Academic literature on the topic 'Nonlinear first order planar systems'
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Journal articles on the topic "Nonlinear first order planar systems"
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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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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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 (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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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 (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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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 (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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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 (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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Ye, Jiazhen, Yuki Todo, Zheng Tang, Bin Li, and Yu Zhang. "Artificial Visual System for Orientation Detection." Electronics 11, no. 4 (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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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 (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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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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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 (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.
Dissertations / Theses on the topic "Nonlinear first order planar systems"
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Rejoub, Riad A. "Projective and non-projective systems of first order nonlinear differential equations." Scholarly Commons, 1992. https://scholarlycommons.pacific.edu/uop_etds/2228.
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It is well established that many physical and chemical phenomena such as those in chemical reaction kinetics, laser cavities, rotating fluids, and in plasmas and in solid state physics are governed by nonlinear differential equations whose solutions are of variable character and even may lack regularities. Such systems are usually first studied qualitatively by examining their temporal behavior near singular points of their phase portrait.
In this work we will be concerned with systems governed by the time evolution equations [see PDF for mathematical formulas]
The xi may generally be considered to be concentrations of species in a chemical reaction, in which case the k's are rate constants. In some cases the xi may be considered to be position and momentum variables in a mechanical system. We will divide the equations into two classes: those in which the evolution can be carried out by the action of one of Lie's transformation groups of the plane, and those for which this is not possible. Members of the first class can be integrated by quadrature either directly or by use of an integrating factor; those in the second class cannot. Of those in the first class the most interesting evolve by transformations of the projective group, and these, as well as the equations that cannot be integrated by quadrature, we study in some detail. We seek a qualitative analysis of systems which have no linear terms in their evolution equations when the origin from which the xi are measured is a critical point. The standard, linear, phase plane analysis is of course not adequate for our purposes.
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Yang, Lixiang. "Modeling Waves in Linear and Nonlinear Solids by First-Order Hyperbolic Differential Equations." The Ohio State University, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=osu1303846979.
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Rocha, Eugénio Alexandre Miguel. "Uma Abordagem Algébrica à Teoria de Controlo Não Linear." Doctoral thesis, Universidade de Aveiro, 2003. http://hdl.handle.net/10773/21444.
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Doutoramento em Matemática<br>Nesta tese de Doutoramento desenvolve-se principalmente uma abordagem
algébrica à teoria de sistemas de controlo não lineares. No entanto, outros
tópicos são também estudados. Os tópicos tratados são os seguidamente
enunciados: fórmulas para sistemas de controlo sobre álgebras de Lie livres,
estabilidade de um sistema de corpos rolantes, algoritmos para aritmética
digital, e equações integrais de Fredholm não lineares.
No primeiro e principal tópico estudam-se representações para as soluções
de sistemas de controlo lineares no controlo. As suas trajetórias são representadas
pelas chamadas séries de Chen. Estuda-se a representação formal
destas séries através da introdução de várias álgebras não associativas e
técnicas específicas de álgebras de Lie livres. Sistemas de coordenadas para
estes sistemas são estudados, nomeadamente, coordenadas de primeiro tipo
e de segundo tipo. Apresenta-se uma demonstração alternativa para as
coordenadas de segundo tipo e obtêm-se expressões explícitas para as coordenadas
de primeiro tipo. Estas últimas estão intimamente ligadas ao
logaritmo da série de Chen que, por sua vez, tem fortes relações com
uma fórmula designada na literatura por “continuous Baker-Campbell-
Hausdorff formula”. São ainda apresentadas aplicações à teoria de funções
simétricas não comutativas. É, por fim, caracterizado o mapa de monodromia
de um campo de vectores não linear e periódico no tempo em relação
a uma truncatura do logaritmo de Chen.
No segundo tópico é estudada a estabilizabilidade de um sistema de quaisquer
dois corpos que rolem um sobre o outro sem deslizar ou torcer.
Constroem-se controlos fechados e dependentes do tempo que tornam a
origem do sistema de dois corpos num sistema localmente assimptoticamente
estável. Vários exemplos e algumas implementações em Maple°c
são discutidos.
