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Journal articles on the topic 'Nonlinear periodic systems'

Author: Grafiati

Published: 19 October 2024

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1

Gasiński, Leszek, and Nikolaos S. Papageorgiou. "Nonlinear Multivalued Periodic Systems." Journal of Dynamical and Control Systems 25, no. 2 (June 14, 2018): 219–43. http://dx.doi.org/10.1007/s10883-018-9408-9.

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2

Verriest, Erik I. "Balancing for Discrete Periodic Nonlinear Systems." IFAC Proceedings Volumes 34, no. 12 (August 2001): 249–54. http://dx.doi.org/10.1016/s1474-6670(17)34093-4.

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3

Luo, Albert C. J. "Periodic Flows to Chaos Based on Discrete Implicit Mappings of Continuous Nonlinear Systems." International Journal of Bifurcation and Chaos 25, no. 03 (March 2015): 1550044. http://dx.doi.org/10.1142/s0218127415500443.

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This paper presents a semi-analytical method for periodic flows in continuous nonlinear dynamical systems. For the semi-analytical approach, differential equations of nonlinear dynamical systems are discretized to obtain implicit maps, and a mapping structure based on the implicit maps is employed for a periodic flow. From mapping structures, periodic flows in nonlinear dynamical systems are predicted analytically and the corresponding stability and bifurcations of the periodic flows are determined through the eigenvalue analysis. The periodic flows predicted by the single-step implicit maps are discussed first, and the periodic flows predicted by the multistep implicit maps are also presented. Periodic flows in time-delay nonlinear dynamical systems are discussed by the single-step and multistep implicit maps. The time-delay nodes in discretization of time-delay nonlinear systems were treated by both an interpolation and a direct integration. Based on the discrete nodes of periodic flows in nonlinear dynamical systems with/without time-delay, the discrete Fourier series responses of periodic flows are presented. To demonstrate the methodology, the bifurcation tree of period-1 motion to chaos in a Duffing oscillator is presented as a sampled problem. The method presented in this paper can be applied to nonlinear dynamical systems, which cannot be solved directly by analytical methods.

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4

Can, Le Xuan. "On periodic waves of the nonlinear systems." Vietnam Journal of Mechanics 20, no. 4 (December 30, 1998): 11–19. http://dx.doi.org/10.15625/0866-7136/10037.

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The paper is concerned with the solvability and approximate solution of the nonlinear partial differential equation describing the periodic wave propagation. Necessary and sufficient conditions for the existence of the periodic wave solutions are obtained. An approximate method for solving the equation is presented. As an illustrative example, the equation of periodic waves of the electric cables is considered.

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5

Sundararajan, P., and S. T. Noah. "Dynamics of Forced Nonlinear Systems Using Shooting/Arc-Length Continuation Method—Application to Rotor Systems." Journal of Vibration and Acoustics 119, no. 1 (January 1, 1997): 9–20. http://dx.doi.org/10.1115/1.2889694.

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The analysis of systems subjected to periodic excitations can be highly complex in the presence of strong nonlinearities. Nonlinear systems exhibit a variety of dynamic behavior that includes periodic, almost-periodic (quasi-periodic), and chaotic motions. This paper describes a computational algorithm based on the shooting method that calculates the periodic responses of a nonlinear system under periodic excitation. The current algorithm calculates also the stability of periodic solutions and locates system parameter ranges where aperiodic and chaotic responses bifurcate from the periodic response. Once the system response for a parameter is known, the solution for near range of the parameter is calculated efficiently using a pseudo-arc length continuation procedure. Practical procedures for continuation, numerical difficulties and some strategies for overcoming them are also given. The numerical scheme is used to study the imbalance response of a rigid rotor supported on squeeze-film dampers and journal bearings, which have nonlinear stiffness and damping characteristics. Rotor spinning speed is used as the bifurcation parameter, and speed ranges of sub-harmonic, quasi-periodic and chaotic motions are calculated for a set of system parameters of practical interest. The mechanisms of these bifurcations also are explained through Floquet theory, and bifurcation diagrams.

