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On a simple approach to nonlinear oscillators  [PDF]
Francisco M. Fernández
Physics , 2009,
Abstract: We analyze a simple textbook approach to nonlinear oscillators proposed recently, disclose its errors, limitations and misconceptions and complete the calculations that the authors failed to perform.
Parameters and solutions of linear and nonlinear oscillators  [cached]
Rina Ling
International Journal of Mathematics and Mathematical Sciences , 1979, DOI: 10.1155/s0161171279000399
Abstract: Relationship between existence of solutions for certain classes of nonlinear boundary value problems and the smallest or the largest eigenvalue of the corresponding linear problem is obtained. Behavior of the solutions, as the parameter increases, is also studied.
Escape time from potential wells of strongly nonlinear oscillators with slowly varying parameters  [PDF]
Jianping Cai,Y. P. Li,Xiaofeng Wu
Mathematical Problems in Engineering , 2005, DOI: 10.1155/mpe.2005.365
Abstract: The effect of negative damping to an oscillatory system is to force the amplitude to increase gradually and the motion will be out of the potential well of the oscillatory system eventually. In order to deduce the escape time from the potential well of quadratic or cubic nonlinear oscillator, the multiple scales method is firstly used to obtain the asymptotic solutions of strongly nonlinear oscillators with slowly varying parameters, and secondly the character of modulus of Jacobian elliptic function is applied to derive the equations governing the escape time. The approximate potential method, instead of Taylor series expansion, is used to approximate the potential of an oscillation system such that the asymptotic solution can be expressed in terms of Jacobian elliptic function. Numerical examples verify the efficiency of the present method.
Escape time from potential wells of strongly nonlinear oscillators with slowly varying parameters
Cai Jianping,Li Y. P.,Wu Xiaofeng
Mathematical Problems in Engineering , 2005,
Abstract: The effect of negative damping to an oscillatory system is to force the amplitude to increase gradually and the motion will be out of the potential well of the oscillatory system eventually. In order to deduce the escape time from the potential well of quadratic or cubic nonlinear oscillator, the multiple scales method is firstly used to obtain the asymptotic solutions of strongly nonlinear oscillators with slowly varying parameters, and secondly the character of modulus of Jacobian elliptic function is applied to derive the equations governing the escape time. The approximate potential method, instead of Taylor series expansion, is used to approximate the potential of an oscillation system such that the asymptotic solution can be expressed in terms of Jacobian elliptic function. Numerical examples verify the efficiency of the present method.
Adaptive Control for Modified Projective Synchronization-Based Approach for Estimating All Parameters of a Class of Uncertain Systems: Case of Modified Colpitts Oscillators  [PDF]
Soup Tewa Kammogne,Hilaire Bertrand Fotsin
Journal of Chaos , 2014, DOI: 10.1155/2014/659647
Abstract: A method of estimation of all parameters of a class of nonlinear uncertain dynamical systems is considered, based on the modified projective synchronization (MPS). The case of modified Colpitts oscillators is investigated. Through a suitable transformation of the dynamical system, sufficient conditions for achieving synchronization are derived based on Lyapunov stability theory. Global stability and asymptotic robust synchronization of the considered system are investigated. The proposed approach offers a systematic design procedure for robust adaptive synchronization of a large class of chaotic systems. The combined effect of both an additive white Gaussian noise (AWGN) and an artificial perturbation is numerically investigated. Results of numerical simulations confirm the effectiveness of the proposed control strategy. 1. Introduction Synchronization of chaotic systems and their potential applications in wide areas of physics and engineering sciences is currently a field of great interest ([1, 2] and references cited therein). The first idea of synchronizing two identical chaotic systems with different initial conditions is introduced by Pecora and Carroll [3] and the method is realized in electronic circuits. Synchronization techniques have been improved in recent years, and many different methods are applied theoretically and experimentally to synchronize the chaotic systems which include back stepping design technique [4], projective synchronization (PS) [5], modified projective synchronization (MPS) [6, 7], generalized synchronization [8], adaptive modified projective synchronization [9], lag synchronization [10], anticipating synchronization [11], phase synchronization [12], and their combinations [13]. Synchronization may involve several systems without a prescribed hierarchy (bidirectional) as it is the case in synchronization of networks of systems [14, 15], often happening naturally, for instance, in certain biological systems. Another intensive area of research to emphasize within bidirectional synchronization is the study of the consensus paradigm (see an excellent text in [16]). Amongst all kinds of chaos synchronization, MPS is the state-of-the-art of synchronization schemes. MPS means that the master and slave systems could be synchronized up to a constant scaling matrix. Recently, various control methods which include adaptive control [17, 18] and active control [7, 19, 20] have been introduced. Most of the works done on MPS of chaotic systems have used active control method since it is easy to design a control input and to deal with
Multiple-Parameter Hamiltonian Approach for Higher Accurate Approximations of a Nonlinear Oscillator with Discontinuity  [PDF]
Najeeb Alam Khan,Muhammad Jamil,Asmat Ara
International Journal of Differential Equations , 2011, DOI: 10.1155/2011/649748