No terceiro tópico, em apêndice, constroem-se algoritmos para calcular o
valor de várias funções fundamentais na aritmética digital, sendo possível
a sua implementação em microprocessadores. São também obtidos os seus
domínios de convergência.
No último tópico, também em apêndice, demonstra-se a existência e unicidade
de solução para uma classe de equações integrais não lineares com
atraso. O atraso tem um carácter funcional, mostrando-se ainda a diferenciabilidade
no sentido de Fréchet da solução em relação à função de atraso.<br>In this PhD thesis several subjects are studied regarding the following topics:
formulas for nonlinear control systems on free Lie algebras, stabilizability
of nonlinear control systems, digital arithmetic algorithms, and nonlinear
Fredholm integral equations with delay.
The first and principal topic is mainly related with a problem known as the
continuous Baker-Campbell-Hausdorff exponents. We propose a calculus
to deal with formal nonautonomous ordinary differential equations evolving
on the algebra of formal series defined on an alphabet. We introduce and
connect several (non)associative algebras as Lie, shuffle, zinbiel, pre-zinbiel,
chronological (pre-Lie), pre-chronological, dendriform, D-I, and I-D. Most of
those notions were also introduced into the universal enveloping algebra of
a free Lie algebra. We study Chen series and iterated integrals by relating
them with nonlinear control systems linear in control. At the heart of
all the theory of Chen series resides a zinbiel and shuffle homomorphism
that allows us to construct a purely formal representation of Chen series
on algebras of words. It is also given a pre-zinbiel representation of the
chronological exponential, introduced by A.Agrachev and R.Gamkrelidze
on the context of a tool to deal with nonlinear nonautonomous ordinary
differential equations over a manifold, the so-called chronological calculus.
An extensive description of that calculus is made, collecting some
fragmented results on several publications. It is a fundamental tool of
study along the thesis. We also present an alternative demonstration
of the result of H.Sussmann about coordinates of second kind using
the mentioned tools. This simple and comprehensive proof shows that
coordinates of second kind are exactly the image of elements of the dual
basis of a Hall basis, under the above discussed homomorphism. We obtain
explicit expressions for the logarithm of Chen series and the respective
coordinates of first kind, by defining several operations on a forest of
leaf-labelled trees. It is the same as saying that we have an explicit
formula for the functional coefficients of the Lie brackets on a continuous
Baker-Campbell-Hausdorff-Dynkin formula when a Hall basis is used. We
apply those formulas to relate some noncommutative symmetric functions,
and we also connect the monodromy map of a time-periodic nonlinear
vector field with a truncation of the Chen logarithm.
On the second topic, we study any system of two bodies rolling one
over the other without twisting or slipping. By using the Chen logarithm
expressions, the monodromy map of a flow and Lyapunov functions, we
construct time-variant controls that turn the origin of a control system
linear in control into a locally asymptotically stable equilibrium point.
Stabilizers for control systems whose vector fields generate a nilpotent Lie
algebra with degree of nilpotency · 3 are also given. Some examples are
presented and Maple°c were implemented.
The third topic, on appendix, concerns the construction of efficient
algorithms for Digital Arithmetic, potentially for the implementation in
microprocessors. The algorithms are intended for the computation of
several functions as the division, square root, sines, cosines, exponential,
logarithm, etc. By using redundant number representations and methods
of Lyapunov stability for discrete dynamical systems, we obtain several
algorithms (that can be glued together into an algorithm for parallel execution)
having the same core and selection scheme in each iteration. We also prove
their domains of convergence and discuss possible extensions.
The last topic, also on appendix, studies the set of solutions of a class
of nonlinear Fredholm integral equations with general delay. The delay
is of functional character modelled by a continuous lag function. We
ensure existence and uniqueness of a continuous (positive) solution of such
equation. Moreover, under additional conditions, it is obtained the Fr´echet
differentiability of the solution with respect to the lag function.
