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Journal articles on the topic 'Nonlinear first order planar systems'

Author: Grafiati

Published: 14 March 2023

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1

El-Ganaini, Shoukry Ibrahim Atia. "The First Integral Method to the Nonlinear Schrodinger Equations in Higher Dimensions." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/349173.

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The first integral method introduced by Feng is adopted for solving some important nonlinear partial differential equations, including the (2 + 1)-dimensional hyperbolic nonlinear Schrodinger (HNLS) equation, the generalized nonlinear Schrodinger (GNLS) equation with a source, and the higher-order nonlinear Schrodinger equation in nonlinear optical fibers. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner.

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2

Wang, Hong Qi. "Dynamics Modeling of the Planar Double Inverted Pendulum." Applied Mechanics and Materials 195-196 (August 2012): 17–22. http://dx.doi.org/10.4028/www.scientific.net/amm.195-196.17.

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planar double inverted pendulum is a strong coupling, uncertain and complex nonlinear system, and the dynamics model of which is the basis of control, simulation and analysis. In the paper coordinate systems of the planar double inverted pendulum were first defined, and then the dynamics model of which was built up based on screw theory and the Lagrange principle. The modeling method used being systematic and standardized, it is easy to extend to dynamics modeling of higher order planar inverted pendulums or other multi-body systems.

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3

Raaghul, B., M. R. Kannan, and T. Vijayakumar. "First Order Hyper polarizability and Intramolecular Charge Transfer of N-Ethyl-N-(2-Hydroxyethyl)-4-(4-Nitrophenylazo) Aniline for Photonic Applications." IOP Conference Series: Materials Science and Engineering 1219, no. 1 (January 1, 2022): 012035. http://dx.doi.org/10.1088/1757-899x/1219/1/012035.

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Abstract We report a computational methodology for estimating the first order hyperpolarizability ((3) of N-ethyl-N-(2-hydroxyethyl)-4-(4-nitrophenylazo) aniline, a nonlinear optical (NLO) chromophore being used as anazobenzene dye. The planar and non-planar structures of N-ethyl-N-(2-hydroxyethyl)-4-(4-nitrophenylazo) aniline were optimized by B3LYP/6-311++G (D,P) basis set using GAUSSIAN’09W program package. The dipole moment, static polarizability, first order hyperpolarizability were studied using HF/6-31G (D) basis set. The (3 value of the planar and non-planar structure of N-ethyl-N-(2-hydroxyethyl)-4-(4-nitrophenylazo) aniline is 196 and 124 times higher than that of urea standard, respectively, which qualify both of the novel molecular systems as highly efficient for second order NLO applications and adds evidence to the importance of derealization of ^-electrons in a system for effective second order NLO processes. Experiments were also performed using an Nd-YAG laser of pulses in the range of 5-10 nanoseconds with a tunable peak power of 4.7 W.

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4

Li, Jibin. "Exact Solutions and Bifurcations in Invariant Manifolds for a Nonic Derivative Nonlinear Schrödinger Equation." International Journal of Bifurcation and Chaos 26, no. 08 (July 2016): 1650136. http://dx.doi.org/10.1142/s0218127416501364.

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Propagating modes in a class of nonic derivative nonlinear Schrödinger equations incorporating ninth order nonlinearity are investigated by the method of dynamical systems. Because the functions [Formula: see text] and [Formula: see text] in the solutions [Formula: see text], [Formula: see text] satisfy a four-dimensional integral system having two first integrals (i.e. the invariants of motion), a planar dynamical system for the squared wave amplitude [Formula: see text] can be derived in the invariant manifold of the four-dimensional integrable system. By using the bifurcation theory of dynamical systems, under different parameter conditions, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions for this planar dynamical system can be given. Therefore, under some parameter conditions, solutions [Formula: see text] and [Formula: see text] can be exactly obtained. Thirty six exact explicit solutions of equation are derived.

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5

Fan, Zhihui, and Zhengdong Du. "Bifurcation of Periodic Orbits Crossing Switching Manifolds Multiple Times in Planar Piecewise Smooth Systems." International Journal of Bifurcation and Chaos 29, no. 12 (November 2019): 1950160. http://dx.doi.org/10.1142/s0218127419501608.