Abstract: We applied a new approach to obtain natural frequency of the nonlinear oscillator with discontinuity. He's Hamiltonian approach is modified for nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(u). We employed this method for higher-order approximate solution of the nonlinear oscillator equation. This property is used to obtain approximate frequency-amplitude relationship of a nonlinear oscillator with high accuracy. Many numerical results are given to prove the efficiency of the suggested technique. 1. Introduction The study of nonlinear oscillator problems is of crucial importance not only in all areas of physics but also in engineering and other disciplines. It is of great importance to study analytically nonlinear oscillators to obtain approximate frequency-amplitude relationship because of their wide applications. Traditional perturbation method provides us with a simple approach to the determination of the frequency-amplitude relationship, but the results are valid only for special cases, that is, for weakly nonlinear systems or for the case when the amplitude is very small. In order to overcome the shortcomings arising in traditional perturbation methods, various alternative approaches have been proposed, for example, variational iteration method [1–3], homotopy perturbation method [4–7], Lindstedt-Poincare method [8], variational approach [9, 10], parameter-expanding method [11] and max-min approach [12], harmonic balance method [13], and Hamiltonian approach [14]. In the present study, the mentioned parameters are those undetermined values in the assumed solution. In the three-parameters technique, the motion is assumed as , where , , are the angular frequency of motion and Fourier coefficients, respectively. The three undetermined parameters are determined by using the governing equation of motion and the initial conditions imposed. The way for obtaining the parameters in He’s Hamiltonian technique is quite different from that in the harmonic balance method. Therefore, the present technique is not the same as the harmonic balance method. Finally, the paper provides a lot of higher accurate results for the angular frequency of the motion. 2. Analysis In this paper, we consider a general form of nonlinear oscillator with initial conditions and . The variational principle for (2.1) suggested by He [9] can be written as where is period of the nonlinear oscillator, . In the functional (2.2), is the kinetic energy, so that the functional (2.2) is the least Lagrangian action, from which we can write the
A Simple and Unified Approach to Identify Integrable Nonlinear Oscillators and Systems  [PDF]
V. K. Chandrasekar,S. N. Pandey,M. Senthilvelan,M. Lakshmanan
Physics , 2005, DOI: 10.1063/1.2171520
Abstract: In this paper, we consider a generalized second order nonlinear ordinary differential equation of the form $\ddot{x}+(k_1x^q+k_2)\dot{x}+k_3x^{2q+1}+k_4x^{q+1}+\lambda_1x=0$, where $k_i$'s, $i=1,2,3,4$, $\lambda_1$ and $q$ are arbitrary parameters, which includes several physically important nonlinear oscillators such as the simple harmonic oscillator, anharmonic oscillator, force-free Helmholtz oscillator, force-free Duffing and Duffing-van der Pol oscillators, modified Emden type equation and its hierarchy, generalized Duffing-van der Pol oscillator equation hierarchy and so on and investigate the integrability properties of this rather general equation. We identify several new integrable cases for arbitrary value of the exponent $q, q\in R$. The $q=1$ and $q=2$ cases are analyzed in detail and the results are generalized to arbitrary $q$. Our results show that many classical integrable nonlinear oscillators can be derived as sub-cases of our results and significantly enlarge the list of integrable equations that exist in the contemporary literature. To explore the above underlying results we use the recently introduced generalized extended Prelle-Singer procedure applicable to second order ODEs. As an added advantage of the method we not only identify integrable regimes but also construct integrating factors, integrals of motion and general solutions for the integrable cases, wherever possible, and bring out the mathematical structures associated with each of the integrable cases.
Squeezing in coupled oscillators having neither nonlinear terms nor time-dependent parameters
Rodrigues, H.;Portes Jr., D.;Duarte, S.B.;Baseia, B.;
Brazilian Journal of Physics , 2001, DOI: 10.1590/S0103-97332001000400006
Abstract: the occurrence of squeezing effects in coupled oscillators, and the transference between them, has been studied in various situations using hamiltonians involving either nonlinear terms or time-dependent parameters. we consider a simplified scheme generating this effect, and discuss its origin.
Squeezing in coupled oscillators having neither nonlinear terms nor time-dependent parameters  [cached]
Rodrigues H.,Portes Jr. D.,Duarte S.B.,Baseia B.
Brazilian Journal of Physics , 2001,
Abstract: The occurrence of squeezing effects in coupled oscillators, and the transference between them, has been studied in various situations using Hamiltonians involving either nonlinear terms or time-dependent parameters. We consider a simplified scheme generating this effect, and discuss its origin.
A proposal for factorization using Kerr nonlinearities between three harmonic oscillators  [PDF]
H. T. Ng,Franco Nori
Computer Science , 2010,
Abstract: We propose an alternative method to factorize an integer by using three harmonic oscillators. These oscillators are coupled together via specific Kerr nonlinear interactions. This method can be applied even if two harmonic oscillators are prepared in mixed states. As simple examples, we show how to factorize N=15 and 35 using this approach. The effect of dissipation of the harmonic oscillators on the performance of this method is studied. We also study the realization of nonlinear interactions between the coupled oscillators. However, the probability of finding the factors of a number is inversely proportional to its input size. The probability becomes low when this number is large. We discuss the limitations of this approach.
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