Books on the topic "Nonlinear first order planar systems"
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Southeast Geometry Seminar (15th 2009 University of Alabama at Birmingham). Geometric analysis, mathematical relativity, and nonlinear partial differential equations: Southeast Geometry Seminars Emory University, Georgia Institute of Technology, University of Alabama, Birmingham, and the University of Tennessee, 2009-2011. Edited by Ghomi Mohammad 1969-. American Mathematical Society, 2013.
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Litvinov, G. L. (Grigoriĭ Lazarevich), 1944- editor of compilation and Sergeev, S. N., 1981- editor of compilation, eds. Tropical and idempotent mathematics and applications: International Workshop on Tropical and Idempotent Mathematics, August 26-31, 2012, Independent University, Moscow, Russia. American Mathematical Society, 2014.
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Ducruix, Arnaud, and Richard Giegé, eds. Crystallization of Nucleic Acids and Proteins. Oxford University Press, 1999. http://dx.doi.org/10.1093/oso/9780199636792.001.0001.
Full textAbstract:
Crystallography is the major method of determining structures of biological macromolecules yet crystallization techniques are still regarded as difficult to perform. This new edition of Crystallization of Nucleic Acids and Proteins: A Practical Approach continues in the vein of the first edition by providing a detailed and rational guide to producing crystals of proteins and nucleic acids of sufficient quantity and quality for diffraction studies. It has been thoroughly updated to include all the major new techniques such as the uses of molecular biology in structural biology (maximizing expression systems, sequence modifications to enable crystallization, and the introduction of anomalous scatterers); diagnostic analysis of prenucleation and nucleation by spectroscopic methods; and the two- dimensional electron crystallography of soluble proteins on planar lipid films. As well as an introduction to crystallogenesis, the other topics covered are: Handling macromolecular solutions, experimental design, seeding, proceeding from solutions to crystals Crystallization in gels Crystallization of nucleic acid complexes and membrane proteins Soaking techniques Preliminary characterization of crystals in order to tell whether they are suitable for diffraction studies. As with all Practical Approach books the protocols have been written by experienced researchers and are tried an tested methods. The underlying theory is brought together with the laboratory protocols to provide researchers with the conceptual and methodological tools necessary to exploit these powerful techniques. Crystallization of Nucleic Acids and Proteins: A Practical Approach 2e will be an invaluable manual of practical crystallization methods to researchers in molecular biology, crystallography, protein engineering, and biological chemistry.
Book chapters on the topic "Nonlinear first order planar systems"
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Mouyon, Ph. "First-order control of nonlinear systems." In Nonlinear Systems. Springer US, 1997. http://dx.doi.org/10.1007/978-1-4615-6395-2_2.
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Galor, Oded. "Multi-Dimensional, First-Order, Nonlinear Systems." In Discrete Dynamical Systems. Springer Berlin Heidelberg, 2007. http://dx.doi.org/10.1007/3-540-36776-4_4.
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Yong, Wen-An. "Singular Perturbations of First-Order Hyperbolic Systems." In Nonlinear Hyperbolic Problems: Theoretical, Applied, and Computational Aspects. Vieweg+Teubner Verlag, 1993. http://dx.doi.org/10.1007/978-3-322-87871-7_72.
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Ipek, Pembe, and Zameddin I. Ismailov. "The General Form of Maximally Accretive Quasi-differential Operators for First Order." In Nonlinear Systems and Complexity. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-90972-1_6.
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Xing, Jing Tang. "First Order Approximations and Matrix Spaces." In Energy Flow Theory of Nonlinear Dynamical Systems with Applications. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-17741-0_5.
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Liu, Yan, Kai Ma, Hao He, and Jun Xiao. "Analytical Periodic Motions for a First-Order Nonlinear Circuit System Under Different Excitations." In Nonlinear Systems and Complexity. Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-030-94301-1_10.
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Levrie, Paul, Marc Van Barel, and Adhemar Bultheel. "First-Order Linear Recurrence Systems and General N-Fractions." In Nonlinear Numerical Methods and Rational Approximation II. Springer Netherlands, 1994. http://dx.doi.org/10.1007/978-94-011-0970-3_33.