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In this paper, we discuss the bifurcation of periodic orbits in planar piecewise smooth systems with discontinuities on finitely many smooth curves intersecting at the origin. We assume that the unperturbed system has either a limit cycle or a periodic annulus such that the limit cycle or each periodic orbit in the periodic annulus crosses every switching curve transversally multiple times. When the unperturbed system has a limit cycle, we give the conditions for its stability and persistence. When the unperturbed system has a periodic annulus, we obtain the expression of the first order Melnikov function and establish sufficient conditions under which limit cycles can bifurcate from the annulus. As an example, we construct a concrete nonlinear planar piecewise smooth system with three zones with 11 limit cycles bifurcated from the periodic annulus.

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6

Aldhafeeri, Anwar, and Muneerah Al Nuwairan. "Bifurcation of Some Novel Wave Solutions for Modified Nonlinear Schrödinger Equation with Time M-Fractional Derivative." Mathematics 11, no. 5 (March 2, 2023): 1219. http://dx.doi.org/10.3390/math11051219.

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In this paper, we investigate the time M-fractional modified nonlinear Schrödinger equation that describes the propagation of rogue waves in deep water. Periodic, solitary, and kink (or anti-kink) wave solutions are discussed using the bifurcation theory for planar integrable systems. Some new wave solutions are constructed using the first integral for the traveling wave system. The degeneracy of the obtained solutions is investigated by using the transition between orbits. We visually explore some of the solutions using graphical representations for different values of the fractional order.

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7

Ye, Jiazhen, Yuki Todo, Zheng Tang, Bin Li, and Yu Zhang. "Artificial Visual System for Orientation Detection." Electronics 11, no. 4 (February 13, 2022): 568. http://dx.doi.org/10.3390/electronics11040568.

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The human visual system is one of the most important components of the nervous system, responsible for visual perception. The research on orientation detection, in which neurons of the visual cortex respond only to a line stimulus in a particular orientation, is an important driving force of computer vision and biological vision. However, the principle underlying orientation detection remains a mystery. In order to solve this mystery, we first propose a completely new mechanism that explains planar orientation detection in a quantitative manner. First, we assume that there are planar orientation-detective neurons which respond only to a particular planar orientation locally and that these neurons detect local planar orientation information based on nonlinear interactions that take place on the dendrites. Then, we propose an implementation of these local planar orientation-detective neurons based on their dendritic computations, use them to extract the local planar orientation information, and infer the global planar orientation information from the local planar orientation information. Furthermore, based on this mechanism, we propose an artificial visual system (AVS) for planar orientation detection and other visual information processing. In order to prove the effectiveness of our mechanism and the AVS, we conducted a series of experiments on rectangular images which included rectangles of various sizes, shapes and positions. Computer simulations show that the mechanism can perfectly perform planar orientation detection regardless of their sizes, shapes and positions in all experiments. Furthermore, we compared the performance of both AVS and a traditional convolution neural network (CNN) on planar orientation detection and found that AVS completely outperformed CNN in planar orientation detection in terms of identification accuracy, noise resistance, computation and learning cost, hardware implementation and reasonability.

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8

Konstandakopoulou, Foteini, George Hatzigeorgiou, Konstantinos Evangelinos, Thomas Tsalis, and Ioannis Nikolaou. "A New Method to Evaluate the Post-Earthquake Performance and Safety of Reinforced Concrete Structural Frame Systems." Infrastructures 5, no. 2 (February 1, 2020): 16. http://dx.doi.org/10.3390/infrastructures5020016.

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This study examines the relation between maximum seismic displacements and residual displacements for reinforced concrete building structures. In order to achieve a reliable relationship between these critical structural parameters for the seismic performance of concrete buildings, an extensive parametric study is conducted by examining the nonlinear behavior of numerous planar framed structures. In this work, dynamic inelastic analyses are executed to investigate the seismic behavior of two sets of frames. The first group consists of four planar frames which have been designed for seismic and vertical loads according to modern structural codes while the second group also consists of four frames, which have been designed for vertical loads only, in order to examine older structures that have been designed using codes with inadequate seismic provisions. These two sets of buildings are subjected to various earthquakes with different amplitudes in order to develop a large structural response databank. On the basis of this wide-ranging parametric investigation, after an appropriate statistical analysis, simple empirical expressions are proposed for a straightforward and efficient evaluation of maximum seismic displacements of reinforced concrete buildings structures from their permanent deformation. Permanent displacements can be measured in-situ after strong ground motions as a post-earthquake assessment. It can be concluded that the measure of permanent deformation can be efficiently used to estimate the post-seismic performance level of reinforced concrete buildings.

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9

Rissel, Manuel, and Ya-Guang Wang. "Global exact controllability of ideal incompressible magnetohydrodynamic flows through a planar duct." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 103. http://dx.doi.org/10.1051/cocv/2021099.

