Статті в журналах: "Nonlinearitie"

1

Bahrouni, Anouar. "A Note on the Existence Results for Schrödinger–Maxwell System with Super-Critical Nonlinearitie." Acta Applicandae Mathematicae 166, no. 1 (May 9, 2019): 215–21. http://dx.doi.org/10.1007/s10440-019-00263-3.

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2

Śliwiński, Przemysław. "On-line wavelet estimation of Hammerstein system nonlinearity." International Journal of Applied Mathematics and Computer Science 20, no. 3 (September 1, 2010): 513–23. http://dx.doi.org/10.2478/v10006-010-0038-y.

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On-line wavelet estimation of Hammerstein system nonlinearityA new algorithm for nonparametric wavelet estimation of Hammerstein system nonlinearity is proposed. The algorithm works in the on-line regime (viz., past measurements are not available) and offers a convenient uniform routine for nonlinearity estimation at an arbitrary point and at any moment of the identification process. The pointwise convergence of the estimate to locally bounded nonlinearities and the rate of this convergence are both established.

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3

Altuğ, Sumru, Richard A. Ashley, and Douglas M. Patterson. "ARE TECHNOLOGY SHOCKS NONLINEAR?" Macroeconomic Dynamics 3, no. 4 (December 1999): 506–33. http://dx.doi.org/10.1017/s1365100599013036.

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The behavior of postwar real U.S. GNP, the inputs to an aggregate production function, and several formulations of the associated Solow residuals for the presence of nonlinearities in their generating mechanisms are examined. Three different statistical tests for nonlinearity are implemented: the McLeod-Li test, the BDS test, and the Hinich bicovariance test. We find substantial evidence for nonlinearity in the generating mechanism of real GNP growth but no evidence for nonlinearity in the Solow residuals. We further find that the generating mechanism of the labor input series is nonlinear, whereas that of the capital services input appears to be linear. We therefore conclude that the observed nonlinearity in real output arises from nonlinearities in the labor markets, not from nonlinearities in the technical shocks driving the system. Finally, we investigate the source of the nonlinearities in the labor markets by examining simulated data from a model of the Dutch economy with asymmetric adjustment costs.

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4

Schäfer, Dominik. "Influence of Fluid Viscosity and Compressibility on Nonlinearities in Generalized Aerodynamic Forces for T-Tail Flutter." Aerospace 9, no. 5 (May 9, 2022): 256. http://dx.doi.org/10.3390/aerospace9050256.

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The numerical assessment of T-tail flutter requires a nonlinear description of the structural deformations when the unsteady aerodynamic forces comprise terms from lifting surface roll motion. For linear flutter, a linear deformation description of the vertical tail plane (VTP) out-of-plane bending results in a spurious stiffening proportional to the steady lift forces, which is corrected by incorporating second-order deformation terms in the equations of motion. While the effect of these nonlinear deformation components on the stiffness of the VTP out-of-plane bending mode shape is known from the literature, their impact on the aerodynamic coupling terms involved in T-tail flutter has not been studied so far, especially regarding amplitude-dependent characteristics. This term affects numerical results targeting common flutter analysis, as well as the study of amplitude-dependent dynamic aeroelastic stability phenomena, e.g., Limit Cycle Oscillations (LCOs). As LCOs might occur below the linear flutter boundary, fundamental knowledge about the structural and aerodynamic nonlinearities occurring in the dynamical system is essential. This paper gives an insight into the aerodynamic nonlinearities for representative structural deformations usually encountered in T-tail flutter mechanisms using a CFD approach in the time domain. It further outlines the impact of geometrically nonlinear deformations on the aerodynamic nonlinearities. For this, the horizontal tail plane (HTP) is considered in isolated form to exclude aerodynamic interference effects from the studies and subjected to rigid body roll and yaw motion as an approximation to the structural mode shapes. The complexity of the aerodynamics is increased successively from subsonic inviscid flow to transonic viscous flow. At a subsonic Mach number, a distinct aerodynamic nonlinearity in stiffness and damping in the aerodynamic coupling term HTP roll on yaw is shown. Geometric nonlinearities result in an almost entire cancellation of the stiffness nonlinearity and an increase in damping nonlinearity. The viscous forces result in a stiffness offset with respect to the inviscid results, but do not alter the observed nonlinearities, as well as the impact of geometric nonlinearities. At a transonic Mach number, the aerodynamic stiffness nonlinearity is amplified further and the damping nonlinearity is reduced considerably. Here, the geometrically nonlinear motion description reduces the aerodynamic stiffness nonlinearity as well, but does not cancel it.

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5

He, Shuai, Ching-Tai Ng, and Carman Yeung. "Time-Domain Spectral Finite Element Method for Modeling Second Harmonic Generation of Guided Waves Induced by Material, Geometric and Contact Nonlinearities in Beams." International Journal of Structural Stability and Dynamics 20, no. 10 (August 31, 2020): 2042005. http://dx.doi.org/10.1142/s0219455420420055.

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This study proposes a time-domain spectral finite element (SFE) method for simulating the second harmonic generation (SHG) of nonlinear guided wave due to material, geometric and contact nonlinearities in beams. The time-domain SFE method is developed based on the Mindlin–Hermann rod and Timoshenko beam theory. The material and geometric nonlinearities are modeled by adapting the constitutive relation between stress and strain using a second-order approximation. The contact nonlinearity induced by breathing crack is simulated by bilinear crack mechanism. The material and geometric nonlinearities of the SFE model are validated analytically and the contact nonlinearity is verified numerically using three-dimensional (3D) finite element (FE) simulation. There is good agreement between the analytical, numerical and SFE results, demonstrating the accuracy of the proposed method. Numerical case studies are conducted to investigate the influence of number of cycles and amplitude of the excitation signal on the SHG and its performance in damage detection. The results show that the amplitude of the SHG increases with the numbers of cycles and amplitude of the excitation signal. The amplitudes of the SHG due to material and geometric nonlinearities are also compared with the contact nonlinearity when a breathing crack exists in the beam. It shows that the material and geometric nonlinearities have much less contribution to the SHG than the contact nonlinearity. In addition, the SHG can accurately determine the crack location without using the reference data. Overall, the findings of this study help further advance the use of SHG for damage detection.

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6

Quintana, Anthony, Rui Vasconcellos, Glen Throneberry, and Abdessattar Abdelkefi. "Nonlinear Analysis and Bifurcation Characteristics of Whirl Flutter in Unmanned Aerial Systems." Drones 5, no. 4 (October 21, 2021): 122. http://dx.doi.org/10.3390/drones5040122.