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Song, Yongduan, and Yujuan Wang. "Finite-Time Leaderless Consensus Control for Systems with First-Order Uncertain Dynamics." In Cooperative Control of Nonlinear Networked Systems. Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-04972-0_6.
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Lewis, Frank L., Hongwei Zhang, Kristian Hengster-Movric, and Abhijit Das. "Cooperative Adaptive Control for Systems with First-Order Nonlinear Dynamics." In Communications and Control Engineering. Springer London, 2014. http://dx.doi.org/10.1007/978-1-4471-5574-4_8.
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Berhanu, S. "On Involutive Systems of First-order Nonlinear Partial Differential Equations." In Complex Analysis. Birkhäuser Basel, 2010. http://dx.doi.org/10.1007/978-3-0346-0009-5_2.
Full textConference papers on the topic "Nonlinear first order planar systems"
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Tripathi, Astitva, and Anil K. Bajaj. "Computational Synthesis for Nonlinear Dynamics Based Design of Planar Resonant Structures." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-71463.
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Nonlinear phenomena such as internal resonances have significant potential applications in Micro Electro Mechanical Systems (MEMS) for increasing the sensitivity of biological and chemical sensors and signal processing elements in circuits. While several theoretical systems are known which exhibit 1:2 or 1:3 internal resonances, designing systems that have the desired properties required for internal resonance as well as are physically realizable as MEMS devices is a significant challenge. Traditionally, the design process for obtaining resonant structures exhibiting an internal resonance has relied heavily on the designer’s prior knowledge and experience. However, with advances in computing power and topology optimization techniques, it should be possible to synthesize structures with the required nonlinear properties (such as having modal interactions) computationally. In this work, a preliminary method for computer based synthesis of structures consisting of beams for desired internal resonance is presented. The linear structural design is accompalished by a Finite Element Method (FEM) formulation implemented in Matlab to start with a base structure and iteratively modify it to obtain a structure with the desired properties. Possible design criteria are having the first two natural frequencies of the structure in some required ratio (such as 1:2 or 1:3). Once a topology of the structure is achieved that meets the desired criterion (i.e., the program converges to a definite structure), the linear mode shapes of the structure can be extracted from the finite element analysis, and a more complete Lagrangian formulation of the nonlinear elastic structure can be used to develop a nonlinear two-mode model of the structure. The reduced-order model is expected to capture the appropriate resonant dynamics associated with modal interactions between the two modes. The nonlinear response of the structure can be obtained by application of perturbation methods such as averaging on the two-mode model. Many candidate structures are synthesized that meet the desired modal frequency criterion and their nonlinear responses are compared.
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Brenn, Günter, Marie-Charlotte Renoult, and Innocent Mutabazi. "Weakly nonlinear instability of a viscous liquid jet." In ILASS2017 - 28th European Conference on Liquid Atomization and Spray Systems. Universitat Politècnica València, 2017. http://dx.doi.org/10.4995/ilass2017.2017.4711.
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Aweakly nonlinear stability analysis of an axisymmetric viscous liquid jet is performed. The calculation is based ona small-amplitude perturbation method and restricted to second order. Contrary to the inviscid jet and the planar viscous sheet cases studied by Yuen in 1968 [1] and Yang et al. in 2013 [2], respectively, a part of the solution results from a polynomial approximation of Bessel functions. Results on interface shapes for a small wave number and initial perturbation amplitude, four different Ohnesorge numbers, taking into account the approximate part or not, are used to predict the influence of liquid viscosity on satellite drop formation and evaluate the influence of the approximation. It is observed that the liquid viscosity has a retarding effect on satellite drop formation, in agreement with previous experimental and numerical work. In addition, it is found that the approximate terms can be reasonably ignored, providing a simpler viscous weakly nonlinear model for the description of the first nonlinearity growth in liquid jets.The present work replaces the ILASS 2016 paper [3] by the authors on the same subject.DOI: http://dx.doi.org/10.4995/ILASS2017.2017.4711
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Dörfle, M., and R. Graham. "Semi-classical Limit of Chaos and Quantum Noise in Second Harmonic Generation." In Instabilities and Dynamics of Lasers and Nonlinear Optical Systems. Optica Publishing Group, 1985. http://dx.doi.org/10.1364/idlnos.1985.thc6.