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This article is concerned with the global exact controllability for ideal incompressible magnetohydrodynamics in a rectangular domain where the controls are situated in both vertical walls. First, global exact controllability via boundary controls is established for a related Elsässer type system by applying the return method, introduced in Coron [Math. Control Signals Syst. 5 (1992) 295–312]. Similar results are then inferred for the original magnetohydrodynamics system with the help of a special pressure-like corrector in the induction equation. Overall, the main difficulties stem from the nonlinear coupling between the fluid velocity and the magnetic field in combination with the aim of exactly controlling the system. In order to overcome some of the obstacles, we introduce ad-hoc constructions, such as suitable initial data extensions outside of the physical part of the domain and a certain weighted space.

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10

Steckiewicz, Adam, Kornelia Konopka, Agnieszka Choroszucho, and Jacek Maciej Stankiewicz. "Temperature Measurement at Curved Surfaces Using 3D Printed Planar Resistance Temperature Detectors." Electronics 10, no. 9 (May 7, 2021): 1100. http://dx.doi.org/10.3390/electronics10091100.

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In this article, novel 3D printed sensors for temperature measurement are presented. A planar structure of the resistive element is made, utilizing paths of a conductive filament embedded in an elastic base. Both electrically conductive and flexible filaments are used simultaneously during the 3D printing procedure, to form a ready–to–use measuring device. Due to the achieved flexibility, the detectors may be used on curved and irregular surfaces, with no concern for their possible damage. The geometry and properties of the proposed resistance detectors are discussed, along with a printing procedure. Numerical models of considered sensors are characterized, and the calculated current distributions as well as equivalent resistances of the different structures are compared. Then, a nonlinear influence of temperature on the resistance is experimentally determined for the exemplary planar sensors. Based on these results, using first–order and hybrid linear–exponential approximations, the analytical formulae are derived. Additionally, the device to measure an average temperature from several measuring surfaces is considered. Since geometry of the sensor can be designed utilizing presented approach and printed by applying fused deposition modeling, the functional device can be customized to individual needs.

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11

Singh, Ram Binoy. "Planar Oscillation of the Satellite Near Parametric Resonance in the Elliptic Orbit." International Astronomical Union Colloquium 132 (1993): 391–97. http://dx.doi.org/10.1017/s0252921100066288.

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AbstractThis paper is devoted to the study of motion of an artificial satellite relative to its centre of mass near parametric resonance in elliptic orbit. It is well known fact that the satellite, of the form of an ellipsoid with three unequal axes, while moving about the central planet, oscillates about the stable position of equilibrium (the longest axis of the satellite coinciding with the radius vector of its centre of mass). The oscillation of the satellite about this position of equilibrium in the orbital plane of its centre of mass is described by a well known second order nonlinear differential equation with a periodic sine force. Naturally there will be resonance cases (main as well as parametric) for such a systems. In the previous author’s work, it was discovered a series of parametric resonances for the system, which corresponds to n = ½k where k is a non-zero integer and n is a parameter depending on the shape of the satellite. The parametric resonance, for k = 1, has been considered here. The first approximate solution of the equation of motion has been obtained by Eogoliubov-Krilov method with e (the eccentricity of the orbit of the centre of mass of the satellite) as the small parameter. This method enables us to visualise the oscillation of the satellite for the resonance case as well as near the resonance. Three stationary values of the amplitudes and phase of oscillation have been obtained, out of which only one is stable near this particular parametric resonance. At the resonance there appear only one stationary regime of oscillation with a very small amplitude.

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12

Biagioni, H. A. "Generalized solutions to nonlinear first-order systems." Monatshefte f�r Mathematik 118, no. 1-2 (March 1994): 7–20. http://dx.doi.org/10.1007/bf01305770.

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13

Xu, R., and K. Komvopoulos. "A Reduced-Order Dynamic Model of Nonlinear Oscillating Devices." Journal of Dynamic Systems, Measurement, and Control 129, no. 4 (January 22, 2007): 514–21. http://dx.doi.org/10.1115/1.2745858.

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A reduced-order dynamic model is presented for nonlinear devices subjected to in-plane oscillatory motion. Comparisons between numerical and finite element results demonstrate that the nonlinear behavior of a planar resonator can be predicted accurately by the derived dynamic model with significantly less computation. Simulation results illustrate the effects of nonlinear stiffness, damping ratio, electrostatic driving force, and device dimensions on the nonlinear dynamic behavior. The analysis yields two possible stable responses, depending on the initial rotation angle and rotation rate. The present dynamic model can be easily modified to analyze the nonlinear response of various planar resonators.