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Aerial drones have improved significantly over the recent decades with stronger and smaller motors, more powerful propellers, and overall optimization of systems. These improvements have consequently increased top speeds and improved a variety of performance aspects, along with introducing new structural challenges, such as whirl flutter. Whirl flutter is an aeroelastic instability that can be affected by structural or aerodynamic nonlinearities. This instability may affect the prediction of potentially dangerous behaviors. In this work, a nonlinear reduced-order model for a nacelle-rotor system, considering quasi-steady aerodynamics, is implemented. First, a parametric study for the linear system is performed to determine the main aerodynamic and structural characteristics that affect the onset of instability. Multiple polynomial nonlinearities in the two degrees of freedom nacelle-rotor model are tested to simulate possible structural nonlinear effects including symmetric cubic hardening nonlinearities for the pitch and yaw degrees of freedom; purely yaw nonlinearity; purely pitch nonlinearity; and a combination of quadratic, cubic, and fifth-order nonlinearities for both degrees of freedom. Results show that the presence of hardening structural nonlinearities introduces limit cycle oscillations to the system in the post-flutter regime. Moreover, it is demonstrated that the inclusion of quadratic nonlinearity introduces asymmetric oscillations and subcritical behavior, where large and potentially dangerous deformations can be reached before the predicted linear flutter speed.

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7

Mu, Rongjun, and Chao Cheng. "Controller Design of Complex System Based on Nonlinear Strength." Mathematical Problems in Engineering 2015 (2015): 1–12. http://dx.doi.org/10.1155/2015/523197.

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This paper presents a new idea of controller design for complex systems. The nonlinearity index method was first developed for error propagation of nonlinear system. The nonlinearity indices access the boundary between the strong and the weak nonlinearities of the system model. The algorithm of nonlinearity index according to engineering application is first proposed in this paper. Applying this method on nonlinear systems is an effective way to measure the nonlinear strength of dynamics model over the full flight envelope. The nonlinearity indices access the boundary between the strong and the weak nonlinearities of system model. According to the different nonlinear strength of dynamical model, the control system is designed. The simulation time of dynamical complex system is selected by the maximum value of dynamic nonlinearity indices. Take a missile as example; dynamical system and control characteristic of missile are simulated. The simulation results show that the method is correct and appropriate.

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8

Kristály, Alexandru, and Nikolaos S. Papageorgiou. "Multiplicity theorems for semilinear elliptic problems depending on a parameter." Proceedings of the Edinburgh Mathematical Society 52, no. 1 (February 2009): 171–80. http://dx.doi.org/10.1017/s0013091507000665.

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AbstractWe consider semilinear elliptic problems in which the right-hand-side nonlinearity depends on a parameter λ > 0. Two multiplicity results are presented, guaranteeing the existence of at least three non-trivial solutions for this kind of problem, when the parameter λ belongs to an interval (0,λ*). Our approach is based on variational techniques, truncation methods and critical groups. The first result incorporates as a special case problems with concave–convex nonlinearities, while the second one involves concave nonlinearities perturbed by an asymptotically linear nonlinearity at infinity.

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9

Wu, Zhijun, Lifeng Fan, and Shihe Zhao. "Effects of Hydraulic Gradient, Intersecting Angle, Aperture, and Fracture Length on the Nonlinearity of Fluid Flow in Smooth Intersecting Fractures: An Experimental Investigation." Geofluids 2018 (2018): 1–14. http://dx.doi.org/10.1155/2018/9352608.

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This study experimentally investigated the nonlinearity of fluid flow in smooth intersecting fractures with a high Reynolds number and high hydraulic gradient. A series of fluid flow tests were conducted on one-inlet-two-outlet fracture patterns with a single intersection. During the experimental tests, the syringe pressure gradient was controlled and varied within the range of 0.20–1.80 MPa/m. Since the syringe pump used in the tests provided a stable flow rate for each hydraulic gradient, the effects of hydraulic gradient, intersecting angle, aperture, and fracture length on the nonlinearities of fluid flow have been analysed for both effluent fractures. The results showed that as the hydraulic gradient or aperture increases, the nonlinearities of fluid flow in both the effluent fractures and the influent fracture increase. However, the nonlinearity of fluid flow in one effluent fracture decreased with increasing intersecting angle or increasing fracture length, as the nonlinearity of fluid flow in the other effluent fracture simultaneously increased. In addition, the nonlinearities of fluid flow in each of the effluent fractures exceed that of the influent fracture.

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10

Kashchenko, Alexandra. "Asymptotics of Solutions to a Differential Equation with Delay and Nonlinearity Having Simple Behaviour at Infinity." Mathematics 10, no. 18 (September 16, 2022): 3360. http://dx.doi.org/10.3390/math10183360.

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In this paper, we study nonlocal dynamics of a nonlinear delay differential equation. This equation with different types of nonlinearities appears in medical, physical, biological, and ecological applications. The type of nonlinearity in this paper is a generalization of two important for applications types of nonlinearities: piecewise constant and compactly supported functions. We study asymptotics of solutions under the condition that nonlinearity is multiplied by a large parameter. We construct all solutions of the equation with initial conditions from a wide subset of the phase space and find conditions on the parameters of equations for having periodic solutions.

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11

Liu, Xiao Wei, Xue Bin Lu, Rong Yan Chuai, Chang Zhi Shi, Ming Xue Huo, and Wei Ping Chen. "Gauge Factor and Nonlinearity of P-Type Polysilicon Nanofilms." Advanced Materials Research 60-61 (January 2009): 84–88. http://dx.doi.org/10.4028/www.scientific.net/amr.60-61.84.

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The gauge factor and nonlinearity of 80nm polysilicon nanofilms with different doping concentration were tested. The experimental results show that, from 8.1×1018cm-3 to 2.0×1020cm-3, the gauge factors first increase then decrease, which like the common polysilicon films (thickness is larger than 100nm). From 2.0×1020cm-3 to 7.1×1020cm-3, the gauge factors do not change with doping concentration almost, which can be explained by tunneling piezoresistive theory. When doping concentration is low than 4.1×1019cm-3, the nonlinearities are big, and the nonlinearities become small when doping concentration is high than 4.1×1019cm-3. The nonlinearity is related to the occupied condition of trapping states in grain boundary. The longitudinal gauge factor and nonlinearity are smaller than transverse ones. Take the gauge factor and nonlinearity both into consideration, the optimal doping concentration should be 4.1×1019cm-3. The conclusions are very useful for design and fabrication of polysilicon nanofilms piezoresistive sensor.

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12

Brito, Alexandro G., Elder M. Hemerly, and Waldemar C. Leite Filho. "On the Relation between NARX Clusters and Even/Odd Nonlinearities through Frequency-Domain Analysis." Mathematical Problems in Engineering 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/650737.