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Second harmonic generation and subharmonic generation in a cavity are described by the usual master equation (1) for the statistical operator of the system of two modes (fundamental and second harmonic). The master equation is equivalent to a partial differential equation for the Wigner function which is a generalized Fokker-Planck equation involving partial derivatives of the first, second and third order (2). In the semi-classical limit the third order derivatives are negligible and the Wigner distribution satisfies the Fokker-Planck equation equivalent to the Langevin equation with the formally classical Gaussian white noise ξ1, ξ2 with the only non-vanishing correlation coefficients β1, β2 are the mode amplitudes (normalized to photon numbers), g is the coupling constant, Δ1, 2 the frequency mismatch, x is the damping rate, Fp the amplitude of the pump field.
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Li, Zhi. "On the Hamiltonian Formulation of Thin Free Liquid Sheets." In ASME 2001 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/imece2001/de-23248.
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Abstract A Hamiltonian model is used to study the nonlinear distortion of thin free planar liquid sheets. A systematic asymptotic expansion procedure with the fluctuating value of the sheet velocity as the small parameter is developed for the propagation of long waves on liquid sheets. The first and second order systems of nonlinear evolution equations are derived.
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Wang, Bo, Sergey Nersesov, and Hashem Ashrafiuon. "Formation Control for Underactuated Surface Vessel Networks." In ASME 2020 Dynamic Systems and Control Conference. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/dscc2020-3178.
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Abstract Developing distributed control algorithms for multi-agent systems is difficult when each agent is modeled as a nonlinear dynamical system. Moreover, the problem becomes far more complex if the agents do not have sufficient number of actuators to track any arbitrary trajectory. In this paper, we present the first fully decentralized approach to formation control for networks of underactuated surface vessels. The vessels are modeled as three degree of freedom planar rigid bodies with two actuators. Algebraic graph theory is used to model the network as a directed graph and employing a leader-follower model. We take advantage of the cascade structure of the combined nonlinear kinematic and dynamic model of surface vessels and develop a reduced-order error dynamic model using a state transformation definition. The error dynamics and consequently all system states are then stabilized using sliding mode control approach. It is shown that the stabilization of the reduced-order error dynamics guarantees uniform global asymptotic stability of the closed-loop system subject to bounded uncertainties. The proposed control method can be implemented in directed time-invariant communication networks without the availability of global position measurements for any of the vehicles participating in the network. An example of a a network of five surface vessels is simulated to verify the effective performance of the proposed control approach.
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Bošković, Jovan D. "A Nonlinear Parametrization for Stable Neural Network-Based Identification and Control." In ASME 1997 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 1997. http://dx.doi.org/10.1115/imece1997-0399.
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Abstract In this paper a stable neural network-based identification and control method for a class of first-order nonlinear plants is suggested. The method is based on neural networks built using a rational activation function. The paper discusses a method for stable adjustment of the network parameters and robustness issues in the resulting closed-loop system. The overall control scheme also includes a robust global controller that guarantees that the state of the system is confined to a region in the state space. The combination of robust global and neural network-based controllers appears to be well suited for performance improvement in complex nonlinear systems.
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Li, Shuangbao, Wei Zhang, and Minghui Yao. "Global Bifurcations and Multipulse-Type Chaotic Dynamics in the Interactions of Two Flexural Modes of a Cantilever Beam." In ASME 2009 International Mechanical Engineering Congress and Exposition. ASMEDC, 2009. http://dx.doi.org/10.1115/imece2009-13225.