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14

Dacorogna, Bernard, and Paolo Marcellini. "Cauchy–Dirichlet Problem for First Order Nonlinear Systems." Journal of Functional Analysis 152, no. 2 (February 1998): 404–46. http://dx.doi.org/10.1006/jfan.1997.3172.

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15

Yang, Lijun, Chen Wang, Qingfei Fu, Minglong Du, and Mingxi Tong. "Weakly nonlinear instability of planar viscous sheets." Journal of Fluid Mechanics 735 (October 23, 2013): 249–87. http://dx.doi.org/10.1017/jfm.2013.502.

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AbstractA second-order instability analysis has been performed for sinuous disturbances on two-dimensional planar viscous sheets moving in a stationary gas medium using a perturbation technique. The solutions of second-order interface disturbances have been derived for both temporal instability and spatial instability. It has been found that the second-order interface deformation of the fundamental sinuous wave is varicose or dilational, causing disintegration and resulting in ligaments which are interspaced by half a wavelength. The interface deformation has been presented; the breakup time for temporal instability and breakup length for spatial instability have been calculated. An increase in Weber number and gas-to-liquid density ratio extensively increases both the temporal or spatial growth rate and the second-order initial disturbance amplitude, resulting in a shorter breakup time or length, and a more distorted surface deformation. Under normal conditions, viscosity has a stabilizing effect on the first-order temporal or spatial growth rate, but it plays a dual role in the second-order disturbance amplitude. The overall effect of viscosity is minor and complicated. In the typical condition, in which the Weber number is 400 and the gas-to-liquid density ratio is 0.001, viscosity has a weak stabilizing effect when the Reynolds number is larger than 150 or smaller than 10; when the Reynolds number is between 150 and 10, viscosity has a weak destabilizing effect.

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16

Jiao, Jia, Wenlei Li, and Qingjian Zhou. "Formal First Integrals of General Dynamical Systems." Advances in Mathematical Physics 2016 (2016): 1–9. http://dx.doi.org/10.1155/2016/1036089.

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The goal of this paper is trying to make a complete study on the integrability for general analytic nonlinear systems by first integrals. We will firstly give an exhaustive discussion on analytic planar systems. Then a class of higher dimensional systems with invariant manifolds will be considered; we will develop several criteria for existence of formal integrals and give some applications to illustrate our results at last.

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17

Wang, Chen, Lijun Yang, and Hanyu Ye. "Nonlinear dual-mode instability of planar liquid sheets." Journal of Fluid Mechanics 778 (August 6, 2015): 621–52. http://dx.doi.org/10.1017/jfm.2015.407.

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The nonlinear temporal instability of gas-surrounded planar liquid sheets, whose linear instability contains both sinuous and varicose modes, is studied. Both the weakly nonlinear analysis using a second-order perturbation expansion and the numerical simulation using a boundary integral method have been applied. Their comparison shows that the weakly nonlinear analysis can precisely predict the shapes of sheets for most of the time of disturbance evolution and qualitatively explain the instability mechanism when sheets break up. Both the first harmonics of the linear sinuous mode and linear varicose mode are varicose; they contribute to the breakup of sheets, but the first harmonic generated by the coupling between the linear sinuous and varicose modes is sinuous; it plays an important role in modulating the wave profile. The instability with various initial phase differences between the upper and lower interfaces is examined. Except for the varicose initial disturbance, the linear sinuous mode dominates in the shapes of sheets when their amplitudes grow large. Within the second-order analysis, the major modes that can cause the breakup include the linear varicose mode, the first harmonic of the linear sinuous mode and the first harmonic of the linear varicose mode. The effects of various flow parameters have been investigated. At relatively large wavenumbers where approximate analytical and numerical results agree well when sheets break up, increasing the wavenumber reduces the wave amplitude. Reducing the initial disturbance amplitude makes the first harmonic of the linear sinuous mode the dominant mode in causing the breakup. Increasing the Weber number or gas-to-liquid density ratio significantly reduces breakup time and enhances instability.

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18

Yankov, Kaloyan. "PHASE BEHAVIOUR OF FIRST - ORDER SYSTEMS." Applied Researches in Technics, Technologies and Education 16, no. 2 (2018): 131–37. http://dx.doi.org/10.15547/artte.2018.02.008.

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The phase-plane method gives possibility to study the stability of systems described by linear and nonlinear differential equations. The article is devoted to the capabilities of MathCad for analysis of first order differential equations. An algorithm is proposed and Mathcad's specific operators for the construction and analysis of phase trajectories are described. Approaches for calculation of equilibrium points and determination the type of bifurcation in function of parameter are described. The proposed algorithm is applied to the dose-response curve of the antibiotic tubazid.