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Although polynomial NARX models have been intensively used in nonlinear system identification, few papers discussed how to relate the inner nonlinearities to specific types of clusters and regressors. The objective of this paper is to discuss this relationship for a class of systems that contain even or odd nonlinearities. This class covers block-structured models (Hammerstein, Wiener, and others) and systems with dynamic nonlinearities. To achieve the paper’s aim, a deep frequency-domain analysis is performed. For each type of nonlinearity, all the NARX clusters are investigated and the results show that each regressor type provides specific nonlinear contribution. The investigation is based on an output power spectra analysis when a specific multisinusoidal excitation is applied. According to the spectral contributions in some of the frequency lines, the nonlinearity classification is possible. By applying the same procedure to the clusters, one interprets how these clusters can (or not) contribute to explain the system nonlinearity. The paper findings have two major impacts: (i) one gains deep knowledge on how the nonlinearities are coded by the clusters, and (ii) this information can be used, for instance, to aid a structure selection procedure (ERR, term clustering, etc.) during the discarding of the clusters which are not able to explain the system nonlinear behavior. Some practical and experimental aspects are discussed, while numerical examples are presented to show the validity of the theoretical analysis.

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13

Ngapasare, Arnold, Georgios Theocharis, Olivier Richoux, Vassos Achilleos, and Charalampos Skokos. "Energy spreading, equipartition, and chaos in lattices with non-central forces." Chinese Physics B 31, no. 2 (February 1, 2022): 020506. http://dx.doi.org/10.1088/1674-1056/ac3a5e.

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We numerically study a one-dimensional, nonlinear lattice model which in the linear limit is relevant to the study of bending (flexural) waves. In contrast with the classic one-dimensional mass-spring system, the linear dispersion relation of the considered model has different characteristics in the low frequency limit. By introducing disorder in the masses of the lattice particles, we investigate how different nonlinearities in the potential (cubic, quadratic, and their combination) lead to energy delocalization, equipartition, and chaotic dynamics. We excite the lattice using single site initial momentum excitations corresponding to a strongly localized linear mode and increase the initial energy of excitation. Beyond a certain energy threshold, when the cubic nonlinearity is present, the system is found to reach energy equipartition and total delocalization. On the other hand, when only the quartic nonlinearity is activated, the system remains localized and away from equipartition at least for the energies and evolution times considered here. However, for large enough energies for all types of nonlinearities we observe chaos. This chaotic behavior is combined with energy delocalization when cubic nonlinearities are present, while the appearance of only quadratic nonlinearity leads to energy localization. Our results reveal a rich dynamical behavior and show differences with the relevant Fermi–Pasta–Ulam–Tsingou model. Our findings pave the way for the study of models relevant to bending (flexural) waves in the presence of nonlinearity and disorder, anticipating different energy transport behaviors.

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14

Calza, Alessandro, and Andrea Zaghini. "NONLINEARITIES IN THE DYNAMICS OF THE EURO AREA DEMAND FOR M1." Macroeconomic Dynamics 13, no. 1 (February 2009): 1–19. http://dx.doi.org/10.1017/s1365100508070405.

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This paper finds evidence of nonlinearities in the dynamics of the euro area demand for the narrow aggregate M1. A long-run money demand relationship is first estimated over a sample period covering the past three decades. Although the parameters of the relationship are jointly stable, there are indications of nonlinearity in the residuals of the error-correction model. This nonlinearity is explicitly modeled using a fairly general Markov switching error-correction model with satisfactory results. The empirical findings of the paper are consistent with theoretical predictions of nonlinearities in the dynamics of adjustment to equilibrium stemming from “buffer stock” and “target-threshold” models and with analogous empirical evidence for European countries and the United States.

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15

Jaffal, Issa, and Christian Inard. "A study of the nonlinearity of a building thermal behavior based on metamodeling." E3S Web of Conferences 111 (2019): 04039. http://dx.doi.org/10.1051/e3sconf/201911104039.

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In this work, we present a study of the nonlinearity of building thermal behavior based on a metamodel for cooling energy needs. We studied the nonlinearity of the thermal behavior of an office. The building quadratic behavior and interactions between its components were analyzed based on the metamodel coefficients. The metamodel was fitted with a reduced number of dynamic simulations. The nonlinearity was first assessed as function of the mean outdoor air temperature in fifteen typical European climates and then as function of the internal heat gains for the coldest and hottest climates. The metamodel provided highly accurate results with fast calculation time. However, a higher accuracy was generally obtained for hot climates, high internal heat gains and lightweight thermal mass. Conversely, the nonlinearity of thermal behavior was accentuated in cold climates and with low internal heat gains. Moreover, the interactions between the building components were found to be more influential on cooling energy needs than quadratic behavior. We propose a classification of thermal behavior into three regimes: Highly nonlinear when the energy needs are close to zero; intermediate with decreasing nonlinearities that can be expressed by power functions; and finally, a quasi-linear regime with almost-steady nonlinearities.

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16

Khurshudyan, Asatur Zh. "Nonlinear Green’s Functions for Wave Equation with Quadratic and Hyperbolic Potentials." Advances in Mathematical Physics 2018 (June 3, 2018): 1–9. http://dx.doi.org/10.1155/2018/7179160.

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The advantageous Green’s function method that originally has been developed for nonhomogeneous linear equations has been recently extended to nonlinear equations by Frasca. This article is devoted to rigorous and numerical analysis of some second-order differential equations new nonlinearities by means of Frasca’s method. More specifically, we consider one-dimensional wave equation with quadratic and hyperbolic nonlinearities. The case of exponential nonlinearity has been reported earlier. Using the method of generalized separation of variables, it is shown that a hierarchy of nonlinear wave equations can be reduced to second-order nonlinear ordinary differential equations, to which Frasca’s method is applicable. Numerical error analysis in both cases of nonlinearity is carried out for various source functions supporting the advantage of the method.

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17

Szabolcsi, Róbert. "Time Invariant Static Nonlinearities of the Dynamical Systems." Land Forces Academy Review 27, no. 3 (September 1, 2022): 275–86. http://dx.doi.org/10.2478/raft-2022-0035.

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Abstract Dynamical technical systems are famous for large scale of time invariant nonlinearities applied inside. Some kinds of nonlinearities describe physical properties of devices used in the technical system. Some special kind of time invariant static nonlinearities are for ensure stability limiting and truncating signals inside the dynamical systems. Moreover, nonlinearity is a property of materials used both in static or dynamical induction machines. One of the widely spread and applied method to handle static nonlinearities is dynamical technical systems is the describing function method (DFM). The purpose of the author is to introduce and apply this technique to evaluate stability conditions of the automatic flight control system of the unmanned aerial vehicles (UAVs).