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6

Ortega, Juan-Pablo. "Relative normal modes for nonlinear Hamiltonian systems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 133, no. 3 (June 2003): 665–704. http://dx.doi.org/10.1017/s0308210500002602.

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An estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a Hamiltonian system with continuous symmetry is given. This result constitutes a generalization to the Hamiltonian symmetric framework of a classical result by Weinstein and Moser on the existence of periodic orbits in the energy levels surrounding a stable equilibrium. The estimate obtained is very precise in the sense that it provides a lower bound for the number of relative periodic orbits at each prescribed energy and momentum values neighbouring the stable relative equilibrium in question and with any prefixed (spatio-temporal) isotropy subgroup. Moreover, it is easily computable in particular examples. It is interesting to see how, in our result, the existence of non-trivial relative periodic orbits requires (generic) conditions on the higher-order terms of the Taylor expansion of the Hamiltonian function, in contrast with the purely quadratic requirements of the Weinstein–Moser theorem, which emphasizes the highly nonlinear character of the relatively periodic dynamical objects.

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7

Grigoraş, Victor, and Carmen Grigoraş. "Connecting Analog and Discrete Nonlinear Systems for Noise Generation." Bulletin of the Polytechnic Institute of Iași. Electrical Engineering, Power Engineering, Electronics Section 68, no. 1 (March 1, 2022): 81–90. http://dx.doi.org/10.2478/bipie-2022-0005.

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Abstract Nonlinear systems exhibit complex dynamic behaviour, including quasi-periodic and chaotic. The present contribution presents a composed analogue and discrete-time structure, based on second-order nonlinear building blocks with periodic oscillatory behaviour, that can be used for complex signal generation. The chosen feedback connection of the two modules aims at obtaining a more complex nonlinear dynamic behaviour than that of the building blocks. Performing a parameter scan, it is highlighted that the resulting nonlinear system has a quasi-periodic behaviour for large ranges of parameter values. The nonlinear system attractor projections are obtained by simulation and statistical numerical results are presented, both confirming the possible use of the designed system as a noise generator.

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8

Abbas, Saïd, Mouffak Benchohra, Soufyane Bouriah, and Juan J. Nieto. "Periodic solutions for nonlinear fractional differential systems." Differential Equations & Applications, no. 3 (2018): 299–316. http://dx.doi.org/10.7153/dea-2018-10-21.

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9

Kamenskii, Mikhail, Oleg Makarenkov, and Paolo Nistri. "Small parameter perturbations of nonlinear periodic systems." Nonlinearity 17, no. 1 (October 17, 2003): 193–205. http://dx.doi.org/10.1088/0951-7715/17/1/012.

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10

Ghadimi, M., A. Barari, H. D. Kaliji, and G. Domairry. "Periodic solutions for highly nonlinear oscillation systems." Archives of Civil and Mechanical Engineering 12, no. 3 (September 2012): 389–95. http://dx.doi.org/10.1016/j.acme.2012.06.014.

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11

Ćwiszewski, Aleksander, and Piotr Kokocki. "Periodic solutions of nonlinear hyperbolic evolution systems." Journal of Evolution Equations 10, no. 3 (March 24, 2010): 677–710. http://dx.doi.org/10.1007/s00028-010-0066-y.

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12

Akhmetov, M. U., and N. A. Perestyuk. "Almost-periodic solutions of nonlinear impulse systems." Ukrainian Mathematical Journal 41, no. 3 (March 1989): 259–63. http://dx.doi.org/10.1007/bf01060307.

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13

Zhang, Yu, Xue Yang, and Yong Li. "Affine-Periodic Solutions for Dissipative Systems." Abstract and Applied Analysis 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/157140.

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As generalizations of Yoshizawa’s theorem, it is proved that a dissipative affine-periodic system admits affine-periodic solutions. This result reveals some oscillation mechanism in nonlinear systems.