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19

Shahmansoorian, Aref. "Continuous Stabilizing of First Order Single Input Nonlinear Systems." Intelligent Control and Automation 02, no. 03 (2011): 182–85. http://dx.doi.org/10.4236/ica.2011.23022.

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20

Wang, Yongzhao, Qian Liu, and Qiansheng Feng. "Periodic problem of first order nonlinear uncertain dynamic systems." Journal of Nonlinear Sciences and Applications 10, no. 12 (December 9, 2017): 6288–97. http://dx.doi.org/10.22436/jnsa.010.12.13.

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21

Lattanzio, Corrado, and Wen-An Yong. "HYPERBOLIC-PARABOLIC SINGULAR LIMITS FOR FIRST-ORDER NONLINEAR SYSTEMS." Communications in Partial Differential Equations 26, no. 5-6 (April 30, 2001): 939–64. http://dx.doi.org/10.1081/pde-100002384.

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22

Zhang, T., S. S. Ge, C. C. Hang, and T. Y. Chai. "Adaptive control of first-order systems with nonlinear parameterization." IEEE Transactions on Automatic Control 45, no. 8 (2000): 1512–16. http://dx.doi.org/10.1109/9.871761.

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23

Su, Khac Huan, Jaeyun Yim, Wonhee Kim, and Youngwoo Lee. "Lyapunov-Based Controller Using Nonlinear Observer for Planar Motors." Mathematics 10, no. 13 (June 22, 2022): 2177. http://dx.doi.org/10.3390/math10132177.

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In general, it is not easy work to design controllers and observers for high-order nonlinear systems. Planar motors that are applied to semiconductor wafer-stage processes have 14th-order nonlinear dynamics and require high resolution for position tracking. Thus, many sensors are required to achieve enhanced tracking performance because there are many state variables. To handle these problems, we developed a Lyapunov-based controller to improve the position tracking performance. Consequently, a nonlinear observer (NOB) was also developed to estimate all of the state variables including the position, the velocity, and the phase current using only position feedback. The closed-loop stability is proved through Lyapunov theory and the input-to-state stability (ISS) property. The proposed method was evaluated based on the simulation results and compared with the conventional proportional–integral–derivative (PID) control method to show the improvement in the position tracking performance.

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24

Pavlov, Peter. "Nonlinear damped vibrations of planar discrete systems - numerical and experimental modelling." MATEC Web of Conferences 211 (2018): 02006. http://dx.doi.org/10.1051/matecconf/201821102006.

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The consistently conducted analytical, numerical and experimental studies of nonlinear damped vibrations of planar discrete systems are presented in the paper. The combined methodology is applied to a horizontal vibrating system, consisting of two translational moving bodies connected by three springs. The non-linear nature of the damping is due to the dry friction forces accompanying the vibrating process. The mathematical model of the vibrating system is composed in a matrix form by the second order Lagrange equations. Numerical studies are realized in two ways. Firstly, in the Simulink environment, a simulation model was composed. Then, in the MATLAB environment, an animation model was developed using the third animation method offered by the programming system. The experimental studies were conducted by stand for study the small vibrations of discrete planar systems. The stand is part of the experimental equipment of the Lab for numerical and experimental dynamic modelling, UACEG, Sofia, Bulgaria. (www.dlab-uacg-bg.eu). All models - the dynamic model and its corresponding mathematical, simulation, animation and experimental model are open to additional bodies to obtain discrete vibrating systems with a larger number of degrees of freedom.

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IBRAGIMOV, N. H. "INTEGRATION OF SYSTEMS OF FIRST-ORDER EQUATIONS ADMITTING NONLINEAR SUPERPOSITION." Journal of Nonlinear Mathematical Physics 16, sup1 (January 2009): 137–47. http://dx.doi.org/10.1142/s1402925109000364.

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26

Kossowski, Igor, and Bogdan Przeradzki. "First order systems of odes with nonlinear nonlocal boundary conditions." Electronic Journal of Qualitative Theory of Differential Equations, no. 73 (2015): 1–10. http://dx.doi.org/10.14232/ejqtde.2015.1.73.

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Tang, X. H., Qi-Ming Zhang, and Meirong Zhang. "Lyapunov-type inequalities for the first-order nonlinear Hamiltonian systems." Computers & Mathematics with Applications 62, no. 9 (November 2011): 3603–13. http://dx.doi.org/10.1016/j.camwa.2011.09.011.