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18

Cafagna, Donato, and Giuseppe Grassi. "A Novel Framework for Synchronizing via Scalar Signal Hyperchaotic Systems with One or Several Nonlinearities." International Journal of Bifurcation and Chaos 13, no. 08 (August 2003): 2335–42. http://dx.doi.org/10.1142/s0218127403007990.

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In this paper a novel framework for synchronizing hyperchaotic systems with one or several nonlinearities is presented. The proposed technique exploits observer design and time-division multiplexing of the transmitted signal. A remarkable feature of the approach is that only a scalar signal is required for achieving hyperchaos synchronization. Simulation results are carried out for eighth order circuits with one nonlinearity as well as sixth order circuits with two nonlinearities.

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19

Zaytsev, Y. "СРАВНЕНИЕ ВЛИЯНИЯ НЕЛИНЕЙНОСТЕЙ ПЕРВОГО РОДА И ДВОЙНОЙ НЕЛИНЕЙНОСТИ НА ТЕПЛОПРОВОДНОСТЬ ПЛАСТИН ИЗ СПЛАВОВ АЛЮМИНИЯ, ТИТАНА И СТАЛИ". Transactions of Kremenchuk Mykhailo Ostrohradskyi National University 2 (29 квітня 2019): 127–33. http://dx.doi.org/10.30929/1995-0519.2019.2.127-133.

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20

Bandyopadhyay, Milan, and AtulKrishna Banik. "Numerical modeling and nonlinear analysis of semi-rigid jointed steel frames." Proceedings of the 12th Structural Engineering Convention, SEC 2022: Themes 1-2 1, no. 1 (December 19, 2022): 251–58. http://dx.doi.org/10.38208/acp.v1.505.

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This paper investigates the effect of semi-rigid connection, geometric nonlinearity and material nonlinearity in steel frame analysis. To study the independent and combined effects of different nonlinearities, elastic and inelastic analyses are carried out using general purpose finite element software SAP2000. Connections are modeled as rotational springs and frame as one dimensional beam element. Verification and application of the simplified numerical model developed in the present study is demonstrated by considering different examples and comparing the results of present study with those available in the published literature. Numerical alternatives, key features about modeling and necessary input parameters are discussed in detail. Rotational springs are characterized by linear, bilinear, multi-linear or nonlinear moment-rotation relationships of the connection. Results in terms of beam moment and nodal displacement are presented for both elastic and inelastic analyses. With the increase in connection flexibility, beam mid-span moments and displacements increase, but beam-end moments decrease. It is observed that the influence of connection nonlinearity dominates over material and geometric nonlinearities.

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21

Khovanova, Natalia, Igor Khovanov, and Zarina Davletzhanova. "Nonlinear Energy Harvesting from Random Narrow-Band Excitations." International Journal of Structural Stability and Dynamics 14, no. 08 (November 25, 2014): 1440026. http://dx.doi.org/10.1142/s0219455414400264.

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The efficiency of linear and nonlinear harvesters with different types of nonlinearity is compared. Narrow band ambient vibrations are modeled by harmonic Gaussian noise. We show that the performance of nonlinear harvesters strongly depends on both the form of nonlinearity and the properties of the noise. Particular forms of nonlinearities which can produce a better than linear response are identified, and these depend on the spectral width of the harmonic noise.

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22

NOVITSKY, D. V. "SEARCH FOR THE OPTIMAL PARAMETERS OF RELAXING NONLINEARITY TO OBTAIN SELF-TRAPPING OF AN ULTRASHORT PULSE IN A PHOTONIC CRYSTAL." Journal of Nonlinear Optical Physics & Materials 21, no. 01 (March 2012): 1250010. http://dx.doi.org/10.1142/s0218863512500105.

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We theoretically study the conditions for realization of trapping of a femtosecond pulse inside a one-dimensional photonic crystal with the relaxing cubic nonlinearity. A number of variants is considered: focusing and defocusing nonlinearities of the layers, a half-linear system, structures with differing values of nonlinearity parameters of periodically alternating layers. The results seem to be useful to make the optimal choice of the system characteristics to obtain self-trapping.

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23

Zhou, Zhong Hua, Hiroyuki Nasu, Tadanori Hashimoto, and Kanichi Kamiya. "Third-order nonlinear optical properties of the Na2S–PbS–GeS2 sulfide glasses and the Na2S–PbO–GeS2 oxysulfide glasses." Journal of Materials Research 14, no. 2 (February 1999): 330–33. http://dx.doi.org/10.1557/jmr.1999.0048.

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The third-order optical nonlinearity of glasses of the Na2S–PbS–GeS2 and Na2S–PbO–GeS2 systems was measured by third-harmonic generation method. The third-order nonlinearities of glasses of both systems increase with the increasing lead content. The maximum value of the third-order optical nonlinearity was 3.00 × 10-12 esu. The addition of PbO basically has little influence on third-order optical nonlinearity, and the largest nonlinearity is 1.49 × 10-12 esu. The minimum appearing at 15 mol% PbO can be explained by the decrease of number density of lead and sulfur. Chemical durability of oxysulfide glasses is superior to that of a pure sulfide system; thus the addition of PbO is important in this sense.

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24

McCarty, Rachael, and S. Nima Mahmoodi. "Frequency response analysis of nonlinear tapping-contact mode atomic force microscopy." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 229, no. 2 (May 7, 2014): 377–88. http://dx.doi.org/10.1177/0954406214533676.

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The nonlinear vibrations of the tapping-mode atomic force microscopy probe are investigated due to both nonlinearity in tip–sample contact force and curvature of the microcantilever probe. The nonlinear equations of motion for vibrations of the probe are obtained using Hamilton’s principle. In this work, the contact force is considered to be more dominant while previous works only consider Van der Waals force. The nonlinear contact force is expanded using a Taylor series to provide a polynomial with coefficients that are functions of the tip–sample distance. The outcome of this work allows the proper distance to be chosen before scanning to avoid instability in the response. Instability regions must be avoided for accurate imaging. The results show that initial tip–sample distance has a major effect on the stability of the frequency response and force response curves. For analytical investigation, the mode shapes of the atomic force microscopy probe are derived based on the presence of the nonlinear contact force as a boundary condition at the free end of the probe. The frequency response curve is obtained using the method of multiple scales. The results show that the effects of the nonlinearities due to probe geometry and contact force can be minimized. Minimizing the effects of nonlinearities allows for less cumbersome and calculation intensive software packages for atomic force microscopies. This research shows that one possible method of decreasing the nonlinearity effect is increasing the excitation force; however, this can increase the contact region and is not the best solution for canceling the nonlinearity effect. The superior method which is the major contribution of this paper is to find the optimal initial tip–sample distance and excitation force that minimize the nonlinearity effect. It is shown that at a certain tip–sample distance the quadratic and cubic nonlinearities cancel each other and the system responds linearly.