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Global bifurcations and multipulse-type chaotic dynamics in the interactions of two flexural modes of a cantilever beam are studied using the extended Melnikov method. The cantilever beam studied is subjected to a harmonic axial excitation and transverse excitations at the free end. After the governing nonlinear equations of nonplanar motion with parametric and external excitations are given, the Galerkin’s procedure based on the first flexural mode in each direction is applied to the partial differential governing equation to obtain a two-degree-of-freedom non-autonomous nonlinear system. The resonant case considered here is one-to-one internal resonance, principal parametric resonance-1/2 subharmonic resonance. The method of multiple scales is used to derive four first-order nonlinear ordinary differential equations governing the modulation of the amplitudes and phases of two interacting modes. After transforming the modulations equations into a suitable form, the extended Melnikov method is employed to show the existence of chaotic dynamics by identifying Silnikov-type multipulse jumping orbits in the perturbed phase space. We are able to obtain the explicit restrictions on the damping, forcing, and the detuning parameters, under which multipulse-type chaotic dynamics is to be expected. Physically, such jumping means sudden, large-amplitude departures of the beam from its planar oscillations. Numerical simulations indicate that there chaotic responses and jumping phenomenon in the nonlinear nonplanar oscillations of the cantilever beam.
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Zhang, Wei, Min Sun, Qian Wang, and Jianen Chen. "Subharmonic Orbits of Rectangular Thin Plate With Parametrically and Externally Excitations." In ASME 2012 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/imece2012-85868.
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In this paper, the bifurcations of subharmonic orbits for a four-dimensional rectangular thin plate with parametrically and externally excitations is considered for the first time. The formulas of the rectangular thin plate are derived by using the von Karman-type equation, the Reddy’s third-order shear deformation plate theory and the Galerkin’s approach. The unperturbed system is composed of two independent planar Hamiltonian systems such that the unperturbed system has a family of periodic orbits. The problem addressed here is the determination of sufficient conditions for some of the periodic orbits to generate subharmonic orbits after periodic perturbations. Thus, based on periodic transformations and Poincaré map the subharmonic Melnikov method is improved to enable us to analyze directly the non-autonomous nonlinear dynamical system, which is applied to the non-autonomous governing equations of motion for the parametrically and externally excited rectangular thin plate. The results obtained here indicate that subharmonic motions can occur in the rectangular thin plate. The method succeeds in establishing the existence of subharmonics in perturbed Hamiltonian systems as well as in discussing their bifurcations. Numerical simulation is also employed to find the subharmonic motions of the parametrically and externally excited rectangular thin plate.
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Zhe, Wang, Qiang Tian, and Hiayan Hu. "Dynamics Study and Sensitivity Analysis of Flexible Multibody Systems With Interval Parameters." In ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/detc2016-59349.
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The dynamics of flexible multibody systems with interval parameters is studied based on a non-intrusive computation methodology. The Absolute Nodal Coordinate Formulation (ANCF) is used to model the rigid-flexible multibody system, including the finite elements of the ANCF and the ANCF Reference Nodes (ANCF-RNs). The Chebyshev sampling methods including Chebyshev tensor product (CTP) sampling method and Chebyshev collocation method (CCM), are utilized to generate the Chebyshev surrogate model for Interval Differential Algebraic Equations (IDAEs). For purpose of preventing the interval explosion problem and maintaining computation efficiency, the interval bounds of the IDAEs are determined by scanning the deduced Chebyshev surrogate model. To further improve the computation efficiency, OpenMP directives are also used to parallelize the solving process of the Differential Algebraic Equations (DAEs) by fixing the uncertain interval parameter at the given sampling points. The sensitivity analysis of flexible multibody systems with interval parameters is initially performed by using the direct differentiation method. The direct differentiation method differentiates the dynamic equations with respect to the design variable, which yields the system sensitivity equations governed by DAEs. The generalized alpha method is introduced to integrate the sensitivity DAEs. The sensitivity equations of flexible multibody systems with interval parameters are also described by the IDAEs. Based on the continuum mechanics, the computational efficient analytical formulations for the derivative items of the system sensitivity equations are deduced. Three examples are studied to validate the proposed methodology, including the complicated spatial rigid-flexible multibody systems with a large number of uncertain interval parameters, the flexible system with uncertain interval clearance size joint, and the first order sensitivity analysis of flexible multibody systems with interval parameters. Firstly, the dynamics analysis of a six-arm space robot with six interval parameters is performed. For this case study, the interval dynamics cannot be obtained by directly scanning the IDAEs because extremely huge sets of DAEs with deterministic samples have to be solved. The estimated total computational time for solving the scanned IDAEs will be 1850 days! However, the computational time for solving the scanned Chebyshev surrogate model is 9796.97 seconds. It shows the effectiveness of the proposed computation methodology. Then, the nonlinear dynamics of a planar slider-crank mechanism with uncertain interval clearance size joint is studied in this work. The kinetics model of the revolute clearance joints is formulated under the ANCF-RN framework. Moreover, the influence of the LuGre and the modified Coulomb’s friction force models on the system’s dynamic response is investigated. By analyzing the bounds of dynamic response, the bifurcation diagrams are observed. It must be highlighted that with increasing the size of clearance, it does not automatically lead to unstable behaviors. Finally, the first order sensitivity analysis of flexible multibody systems with interval parameters is also studied in this work. The third one of a flexible mechanism with interval parameters is used to perform the sensitivity analysis.