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28

Palomba, Ilaria, Dario Richiedei, and Alberto Trevisani. "Reduced-Order Observers for Nonlinear State Estimation in Flexible Multibody Systems." Shock and Vibration 2018 (November 1, 2018): 1–12. http://dx.doi.org/10.1155/2018/6538737.

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Modern control schemes adopted in multibody systems take advantage of the knowledge of a large set of measurements of the most important state variables to improve system performances. In the case of flexible-link multibody systems, however, the direct measurement of these state variables is not usually possible or convenient. Hence, it is necessary to estimate them through accurate models and a reduced set of measurements ensuring observability. In order to cope with the large dimension of models adopted for flexible multibody systems, this paper exploits model reduction for synthesizing reduced-order nonlinear state observers. Model reduction is done through a modified Craig-Bampton strategy that handles effectively nonlinearities due to large displacements of the mechanism and through a wise selection of the most important coordinates to be retained in the model. Starting from such a reduced nonlinear model, a nonlinear state observer is developed through the extended Kalman filter (EKF). The method is applied to the numerical test case of a six-bar planar mechanism. The smaller size of the model, compared with the original one, preserves accuracy of the estimates while reducing the computational effort.

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LIN, MU, YUN TANG, GUANRONG CHEN, and YUMING SHI. "SIMPLEST NORMAL FORMS FOR PLANAR SYSTEMS ON EQUILIBRIUM MANIFOLDS." International Journal of Bifurcation and Chaos 19, no. 05 (May 2009): 1695–707. http://dx.doi.org/10.1142/s0218127409023780.

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Equilibrium manifold is a manifold that consists of equilibrium points. Planar systems with one-dimensional equilibrium manifolds are considered in this paper. First, for such planar systems, a unified equation with the horizontal axis as the equilibrium curve is formulated. Then, according to the corresponding linearized systems, different cases are discussed: For the nondegenerate case, the simplest normal form of a system with simplified Bogdanov–Takens singularities is obtained; for the general first-order degenerative case, the simplest normal forms are completely characterized; finally, for the general higher-order degenerative case, deduction of the simplest normal form is illustrated.

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He, Xue Jun, Shi Yun Zhang, and Kai Wei. "The Non-Planar Nonlinear Dynamic Model of the Inclined Cable with Lumped Masses." Applied Mechanics and Materials 353-356 (August 2013): 3243–47. http://dx.doi.org/10.4028/www.scientific.net/amm.353-356.3243.

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A nonlinear dynamic model for non-planar vibration of inclined cable with three lumped masses was developed by Hamilton principle, which considered the influence of the lumped masses, the external excitation and the inclination angle of the cable. The partial differential equations of this system were discretized to four degree of freedom ordinary differential equations by Galerkin method. The results show that there were complex model coupling between in-planar and out-planar model, first and second order model in this nonlinear dynamic system.

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31

Benedikt, Michael, Bart Kuijpers, Christof Löding, Jan Van den Bussche, and Thomas Wilke. "A characterization of first-order topological properties of planar spatial data." Journal of the ACM 53, no. 2 (March 2006): 273–305. http://dx.doi.org/10.1145/1131342.1131346.

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32

Puźniakowska-Gałuch, Elżbieta. "Implicit difference methods for nonlinear first order partial functional differential systems." Applicationes Mathematicae 37, no. 4 (2010): 459–82. http://dx.doi.org/10.4064/am37-4-5.

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33

Y. Abdallah, Ahmed. "Attractors for first order lattice systems with almost periodic nonlinear part." Discrete & Continuous Dynamical Systems - B 25, no. 4 (2020): 1241–55. http://dx.doi.org/10.3934/dcdsb.2019218.

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34

Uslu, Hande, and Murat Sarı. "Monte Carlo based stochastic approach for first order nonlinear ODE systems." Pamukkale University Journal of Engineering Sciences 26, no. 1 (2020): 133–39. http://dx.doi.org/10.5505/pajes.2019.25493.

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Jaroš, Jaroslav, and Takaŝi Kusano. "Extinct Singular Solutions of First Order Systems of Nonlinear Differential Equations." Funkcialaj Ekvacioj 57, no. 3 (2014): 467–75. http://dx.doi.org/10.1619/fesi.57.467.

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36

Zhao, Wenxiao, Han-Fu Chen, Er-Wei Bai, and Kang Li. "Local variable selection of nonlinear nonparametric systems by first order expansion." Systems & Control Letters 111 (January 2018): 1–8. http://dx.doi.org/10.1016/j.sysconle.2017.10.001.