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25

LIPTON, J. M., and K. P. DABKE. "SOFTENING THE NONLINEARITY IN CHUA'S CIRCUIT." International Journal of Bifurcation and Chaos 06, no. 01 (January 1996): 179–83. http://dx.doi.org/10.1142/s0218127496001922.

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The effects of both hard and soft nonlinearities are examined in the frequency domain. Softening the hard nonlinearity in Chua's diode has a similar effect to low pass filtering or reducing the level of high frequency noise components.

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26

Liu, Yaqiong, Yunting Li, Qiuping Liao, and Yunhui Yi. "Classification of nonnegative solutions to fractional Schrödinger-Hatree-Maxwell type system." AIMS Mathematics 6, no. 12 (2021): 13665–88. http://dx.doi.org/10.3934/math.2021794.

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<abstract><p>In this paper, we are concerned with the fractional Schrödinger-Hatree-Maxwell type system. We derive the forms of the nonnegative solution and classify nonlinearities by appling a variant (for nonlocal nonlinearity) of the direct moving spheres method for fractional Laplacians. The main ingredients are the variants (for nonlocal nonlinearity) of the maximum principles, i.e., <italic>narrow region principle</italic> (Theorem 2.3).</p></abstract>

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27

Liu, Ningyu, and Huajiang Ouyang. "Friction-induced vibration considering multiple types of nonlinearities." Nonlinear Dynamics 102, no. 4 (November 3, 2020): 2057–75. http://dx.doi.org/10.1007/s11071-020-06055-x.

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AbstractThe friction-induced vibration of a novel 5-DoF (degree-of-freedom) mass-on-oscillating-belt model considering multiple types of nonlinearities is studied. The first type of nonlinearity in the system is the nonlinear contact stiffness, the second is the non-smooth behaviour including stick, slip and separation, and the third is the geometrical nonlinearity brought about by the moving-load feature of the mass slider on the rigid belt. Both the linear stability of the system and the nonlinear steady-state responses are investigated, and rich dynamic behaviours of the system are revealed. The results of numerical study indicate the necessity of the transient dynamic analysis in the study of friction-induced-vibration problems as the linear stability analysis fails to detect the occurrence of self-excited vibration when two stable solutions coexist in the system. The bifurcation behaviour of the steady-state responses of the system versus some parameters is determined. Additionally, the significant effects of each type of nonlinearity on the linear stability and nonlinear steady-state responses of the system are discovered, which underlie the necessity to take multiple types of nonlinearities into account in the research of friction-induced vibration and noise.

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28

Albulescu, Claudiu Tiberiu, Aviral Kumar Tiwari, and Phouphet Kyophilavong. "Nonlinearities and Chaos: A New Analysis of CEE Stock Markets." Mathematics 9, no. 7 (March 25, 2021): 707. http://dx.doi.org/10.3390/math9070707.

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After a long transition period, the Central and Eastern European (CEE) capital markets have consolidated their place in the financial systems. However, little is known about the price behavior and efficiency of these markets. In this context, using a battery of tests for nonlinear and chaotic behavior, we look for the presence of nonlinearities and chaos in five CEE stock markets. We document, in general, the presence of nonlinearities and chaos which questions the efficient market hypothesis. However, if all tests highlight a chaotic behavior for the analyzed index returns, there are noteworthy differences between the analyzed stock markets underlined by nonlinearity tests, which question, thus, their level of significance. Moreover, the results of nonlinearity tests partially contrast the previous findings reported in the literature on the same group of stock markets, showing, thus, a change in their recent behavior, compared with the 1990s.

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29

Bianucci, Marco, Antonietta Capotondi, Riccardo Mannella, and Silvia Merlino. "Linear or Nonlinear Modeling for ENSO Dynamics?" Atmosphere 9, no. 11 (November 8, 2018): 435. http://dx.doi.org/10.3390/atmos9110435.

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The observed ENSO statistics exhibits a non-Gaussian behavior, which is indicative of the presence of nonlinear processes. In this paper, we use the Recharge Oscillator Model (ROM), a largely used Low-Order Model (LOM) of ENSO, as well as methodologies borrowed from the field of statistical mechanics to identify which aspects of the system may give rise to nonlinearities that are consistent with the observed ENSO statistics. In particular, we are interested in understanding whether the nonlinearities reside in the system dynamics or in the fast atmospheric forcing. Our results indicate that one important dynamical nonlinearity often introduced in the ROM cannot justify a non-Gaussian system behavior, while the nonlinearity in the atmospheric forcing can instead produce a statistics similar to the observed. The implications of the non-Gaussian character of ENSO statistics for the frequency of extreme El Niño events is then examined.

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30

Zvyagintseva, Tatiana E. "On the conditions for cycles existence in a second-order discrete-time system with sector-nonlinearity." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 8, no. 1 (2021): 63–72. http://dx.doi.org/10.21638/spbu01.2021.106.

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In this paper, a second-order discrete-time automatic control system is studied. This work is a continuation of the research presented in the author’s papers “On the Aizerman problem: coefficient conditions for the existence of a four-period cycle in a second-order discrete-time system” and “On the Aizerman problem: coefficient conditions for the existence of threeand six-period cycles in a second-order discrete-time system”, where systems with two- and three-periodic nonlinearities lying in the Hurwitz angle were considered. The systems with nonlinearities subjected to stronger constraints are discussed in this paper. It is assumed that the nonlinearity not only lies in the Hurwitz angle, but also satisfies the additional sector-condition. This formulation of the problem is found in many works devoted to theoretical and applied questions of the automatic control theory. In this paper, a system with such nonlinearity is explored for all possible values of the parameters. It is shown that in this case there are parameter values for which a system with a two-periodic nonlinearity has a family of four-period cycles, and a system with a three-periodic nonlinearity has a family of three- or six-period cycles. The conditions on the parameters under which the system can have a family of periodic solutions are written out explicitly. The proofs of the theorems provide a method for constructing nonlinearity in such a way that any solution of the system with initial data lying on some definite ray is periodic.

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31

Seok, Jinmyoung. "Infinitely Many Standing Waves for the Nonlinear Chern-Simons-Schrödinger Equations." Advances in Mathematical Physics 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/519374.