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Cappelli, M., B. Castillo-Toledo, S. Di Gennaro, F. Memmi, and M. Sepielli. "Digital Nonlinear Control for a Pressurizer in a Pressurized Water Reactor." In 2013 21st International Conference on Nuclear Engineering. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/icone21-16374.
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Studying criteria ensuring better implementations of control laws and logics on digital programmable devices is today a crucial point in order to improve performance and safety in nuclear plants of novel design. To this aim, a mathematical model of a PWR is considered, describing the dynamics of the primary circuit. This model is simple enough to allow the determination of a control policy, but it is also sufficiently accurate to capture the nonlinear, time–varying, and switching nature of the system. A dynamic controller for the pressurizer pressure and water level is hence designed. Its main advantage is the reduction of the control response. This latter is a classical drawback, which may cause some troubles to pressure control, due to the long response of temperature sensors. The digital implementation of the controller has been taken into account to study the important aspects of the influence of the implementation on the performance of the control law. A particularly effective and flexible environment where it is possible to analyze these aspects is Matlab/Simulink, a general–purpose tool for analysis and simulation of multi–domain dynamic systems, practically a standard in control implementations, also in industrial applications. The designed controller ensures a good performance when applied to the model used to derive them, also in the presence of unmodeled uncertainties and disturbances. Its nonlinear switching nature, reflecting the real pressurizer dynamics, ensures better transient behaviors. Since it contains a PI action, it represents an evolution and an improvement with respect to classical PID controllers, usually implemented in standard control actions. To better analyze the control performance, three steps have been considered: first, the designed nonlinear controller has been compared with a linear one. Second, it has been tested on a more detailed model, with more realistic dynamics. Finally, the digital implementation of the controller has been simulated, in order to optimize the electronic implementation on the FPGA–CPU hybrid platform. A simulation study has been conducted with sampling times in ascending order, to further test its robustness against delays. LabVIEW development tool and technology has been chosen to create the electronic layout and to obtain preliminary results in terms of timing, resources availability and power consumption. Due to the same graphical mode of programming, an immediate link between Matlab/Scilab and LabVIEW is possible in order to optimize the use of each language where it guarantees the best performance. The methodology here presented could be very useful to better integrate the different features of control engineering and IC design, thus giving the possibility of realizing more accurate simulations and permitting to perform a hardware in the loop (HIL) testing of a wide variety of control algorithms.
Reports on the topic "Nonlinear first order planar systems"
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Wu, Yingjie, Selim Gunay, and Khalid Mosalam. Hybrid Simulations for the Seismic Evaluation of Resilient Highway Bridge Systems. Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA, 2020. http://dx.doi.org/10.55461/ytgv8834.