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14

Silva, Edcarlos D., J. C. de Albuquerque, and Maxwell L. Silva. "Periodic and asymptotically periodic quasilinear elliptic systems." Journal of Mathematical Physics 61, no. 9 (September 1, 2020): 091501. http://dx.doi.org/10.1063/5.0012134.

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15

Cai, C. W., H. C. Chan, and Y. K. Cheung. "Localized Modes in Periodic Systems With Nonlinear Disorders." Journal of Applied Mechanics 64, no. 4 (December 1, 1997): 940–45. http://dx.doi.org/10.1115/1.2789003.

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The localized modes of periodic systems with infinite degrees-of-freedom and having one or two nonlinear disorders are examined by using the Lindstedt-Poincare (L-P) method. The set of nonlinear algebraic equations with infinite number of variables is derived and solved exactly by the U-transformation technique. It is shown that the localized modes exist for any amount of the ratio between the linear coupling stiffness kc and the coefficient γ of the nonlinear disordered term, and the nonsymmetric localized mode in the periodic system with two nonlinear disorders occurs as the ratio kc/γ, decreasing to a critical value depending on the maximum amplitude.

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16

Da´vid, Alexandra, and S. C. Sinha. "Bifurcation Control of Nonlinear Systems With Time-Periodic Coefficients." Journal of Dynamic Systems, Measurement, and Control 125, no. 4 (December 1, 2003): 541–48. http://dx.doi.org/10.1115/1.1636194.

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In this study, a method for the nonlinear bifurcation control of systems with periodic coefficients is presented. The aim of bifurcation control is to stabilize post bifurcation limit sets or modify other nonlinear characteristics such as stability, amplitude or rate of growth by employing purely nonlinear feedback controllers. The method is based on an application of the Lyapunov-Floquet transformation that converts periodic systems into equivalent forms with time-invariant linear parts. Then, through applications of time-periodic center manifold reduction and time-dependent normal form theory completely time-invariant nonlinear equations are obtained for codimension one bifurcations. The appropriate control gains are chosen in the time-invariant domain and transformed back to the original variables. The control strategy is illustrated through the examples of a parametrically excited simple pendulum undergoing symmetry-breaking bifurcation and a double inverted pendulum subjected to a periodic load in the case of a secondary Hopf bifurcation.

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17

Guttalu, Ramesh S., and Henryk Flashner. "Analysis of Nonlinear Systems by Truncated Point Mappings." Applied Mechanics Reviews 42, no. 11S (November 1, 1989): S83—S92. http://dx.doi.org/10.1115/1.3152412.

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This paper summarizes results obtained by the authors regarding the utility of truncated point mappings which have been recently published in a series of papers. The method described here is applicable to the analysis of multidimensional, multiparameter, periodic nonlinear systems by means of truncated point mappings. Based on multinomial truncation, an explicit analytical expression is determined for the point mapping in terms of the states and parameters of the system to any order of approximation. By combining this approach with analytical techniques, such as the perturbation method employed here, we obtain a powerful tool for finding periodic solutions and for analyzing their stability. The versatility of truncated point mapping method is demonstrated by applying it to study the limit cycles of van der Pol and coupled van der Pol oscillators, the periodic solutions of the forced Duffing’s equation and for a parametric analysis of periodic solutions of Mathieu’s equation.

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18

Sinha, S. C., and Alexandra Dávid. "Control of chaos in nonlinear systems with time-periodic coefficients." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 364, no. 1846 (July 27, 2006): 2417–32. http://dx.doi.org/10.1098/rsta.2006.1832.

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In this study, some techniques for the control of chaotic nonlinear systems with periodic coefficients are presented. First, chaos is eliminated from a given range of the system parameters by driving the system to a desired periodic orbit or to a fixed point using a full-state feedback. One has to deal with the same mathematical problem in the event when an autonomous system exhibiting chaos is desired to be driven to a periodic orbit. This is achieved by employing either a linear or a nonlinear control technique. In the linear method, a linear full-state feedback controller is designed by symbolic computation. The nonlinear technique is based on the idea of feedback linearization. A set of coordinate transformation is introduced, which leads to an equivalent linear system that can be controlled by known methods. Our second idea is to delay the onset of chaos beyond a given parameter range by a purely nonlinear control strategy that employs local bifurcation analysis of time-periodic systems. In this method, nonlinear properties of post-bifurcation dynamics, such as stability or rate of growth of a limit set, are modified by a nonlinear state feedback control. The control strategies are illustrated through examples. All methods are general in the sense that they can be applied to systems with no restrictions on the size of the periodic terms.