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We prove the existence of infinitely many solutions of the nonlinear Chern-Simons-Schrödinger equations under a wide class of nonlinearities. This class includes the standard power-type nonlinearity with exponentp>4. This extends the previous result which covers the exponentp>6.

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32

Butcher, E. A., and R. Lu. "Constant-Gain Linear Feedback Control of Piecewise Linear Structural Systems via Nonlinear Normal Modes." Journal of Vibration and Control 10, no. 10 (October 2004): 1535–58. http://dx.doi.org/10.1177/1077546304042065.

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We present a technique for using constant-gain linear position feedback control to implement eigen-structure assignment of n-degrees-of-freedom conservative structural systems with piecewise linear nonlinearities. We employ three distinct control strategies which utilize methods for approximating the nonlinear normal mode (NNM) frequencies and mode shapes. First, the piecewise modal method (PMM) for approximating NNM frequencies is used to determine n constant actuator gains for eigenvalue (pole) placement. Secondly, eigenvalue placement is accomplished by finding an approximate single-degree-of-freedom reduced model with one actuator gain for the mode to be controlled. The third strategy allows the frequencies and mode shapes (eigenstructure) to be placed by using a full n × n matrix of actuator gains and employing the local equivalent linear stiffness method (LELSM) for approximating NNM frequencies and mode shapes. The techniques are applied to a two-degrees-of-freedom system with two distinct types of nonlinearities: a bilinear clearance nonlinearity and a symmetric deadzone nonlinearity.

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33

Mondal, Goutam, Amit Prashant, and Sudhir K. Jain. "Significance of Interface Nonlinearity on the Seismic Response of a Well-Pier System in Cohesionless Soil." Earthquake Spectra 28, no. 3 (August 2012): 1117–45. http://dx.doi.org/10.1193/1.4000074.

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Interaction between a well foundation (caisson) and the surrounding soil during earthquake shaking involves complicated material and interface nonlinearities such as soil inelasticity, separation, sliding, and uplifting. It is often perceived that the interface nonlinearity has appreciable effects on the seismic response of the well foundation. This paper studies soil-well interface behavior during ground shaking and evaluates the significance of interface nonlinearity on the seismic response of the soil-well-pier (SWP) system. Seismic analysis of the soil-well-pier system was performed using the two-dimensional finite element model considering soil and interface nonlinearities, under both full- and partial-embedment conditions of the well foundation. Soil was assumed to be cohesionless and analyzed under both saturated undrained and dry conditions. Results of this model were compared with those of a model with perfectly-bonded interface. The design displacement and force resultants were found to be marginally overestimated and were on the conservative side in absence of nonlinear interface.

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34

Leamy, M. J., and O. Gottlieb. "Nonlinear Dynamics of a Taut String with Material Nonlinearities." Journal of Vibration and Acoustics 123, no. 1 (August 1, 2000): 53–60. http://dx.doi.org/10.1115/1.1325411.

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A spatial string model incorporating a nonlinear (and nonconservative) material law is proposed using finite deformation continuum mechanics. The resulting model is shown to reduce to the classical nonlinear string when a linear material law is used. The influence of material nonlinearities on the string’s dynamic response to excitation near a transverse natural frequency is shown to be small due to their appearance at high orders only. Material nonlinearities appear at low order in the equations for excitation near a longitudinal natural frequency, and a solution for this case is developed by applying a second order multiple scales method directly to the partial differential equations. The material nonlinearities are found to influence both the degree of nonlinearity in the response and its softening or hardening nature.

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35

Bandyopadhyay, S., M. Chhetri, B. B. Delgado, N. Mavinga, and R. Pardo. "Maximal and minimal weak solutions for elliptic problems with nonlinearity on the boundary." Electronic Research Archive 30, no. 6 (2022): 2121–37. http://dx.doi.org/10.3934/era.2022107.

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<abstract><p>This paper deals with the existence of weak solutions for semilinear elliptic equation with nonlinearity on the boundary. We establish the existence of a maximal and a minimal weak solution between an ordered pair of sub- and supersolution for both monotone and nonmonotone nonlinearities. We use iteration argument when the nonlinearity is monotone. For the nonmonotone case, we utilize the surjectivity of a pseudomonotone and coercive operator, Zorn's lemma and a version of Kato's inequality.</p></abstract>

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36

LUKISHOVA, SVETLANA G. "NONLINEAR OPTICAL RESPONSE OF CYANOBIPHENYL LIQUID CRYSTALS TO HIGH-POWER, NANOSECOND LASER RADIATION." Journal of Nonlinear Optical Physics & Materials 09, no. 03 (September 2000): 365–411. http://dx.doi.org/10.1142/s0218863500000212.

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Results from investigations are summarized into: (1) transient refractive and absorptive (two-photon) nonlinearities at 0.532 μm by the Z-scan method, and (2) reflective nonlinearity in the near-IR, of linearly nonabsorbing cyanobiphenyl liquid crystals under nanosecond laser irradiation. (1) For isotropic liquid crystals at the several-nanosecond time scale and several tens-micrometers beam-waist-diameter, transient molecular-reorientation and thermal/density refractive nonlinearities compete in changing the sign of the total transient refractive nonlinearity. For the different, given pulse durations, the influence of coupled thermal and density effects on nonlinear refraction depends, through buildup time, on the beam-waist diameter. Nonlinear absorption coefficients depend on the incident intensity. For planar nematic layers, cumulative effects in heating (and in refractive nonlinearity) were observed even at low, 2–10 Hz pulse repetition rate. These results are useful for optical power limiting applications, and for intensity and beam-quality sensors of pulsed, high-power lasers. (2) Reflective nonlinearity of chiral-nematic (cholesteric) mirrors near selective reflection conditions for circular polarized light at λ=1.064 μm was studied both under free space irradiation and inside a laser resonator. Specially chosen experimental irradiation conditions make it possible to attribute the observed changing of reflectivity to athermal helix unwinding by the optical field. The results can find applications in laser-resonator mirrors, Q-switches and soft apertures for beam-profile formation, and also in showing the limits of use cholesteric optical elements in high-power laser beams.

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37

Kader, A. H. Abdel, M. S. Abdel Latif, and Dumitru Baleanu. "Modulation instability and some dark and bright optical solitons in weakly nonlocal media with general polynomial law nonlinearity." Modern Physics Letters B 34, no. 04 (December 20, 2019): 2050061. http://dx.doi.org/10.1142/s021798492050061x.

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In this paper, the modulation instability (MI) of a [Formula: see text]-dimensional nonlocal nonlinear Schrödinger equation with general polynomial law nonlinearity and an external potential is investigated. Some new dark and bright soliton solutions are obtained for polynomial law nonlinearities of third and fifth orders.

