deformations of each layer of the plate are accounted for, including both transverse shear deformations and transverse normal deformations where the thickness is concerned, and nonlinear displacement changes. The magnetorheological (MR) identification of an inner layer is carried out using refined plate theory and sandwich bending tests. Using combined methods, the possibility of determining the MRVE parameters robustly, is examined.

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Hu, Haijun, Li Liu, and Jie Mao. "Multiple Nonlinear Oscillations in a𝔻3×𝔻3-Symmetrical Coupled System of Identical Cells with Delays." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/417678.

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A coupled system of nine identical cells with delays and𝔻3×𝔻3-symmetry is considered. The individual cells are modelled by a scalar delay differential equation which includes linear decay and nonlinear delayed feedback. By analyzing the corresponding characteristic equations, the linear stability of the equilibrium is given. We also investigate the simultaneous occurrence of multiple periodic solutions and spatiotemporal patterns of the bifurcating periodic oscillations by using the equivariant bifurcation theory of delay differential equations combined with representation theory of Lie groups. Numerical simulations are carried out to illustrate our theoretical results.

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NIKOLOV, SVETOSLAV, and VALKO PETROV. "NEW RESULTS ABOUT ROUTE TO CHAOS IN ROSSLER SYSTEM." International Journal of Bifurcation and Chaos 14, no. 01 (January 2004): 293–308. http://dx.doi.org/10.1142/s0218127404009065.

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In this paper, the theory of Lyapunov–Andronov is applied to investigate the route to chaos in Rossler system. On the base of a new analytical formula for the first Lyapunov value at the boundary of stability region, we make a detailed bifurcation analysis of this system. From the obtained results the following new conclusions are made: Transition to chaos in the Rossler's system takes place at soft stability loss in the form of a cascade of periodic self-oscillations. Then the occurrence of chaotic self-oscillations in this system takes place under hard stability loss.

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SMITH, WARREN R. "Modulation equations for strongly nonlinear oscillations of an incompressible viscous drop." Journal of Fluid Mechanics 654 (May 11, 2010): 141–59. http://dx.doi.org/10.1017/s0022112010000480.

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Large-amplitude oscillations of incompressible viscous drops are studied at small capillary number. On the long viscous time scale, a formal perturbation scheme is developed to determine original modulation equations. These two ordinary differential equations comprise the averaged condition for conservation of energy and the averaged projection of the Navier–Stokes equations onto the vorticity vector. The modulation equations are applied to the free decay of axisymmetric oblate–prolate spheroid oscillations. On the long time scale, only the modulation equation for energy is required. In this example, the results compare well with linear viscous theory, weakly nonlinear inviscid theory and experimental observations. The new results show that previous experimental observations and numerical simulations are all manifestations of a single-valued relationship between dimensionless decay rate and amplitude. Moreover, if the amplitude of the oscillations does not exceed 30% of the drop radius, this decay rate may be approximated by a quadratic. The new results also show that, when the amplitude of the oscillations exceeds 20% of the drop radius, fluid in the inviscid bulk of the drop is undergoing abrupt changes in its acceleration in comparison to the acceleration during small-amplitude deformations.

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Płociniczak, Łukasz. "Asymptotic analysis of internal relaxation oscillations in a conceptual climate model." IMA Journal of Applied Mathematics 85, no. 3 (May 2, 2020): 467–94. http://dx.doi.org/10.1093/imamat/hxaa014.

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Abstract We construct a dynamical system based on the Källén–Crafoord–Ghil conceptual climate model which includes the ice–albedo and precipitation–temperature feedbacks. Further, we classify the stability of various critical points of the system and identify a parameter which change generates a Hopf bifurcation. This gives rise to a stable limit cycle around a physically interesting critical point. Moreover, it follows from the general theory that the periodic orbit exhibits relaxation-oscillations that are a characteristic feature of the Pleistocene ice ages. We provide an asymptotic analysis of their behaviour and derive a formula for the period along with several estimates. They, in turn, are in a decent agreement with paleoclimatic data and are independent of any parametrization used. Whence, our simple but robust model shows that a climate may exhibit internal relaxation oscillations without any external forcing and for a wide range of parameters.

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Bank, Steven B. "On Complex Oscillation Theory." Applicable Analysis 29, no. 3-4 (January 1988): 209–33. http://dx.doi.org/10.1080/00036818808839782.

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Qi, Lei, Zhiyuan Shen, Jianjian Gao, Guoliang Zhao, Xiang Cui, and Wei Kang. "Wideband modeling and transient analysis of sub-module in modular multilevel converter." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 36, no. 6 (November 6, 2017): 1792–805. http://dx.doi.org/10.1108/compel-12-2016-0582.

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Purpose This paper aims to establish the wideband model of a sub-module in a modular multilevel converter (MMC) and analyze the switch transients of the sub-module. Design/methodology/approach The paper builds an MMC sub-module test circuit and conducts dynamic tests both with and without the bypass thyristor. Then, it builds the wideband model of the MMC sub-module and extracts the model parameters. Finally, based on the wideband model, it simulates the switch transients and analyzes the oscillation mechanism. Findings The dynamic testing shows the bypass thyristor will add oscillations during switch transients, especially during the turn-on process. The thyristor acts like a small capacitor and reduces the total capacitor in the turn-on circuit loop, thus causing under-damped oscillations. Originality/value This paper found that the bypass thyristor will influence the MMC sub-module switch transients under certain circumstances. This paper proposes a partial inductance extraction procedure for the MMC sub-module and builds a wideband model of the sub-module. The wideband model is used to analyze and explain the switch transients, and can be further used for insulated gate bipolar transistor switch oscillation inhibition and sub-module design optimization.

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Sherratt, Jonathan A. "Diffusion-driven instability in oscillating environments." European Journal of Applied Mathematics 6, no. 4 (August 1995): 355–72. http://dx.doi.org/10.1017/s0956792500001893.

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Diffusion-driven instability in systems of reaction-diffusion equations is a commonly used model for pattern formation in both embryology and ecology. In ecological applications, model parameters tend to oscillate in time, because of either daily or seasonal fluctuations in the environment. I investigate the effects of such fluctuations on diffusion-driven instability by considering analytically the possibility of Turing bifurcations when the parameter values (diffusion coefficients and kinetic parameters) oscillate in time between two sets of constant values, with a period that is either very short or very long compared to the time scale of the growth and predation kinetics. I show that oscillations in the kinetics can have quite different effects from oscillations in the dispersal terms. I also discuss the comparison between the solution forms predicted by linear theory and the numerical solutions of a simple nonlinear predator-prey model.

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Jentsch, Lothar, David Natroshvili, and Wolfgang L. Wendland. "General Transmission Problems in the Theory of Elastic Oscillations of Anisotropic Bodies (Mixed Interface Problems)." Journal of Mathematical Analysis and Applications 235, no. 2 (July 1999): 418–34. http://dx.doi.org/10.1006/jmaa.1999.6360.

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Jentsch, Lothar, David Natroshvili, and Wolfgang L. Wendland. "General Transmission Problems in the Theory of Elastic Oscillations of Anisotropic Bodies (Basic Interface Problems)." Journal of Mathematical Analysis and Applications 220, no. 2 (April 1998): 397–433. http://dx.doi.org/10.1006/jmaa.1997.5764.

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Journal articles on the topic 'Applied theory of oscillations'

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