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19

Buică, Adriana, Jean–Pierre Françoise, and Jaume Llibre. "Periodic solutions of nonlinear periodic differential systems with a small parameter." Communications on Pure & Applied Analysis 6, no. 1 (2007): 103–11. http://dx.doi.org/10.3934/cpaa.2007.6.103.

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20

Miyazaki, Rinko, Dohan Kim, Toshiki Naito, and Jong Son Shin. "Fredholm operators, evolution semigroups, and periodic solutions of nonlinear periodic systems." Journal of Differential Equations 257, no. 11 (December 2014): 4214–47. http://dx.doi.org/10.1016/j.jde.2014.08.007.

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21

Luo, Albert C. J. "On Analytical Routes to Chaos in Nonlinear Systems." International Journal of Bifurcation and Chaos 24, no. 04 (April 2014): 1430013. http://dx.doi.org/10.1142/s0218127414300134.

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In this paper, the analytical methods for approximate solutions of periodic motions to chaos in nonlinear dynamical systems are reviewed. Briefly discussed are the traditional analytical methods including the Lagrange stand form, perturbation methods, and method of averaging. A brief literature survey of approximate methods in application is completed, and the weakness of current existing approximate methods is also discussed. Based on the generalized harmonic balance, the analytical solutions of periodic motions in nonlinear dynamical systems with/without time-delay are reviewed, and the analytical solutions for period-m motion to quasi-periodic motion are discussed. The analytical bifurcation trees of period-1 motion to chaos are presented as an application.

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22

LUO, ALBERT C. J., and YU GUO. "PARAMETER CHARACTERISTICS FOR STABLE AND UNSTABLE SOLUTIONS IN NONLINEAR DISCRETE DYNAMICAL SYSTEMS." International Journal of Bifurcation and Chaos 20, no. 10 (October 2010): 3173–91. http://dx.doi.org/10.1142/s0218127410027611.

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This paper studies complete stable and unstable periodic solutions for n-dimensional nonlinear discrete dynamical systems. The positive and negative iterative mappings of discrete systems are used to develop mapping structures of the stable and unstable periodic solutions. The complete bifurcation and stability analysis are presented for the stable and unstable periodic solutions which are based on the positive and negative mapping structures. A comprehensive investigation on the Henon map is carried out for a better understanding of complexity in nonlinear discrete systems. Given is the bifurcation scenario based on positive and negative mappings of the Henon map, and the analytical predictions of the corresponding periodic solutions are achieved. The corresponding eigenvalue analysis of the periodic solutions is presented. The Poincare mapping sections relative to the Neimark bifurcations of periodic solutions are presented. A parameter map for periodic and chaotic solutions is developed. The complete unstable and stable periodic solutions in nonlinear discrete systems are presented for the first time. The results presented in this paper provide a new idea for one to rethink the current existing theories.

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23

Kreuzer, Edwin J. "Stability and Bifurcations of Nonlinear Multibody Systems." Applied Mechanics Reviews 46, no. 11S (November 1, 1993): S156—S159. http://dx.doi.org/10.1115/1.3122631.

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Many technical systems are adequately described only by means of nonlinear mathematical models. Multibody systems became the most important mechanical models for analyzing engineering dynamics problems. The long-term or steady-state behavior of such systems can have a periodic, quasi-periodic, or chaotic character. Changes of the qualitative behavior are characterized by local and global bifurcations. This paper deals with stability problems in multibody system dynamics and explains different bifurcation phenomena as well as methods for analyzing them. Results from a simple oscillator prove the applicability of the methods.