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38

Samet, H. Chachou, M. Benarous, M. Asad-uz-zaman, and U. Al Khawaja. "Effect of Third-Order Dispersion on the Solitonic Solutions of the Schrödinger Equations with Cubic Nonlinearity." Advances in Mathematical Physics 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/323591.

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We derive the solitonic solution of the nonlinear Schrödinger equation with cubic nonlinearity, complex potentials, and time-dependent coefficients using the Darboux transformation. We establish the integrability condition for the most general nonlinear Schrödinger equation with cubic nonlinearity and discuss the effect of the coefficients of the higher-order terms in the solitonic solution. We find that the third-order dispersion term can be used to control the soliton motion without the need for an external potential. We discuss the integrability conditions and find the solitonic solution of some of the well-known nonlinear Schrödinger equations with cubic nonlinearity and time-dependent coefficients. We also investigate the higher-order nonlinear Schrödinger equation with cubic and quintic nonlinearities.

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39

Innocenti, Giacomo, and Paolo Paoletti. "A Novel Dissipativity-Based Control for Inexact Nonlinearity Cancellation Problems." Mathematical Problems in Engineering 2015 (2015): 1–13. http://dx.doi.org/10.1155/2015/319761.

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When dealing with linear systems feedback interconnected with memoryless nonlinearities, a natural control strategy is making the overall dynamics linear at first and then designing a linear controller for the remaining linear dynamics. By canceling the original nonlinearity via a first feedback loop, global linearization can be achieved. However, when the controller is not capable of exactly canceling the nonlinearity, such control strategy may provide unsatisfactory performance or even induce instability. Here, the interplay between accuracy of nonlinearity approximation, quality of state estimation, and robustness of linear controller is investigated and explicit conditions for stability are derived. An alternative controller design based on such conditions is proposed and its effectiveness is compared with standard methods on a benchmark system.

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40

Brake, M. R., and D. J. Segalman. "Modelling localized nonlinearities in continuous systems via the method of augmentation by non-smooth basis functions." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2158 (October 8, 2013): 20130260. http://dx.doi.org/10.1098/rspa.2013.0260.

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Existing solutions for continuous systems with localized, non-smooth nonlinearities (such as impacts) focus on exact methods for satisfying the nonlinear constitutive equations. Exact methods often require that the non-smooth nonlinearities be expressed as piecewise-linear functions, which results in a series of mapping equations between each linear regime of the nonlinearities. This necessitates exact transition times between each linear regime of the nonlinearities, significantly increasing computational time, and limits the analysis to only considering a small number of nonlinearities. A new method is proposed in which the exact, nonlinear constitutive equations are satisfied by augmenting the system's primary basis functions with a set of non-smooth basis functions. Two consequences are that precise contact times are not needed, enabling greater computational efficiency than exact methods, and localized nonlinearities are not limited to piecewise-linear functions. Since each nonlinearity requires only a few non-smooth basis functions, this method is easily expanded to handle large numbers of nonlinearities throughout the domain. To illustrate the application of this method, a pinned–pinned beam example is presented. Results demonstrate that this method requires significantly fewer basis functions to achieve convergence, compared with linear and exact methods, and that this method is orders of magnitude faster than exact methods.

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41

Emile-Geay, J., and M. P. Tingley. "Inferring climate variability from nonlinear proxies: application to paleo-ENSO studies." Climate of the Past Discussions 11, no. 4 (July 8, 2015): 2763–809. http://dx.doi.org/10.5194/cpd-11-2763-2015.

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Abstract. Inferring climate from paleodata frequently assumes a direct, linear relationship between the two, which is seldom met in practice. Here we simulate an idealized proxy characterized by a nonlinear, thresholded relationship with surface temperature, and demonstrate the pitfalls of ignoring nonlinearities in the proxy–climate relationship. We explore three approaches to using this idealized proxy to infer past climate: (i) methods commonly used in the paleoclimate literature, without consideration of nonlinearities, (ii) the same methods, after empirically transforming the data to normality to account for nonlinearities, (iii) using a Bayesian model to invert the mechanistic relationship between the climate and the proxy. We find that neglecting nonlinearity often exaggerates changes in climate variability between different time intervals, and leads to reconstructions with poorly quantified uncertainties. In contrast, explicit recognition of the nonlinear relationship, using either a mechanistic model or an empirical transform, yields significantly better estimates of past climate variations, with more accurate uncertainty quantification. We apply these insights to two paleoclimate settings. Accounting for nonlinearities in the classical sedimentary record from Laguna Pallcacocha leads to quantitative departures from the results of the original study, and markedly affects the detection of variance changes over time. A comparison with the Lake Challa record, also a nonlinear proxy for El Niño–Southern Oscillation, illustrates how inter-proxy comparisons may be altered when accounting for nonlinearity. The results hold implications for how nonlinear recorders of normally distributed climate variables are interpreted, compared to other proxy records, and incorporated into multiproxy reconstructions.

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42

Emile-Geay, J., and M. Tingley. "Inferring climate variability from nonlinear proxies: application to palaeo-ENSO studies." Climate of the Past 12, no. 1 (January 15, 2016): 31–50. http://dx.doi.org/10.5194/cp-12-31-2016.

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Abstract. Inferring climate from palaeodata frequently assumes a direct, linear relationship between the two, which is seldom met in practice. Here we simulate an idealized proxy characterized by a nonlinear, thresholded relationship with surface temperature, and we demonstrate the pitfalls of ignoring nonlinearities in the proxy–climate relationship. We explore three approaches to using this idealized proxy to infer past climate: (i) methods commonly used in the palaeoclimate literature, without consideration of nonlinearities; (ii) the same methods, after empirically transforming the data to normality to account for nonlinearities; and (iii) using a Bayesian model to invert the mechanistic relationship between the climate and the proxy. We find that neglecting nonlinearity often exaggerates changes in climate variability between different time intervals and leads to reconstructions with poorly quantified uncertainties. In contrast, explicit recognition of the nonlinear relationship, using either a mechanistic model or an empirical transform, yields significantly better estimates of past climate variations, with more accurate uncertainty quantification. We apply these insights to two palaeoclimate settings. Accounting for nonlinearities in the classical sedimentary record from Laguna Pallcacocha leads to quantitative departures from the results of the original study, and it markedly affects the detection of variance changes over time. A comparison with the Lake Challa record, also a nonlinear proxy for El Niño–Southern Oscillation, illustrates how inter-proxy comparisons may be altered when accounting for nonlinearity. The results hold implications for how univariate, nonlinear recorders of normally distributed climate variables are interpreted, compared to other proxy records, and incorporated into multiproxy reconstructions.