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Bridges often serve as key links in local and national transportation networks. Bridge closures can result in severe costs, not only in the form of repair or replacement, but also in the form of economic losses related to medium- and long-term interruption of businesses and disruption to surrounding communities. In addition, continuous functionality of bridges is very important after any seismic event for emergency response and recovery purposes. Considering the importance of these structures, the associated structural design philosophy is shifting from collapse prevention to maintaining functionality in the aftermath of moderate to strong earthquakes, referred to as “resiliency” in earthquake engineering research. Moreover, the associated construction philosophy is being modernized with the utilization of accelerated bridge construction (ABC) techniques, which strive to reduce the impact of construction on traffic, society, economy and on-site safety. This report presents two bridge systems that target the aforementioned issues. A study that combined numerical and experimental research was undertaken to characterize the seismic performance of these bridge systems. The first part of the study focuses on the structural system-level response of highway bridges that incorporate a class of innovative connecting devices called the “V-connector,”, which can be used to connect two components in a structural system, e.g., the column and the bridge deck, or the column and its foundation. This device, designed by ACII, Inc., results in an isolation surface at the connection plane via a connector rod placed in a V-shaped tube that is embedded into the concrete. Energy dissipation is provided by friction between a special washer located around the V-shaped tube and a top plate. Because of the period elongation due to the isolation layer and the limited amount of force transferred by the relatively flexible connector rod, bridge columns are protected from experiencing damage, thus leading to improved seismic behavior. The V-connector system also facilitates the ABC by allowing on-site assembly of prefabricated structural parts including those of the V-connector. A single-column, two-span highway bridge located in Northern California was used for the proof-of-concept of the proposed V-connector protective system. The V-connector was designed to result in an elastic bridge response based on nonlinear dynamic analyses of the bridge model with the V-connector. Accordingly, a one-third scale V-connector was fabricated based on a set of selected design parameters. A quasi-static cyclic test was first conducted to characterize the force-displacement relationship of the V-connector, followed by a hybrid simulation (HS) test in the longitudinal direction of the bridge to verify the intended linear elastic response of the bridge system. In the HS test, all bridge components were analytically modeled except for the V-connector, which was simulated as the experimental substructure in a specially designed and constructed test setup. Linear elastic bridge response was confirmed according to the HS results. The response of the bridge with the V-connector was compared against that of the as-built bridge without the V-connector, which experienced significant column damage. These results justified the effectiveness of this innovative device. The second part of the study presents the HS test conducted on a one-third scale two-column bridge bent with self-centering columns (broadly defined as “resilient columns” in this study) to reduce (or ultimately eliminate) any residual drifts. The comparison of the HS test with a previously conducted shaking table test on an identical bridge bent is one of the highlights of this study. The concept of resiliency was incorporated in the design of the bridge bent columns characterized by a well-balanced combination of self-centering, rocking, and energy-dissipating mechanisms. This combination is expected to lead to minimum damage and low levels of residual drifts. The ABC is achieved by utilizing precast columns and end members (cap beam and foundation) through an innovative socket connection. In order to conduct the HS test, a new hybrid simulation system (HSS) was developed, utilizing commonly available software and hardware components in most structural laboratories including: a computational platform using Matlab/Simulink [MathWorks 2015], an interface hardware/software platform dSPACE [2017], and MTS controllers and data acquisition (DAQ) system for the utilized actuators and sensors. Proper operation of the HSS was verified using a trial run without the test specimen before the actual HS test. In the conducted HS test, the two-column bridge bent was simulated as the experimental substructure while modeling the horizontal and vertical inertia masses and corresponding mass proportional damping in the computer. The same ground motions from the shaking table test, consisting of one horizontal component and the vertical component, were applied as input excitations to the equations of motion in the HS. Good matching was obtained between the shaking table and the HS test results, demonstrating the appropriateness of the defined governing equations of motion and the employed damping model, in addition to the reliability of the developed HSS with minimum simulation errors. The small residual drifts and the minimum level of structural damage at large peak drift levels demonstrated the superior seismic response of the innovative design of the bridge bent with self-centering columns. The reliability of the developed HS approach motivated performing a follow-up HS study focusing on the transverse direction of the bridge, where the entire two-span bridge deck and its abutments represented the computational substructure, while the two-column bridge bent was the physical substructure. This investigation was effective in shedding light on the system-level performance of the entire bridge system that incorporated innovative bridge bent design beyond what can be achieved via shaking table tests, which are usually limited by large-scale bridge system testing capacities.