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Brogliato, Bernard, Alexandre Trofino-Neto, and Rogelio Lozano. "Robust adaptive control of a class of nonlinear first order systems." Automatica 28, no. 4 (July 1992): 795–801. http://dx.doi.org/10.1016/0005-1098(92)90039-i.

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38

Kalita, E. A. "On nonlinear higher order elliptic systems positive in the first derivatives." Ukrainian Mathematical Journal 45, no. 7 (July 1993): 1042–48. http://dx.doi.org/10.1007/bf01057451.

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39

Choi, Chan Kyu, and Hong Hee Yoo. "Uncertainty analysis of nonlinear systems employing the first-order reliability method." Journal of Mechanical Science and Technology 26, no. 1 (January 2012): 39–44. http://dx.doi.org/10.1007/s12206-011-1011-x.

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Zhang, Haoran, Jun Huang, and Siyuan He. "Fractional-Order Interval Observer for Multiagent Nonlinear Systems." Fractal and Fractional 6, no. 7 (June 25, 2022): 355. http://dx.doi.org/10.3390/fractalfract6070355.

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A framework of distributed interval observers is introduced for fractional-order multiagent systems in the presence of nonlinearity. First, a frame was designed to construct the upper and lower bounds of the system state. By using monotone system theory, the positivity of the error dynamics could be ensured, which implies that the bounds could trap the original state. Second, a sufficient condition was applied to guarantee the boundedness of distributed interval observers. Then, an extension of Lyapunov function in the fractional calculus field was the basis of the sufficient condition. An algorithm associated with the procedure of the observer design is also provided. Lastly, a numerical simulation is used to demonstrate the effectiveness of the distributed interval observer.

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GAZOR, MAJID, and PEI YU. "FORMAL DECOMPOSITION METHOD AND PARAMETRIC NORMAL FORMS." International Journal of Bifurcation and Chaos 20, no. 11 (November 2010): 3487–515. http://dx.doi.org/10.1142/s0218127410027830.

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We introduce a formal decomposition method for efficiently computing the parametric normal form of nonlinear dynamical systems with multiple parameters. Recently introduced notions of formal basis style and costyle are applied through formal decomposition method to obtain the simplest parametric normal form for degenerate nonlinear parametric center. The necessary formulas are derived and implemented using Maple to compute the simplest parametric normal form of degenerate and nondegenerate nonlinear centers. Our program computes the order of any planar parametric systems associated with this singularity.

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42

Edneral, Victor. "Application of Power Geometry and Normal Form Methods to the Study of Nonlinear ODEs." EPJ Web of Conferences 173 (2018): 01004. http://dx.doi.org/10.1051/epjconf/201817301004.

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This paper describes power transformations of degenerate autonomous polynomial systems of ordinary differential equations which reduce such systems to a non-degenerative form. Example of creating exact first integrals of motion of some planar degenerate system in a closed form is given.

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Liu, Rui, Michal Fečkan, Donal O’Regan, and Jinrong Wang. "Controllability Results for First Order Impulsive Fuzzy Differential Systems." Axioms 11, no. 9 (September 14, 2022): 471. http://dx.doi.org/10.3390/axioms11090471.

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In this paper, we investigate the controllability of first-order impulsive fuzzy differential equations. Using the direct construction method, the controllability of first-order linear impulsive fuzzy differential equations is considered with a<0, the (c1) solution, and a<0, the (c2) solution, respectively.In addition, by employing the Banach fixed-point theorem, the controllability of first-order nonlinear impulsive fuzzy differential equations is studied. Finally, examples are presented to illustrate our theoretical results.

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Ouni, Mohamed Hechmi El, and Nabil Ben Kahla. "NONLINEAR DYNAMIC ANALYSIS OF A CABLE UNDER FIRST AND SECOND ORDER PARAMETRIC EXCITATIONS." Journal of Civil Engineering and Management 18, no. 4 (September 11, 2012): 557–67. http://dx.doi.org/10.3846/13923730.2012.702994.

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It is well known that small periodic vibrations of a cable support through its axial direction produce large spectacular oscillations of the cable. This may occur when the frequency of the anchorage motion is close to the first natural frequency or twice the fundamental frequency of the cable. In this paper, a nonlinear dynamic study of a cable under first and second order parametric excitations is presented. The cable model takes into account sag as well as quadratic and cubic nonlinear couplings between in-plane and out-of-plane motions. As a numerical example, a single-d.o.f. planar model of a horizontal cable is used to study the effect of frequency and amplitude of excitation as well as the natural damping of the cable on its transient and steady state responses with a particular focus on the time needed to trigger first and second order parametric resonance.