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43

CAFAGNA, DONATO, and GIUSEPPE GRASSI. "DECOMPOSITION METHOD FOR STUDYING SMOOTH CHUA'S EQUATION WITH APPLICATION TO HYPERCHAOTIC MULTISCROLL ATTRACTORS." International Journal of Bifurcation and Chaos 17, no. 01 (January 2007): 209–26. http://dx.doi.org/10.1142/s0218127407017276.

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This paper focuses on the numerical study of chaotic dynamics via the Adomian decomposition method. The approach, which provides series solutions of the system equations, is first applied to Chua's circuit and Chua's oscillator, both with cubic nonlinearity. Successively, the method is utilized for obtaining hyperchaotic multiscroll attractors in a ring of three Chua's circuits, where the smooth nonlinearities are Hermite interpolating polynomials. The reported examples show that the approach presents two main features, i.e. the system nonlinearity is preserved and the chaotic solution is provided in a closed form.

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44

Sun, Mingzheng, Jiabao Su, and Hongrui Cai. "Multiple solutions for the p-Laplacian equations with concave nonlinearities via Morse theory." Communications in Contemporary Mathematics 19, no. 03 (April 5, 2017): 1650014. http://dx.doi.org/10.1142/s0219199716500140.

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In this paper, by Morse theory, we study the existence and multiplicity of solutions for the [Formula: see text]-Laplacian equation with a “concave” nonlinearity and a parameter. In our results, we do not need any additional global condition on the nonlinearities, except for a subcritical growth condition.

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45

Candon, M., R. Carrese, H. Ogawa, and P. Marzocca. "Identification of freeplay and aerodynamic nonlinearities in a 2D aerofoil system with via higher-order spectra." Aeronautical Journal 121, no. 1244 (October 2017): 1530–60. http://dx.doi.org/10.1017/aer.2017.88.

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ABSTRACTHigher-Order Spectra (HOS) are used to characterise the nonlinear aeroelastic behaviour of a plunging and pitching 2-degree-of-freedom aerofoil system by diagnosing structural and/or aerodynamic nonlinearities via the nonlinear spectral content of the computed displacement signals. The nonlinear aeroelastic predictions are obtained from high-fidelity viscous fluid-structure interaction simulations. The power spectral, bi-spectral and tri-spectral densities are used to provide insight into the functional form of both freeplay and inviscid/viscous aerodynamic nonlinearities with the system displaying both low- and high-amplitude Limit Cycle Oscillation (LCO). It is shown that in the absence of aerodynamic nonlinearity (low-amplitude LCO) the system is characterised by cubic phase coupling only. Furthermore, when the amplitude of the oscillations becomes large, aerodynamic nonlinearities become prevalent and are characterised by quadratic phase coupling. Physical insights into the nonlinearities are provided in the form of phase-plane diagrams, pressure coefficient distributions and Mach number flowfield contours.

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46

Novotný, Ladislav. "Finite Element Simulation of Bending of Steel Bar Including Plasticity." Applied Mechanics and Materials 816 (November 2015): 182–87. http://dx.doi.org/10.4028/www.scientific.net/amm.816.182.

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The article presents the use of finite element method for the simulation of cold forming process. The numerical simulation of a real technological operation of bending a rod by an industrial bender. Within the simulation, different types of nonlinearities, namely of material nonlinearity, resulting from the flexible plastic material properties of the rod, are considered, geometric nonlinearities result from large displacement and nonlinear contact. This paper briefly describes the elastic – plastic material model. Numerical analysis confirmed the appropriateness of the use of finite element method in the simulation of technological operations and the eventual possible optimization of these processes.

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47

Cooper, Russell, and Jonathan L. Willis. "A Comment on the Economics of Labor Adjustment: Mind the Gap: Evidence from a Monte Carlo Experiment: Reply." American Economic Review 99, no. 5 (December 1, 2009): 2267–76. http://dx.doi.org/10.1257/aer.99.5.2267.

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This note responds to Christian Bayer (2009). Cooper and Willis (2004), hereafter CW, find the aggregate nonlinearities reported in Ricardo Caballero and Eduardo Engel (1993) and Caballero, Engel, and John Haltiwanger (1997) reflect mismeasurement of the employment gap, not nonlinearities in plant-level adjustment. Bayer concludes the CW result is not robust to alternative aggregate shock processes. We concur, but argue that the nonlinearity created by mismeasurement does not disappear. Instead, it is directly related to the level of the aggregate shock. The CW findings are robust for the natural case of unobserved gaps. (JEL E24, J23)

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48

Korman, Philip, Yi Li, and Tiancheng Ouyang. "Exact multiplicity results for boundary value problems with nonlinearities generalising cubic." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 126, no. 3 (1996): 599–616. http://dx.doi.org/10.1017/s0308210500022927.

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Using techniques of bifurcation theory we present two exact multiplicity results for boundary value problems of the typeThe first result concerns the case when the nonlinearity is independent of x and behaves like a cubic in u. The second one deals with a class of nonlinearities with explicit x dependence.

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49

Chernorutskii, Vladimir V., and Mark A. Krasnosel'skii. "Differential inequalities for hysteresis systems." Journal of Applied Mathematics and Stochastic Analysis 9, no. 4 (January 1, 1996): 459–68. http://dx.doi.org/10.1155/s1048953396000408.

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The theory of differential inequalities is extended to functional-differential equations with hysteresis nonlinearities. A key feature is the existence of a semiorder of the state space of nonlinearity and a special monotonicity of the righthand side of differential inequality.This article is dedicated to the memory of Roland L. Dobrushin.

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50

Smirnov, Yury G., Eugenii Yu Smol’kin, and Dmitry V. Valovik. "Nonlinear Double-Layer Bragg Waveguide: Analytical and Numerical Approaches to Investigate Waveguiding Problem." Advances in Numerical Analysis 2014 (January 22, 2014): 1–11. http://dx.doi.org/10.1155/2014/231498.

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The paper is concerned with propagation of surface TE waves in a circular nonhomogeneous two-layered dielectric waveguide filled with nonlinear medium. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of the Green function. The existence of propagating TE waves for chosen nonlinearity (the Kerr law) is proved using the contraction mapping method. Conditions under which k waves can propagate are obtained, and intervals of localization of the corresponding propagation constants are found. For numerical solution of the problem, a method based on solving an auxiliary Cauchy problem (the shooting method) is proposed. In numerical experiment, two types of nonlinearities are considered and compared: the Kerr nonlinearity and nonlinearity with saturation. New propagation regime is found.

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