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Wang, Yun Xia. "Global Stabilization of Nonlinear Control Systems." Applied Mechanics and Materials 543-547 (March 2014): 1447–52. http://dx.doi.org/10.4028/www.scientific.net/amm.543-547.1447.

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This article discusses the global stabilization problem of nonlinear systems, based on the Lyapunov method. First discussed the situation of a class of two order nonlinear stabilization, sufficient conditions for global stabilization of the system. Secondly, we study the global stabilization of a class of three order nonlinear, got a new conclusion stabilization of the global system, and design a feedback control law of the system stabilization, at the end of this article of the global nonlinear system of general types of stabilization problem, we obtain sufficient conditions for stabilization of the system, and design the feedback control law is the system stabilization.

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Özdemir, Mustafa. "High-order singularities of 5R planar parallel robots." Robotica 37, no. 2 (September 20, 2018): 233–45. http://dx.doi.org/10.1017/s0263574718000966.

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SUMMARYSingularity analysis of parallel manipulators is an active research field in robotics. The present article derives for the first time in the literature a condition under which a five-bar parallel robot encounters high-order parallel singularities. In this regard, by focusing on the planar 5R mechanism, a theorem is given in terms of the slope of its coupler curve at the parallel singular configurations. At high-order parallel singularities, the associated determinant vanishes simultaneously with at least its first-order time derivative. The determination of such singularities is quite important since in their presence, some special conditions should be satisfied for bounded inverse dynamic solutions.

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Zhou, Shengfan. "Attractors for first order dissipative lattice dynamical systems." Physica D: Nonlinear Phenomena 178, no. 1-2 (April 2003): 51–61. http://dx.doi.org/10.1016/s0167-2789(02)00807-2.

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Benyassi, Mohamed, Adil Brouri, and Smail Slassi. "Nonlinear systems identification with discontinuous nonlinearity." IAES International Journal of Robotics and Automation (IJRA) 9, no. 1 (March 6, 2019): 34. http://dx.doi.org/10.11591/ijra.v9i1.pp34-41.

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<span>In this paper, nonparametric nonlinear systems identification is proposed. The considered system nonlinearity is nonparametric and is of hard type. This latter can be discontinuous and noninvertible. The entire nonlinear system is structured by Hammerstein model. Furthermore, the linear dynamic block is of any order and can be nonparametric. The problem identification method is done within two stages. In the first stage, the system nonlinearity is identified using simple input signals. In the first stage, the linear dynamic block parameters are estimated using periodic signals. The proposed algorithm can be used of large class of nonlinear systems.</span>

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Udwadia, Firdaus E. "Optimal tracking control of nonlinear dynamical systems." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, no. 2097 (April 24, 2008): 2341–63. http://dx.doi.org/10.1098/rspa.2008.0040.

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This paper presents a simple methodology for obtaining the entire set of continuous controllers that cause a nonlinear dynamical system to exactly track a given trajectory. The trajectory is provided as a set of algebraic and/or differential equations that may or may not be explicitly dependent on time. Closed-form results are also provided for the real-time optimal control of such systems when the control cost to be minimized is any given weighted norm of the control, and the minimization is done not just of the integral of this norm over a span of time but also at each instant of time. The method provided is inspired by results from analytical dynamics and the close connection between nonlinear control and analytical dynamics is explored. The paper progressively moves from mechanical systems that are described by the second-order differential equations of Newton and/or Lagrange to the first-order equations of Poincaré, and then on to general first-order nonlinear dynamical systems. A numerical example illustrates the methodology.

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Bykasov, D. A., A. V. Zubov, and M. G. Mustafin. "Applying Newton’s second order optimization method to define transition keys between planar coordinate systems." E3S Web of Conferences 224 (2020): 01003. http://dx.doi.org/10.1051/e3sconf/202022401003.

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The article considers the theoretical component of Newton’s second-order method, its main advantages and disadvantages when used in geodesy. The algorithm for determining the minimum of target functions by the Newton method of the second order was studied and analyzed in detail. Parameters of connection between flat rectangular coordinate systems are calculated. The task of determining the transition keys is relevant for geodesy. Comparative analysis of Newton’s method with the method of conjugated gradients was carried out. The algorithm for solving this problem was implemented in the Visual Basic for Applications software environment. The obtained data allow us to conclude that the Newton method can be used more widely in geodesy, especially in solving nonlinear optimization problems. However, the successful implementation of the method in geodetic production is possible only if the computational process is automated, by writing software modules in various programming languages to solve a specific problem.

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