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24

KOMINIS, Y., and T. BOUNTIS. "ANALYTICAL SOLUTIONS OF SYSTEMS WITH PIECEWISE LINEAR DYNAMICS." International Journal of Bifurcation and Chaos 20, no. 02 (February 2010): 509–18. http://dx.doi.org/10.1142/s0218127410025570.

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A class of nonautonomous dynamical systems, consisting of an autonomous nonlinear system and an autonomous linear periodic system, each acting by itself at different time intervals, is studied. It is shown that under certain conditions for the durations of the linear and the nonlinear time intervals, the dynamics of the nonautonomous piecewise linear system is closely related to that of its nonlinear autonomous component. As a result, families of explicit periodic, nonperiodic and localized breather-like solutions are analytically obtained for a variety of interesting physical phenomena.

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25

Deshmukh, Venkatesh, Eric A. Butcher, and S. C. Sinha. "Order Reduction of Parametrically Excited Linear and Nonlinear Structural Systems." Journal of Vibration and Acoustics 128, no. 4 (December 21, 2005): 458–68. http://dx.doi.org/10.1115/1.2202151.

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Order reduction of parametrically excited linear and nonlinear structural systems represented by a set of second order equations is considered. First, the system is converted into a second order system with time invariant linear system matrices and (for nonlinear systems) periodically modulated nonlinearities via the Lyapunov-Floquet transformation. Then a master-slave separation of degrees of freedom is used and a relation between the slave coordinates and the master coordinates is constructed. Two possible order reduction techniques are suggested. In the first approach a constant Guyan-like linear kernel which accounts for both stiffness and inertia is employed with a possible periodically modulated nonlinear part for nonlinear systems. The second method for nonlinear systems reduces to finding a time-periodic nonlinear invariant manifold relation in the modal coordinates. In the process, closed form expressions for “true internal” and “true combination” resonances are obtained for various nonlinearities which are generalizations of those previously reported for time-invariant systems. No limits are placed on the size of the time-periodic terms thus making this method extremely general even for strongly excited systems. A four degree-of-freedom mass- spring-damper system with periodic stiffness and damping as well as two and five degree-of-freedom inverted pendula with periodic follower forces are used as illustrative examples. The nonlinear-based reduced models are compared with linear-based reduced models in the presence and absence of nonlinear resonances.

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26

Chiu, Kuo-Shou. "Periodic Solutions for Nonlinear Integro-Differential Systems with Piecewise Constant Argument." Scientific World Journal 2014 (2014): 1–14. http://dx.doi.org/10.1155/2014/514854.

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We investigate the existence of the periodic solutions of a nonlinear integro-differential system with piecewise alternately advanced and retarded argument of generalized type, in short DEPCAG; that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. In the process we use Green’s function for periodic solutions and convert the given DEPCAG into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii’s fixed point theorem to show the existence of a periodic solution of this type of nonlinear differential equations. We also use the contraction mapping principle to show the existence of a unique periodic solution. Appropriate examples are given to show the feasibility of our results.

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27

LUO, ALBERT C. J. "A YING–YANG THEORY IN NONLINEAR DISCRETE DYNAMICAL SYSTEMS." International Journal of Bifurcation and Chaos 20, no. 04 (April 2010): 1085–98. http://dx.doi.org/10.1142/s0218127410026332.

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This paper presents a Ying–Yang theory for nonlinear discrete dynamical systems considering both positive and negative iterations of discrete iterative maps. In the existing analysis, the solutions relative to "Yang" in nonlinear dynamical systems are extensively investigated. However, the solutions pertaining to "Ying" in nonlinear dynamical systems are investigated. A set of concepts on "Ying" and "Yang" in discrete dynamical systems are introduced to help one understand the hidden dynamics in nonlinear discrete dynamical systems. Based on the Ying–Yang theory, the periodic and chaotic solutions in nonlinear discrete dynamical system are discussed, and all possible, stable and unstable periodic solutions can be analytically predicted. A discrete dynamical system with the Henon map is investigated, as an example.

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28

Wu, Rui, Yi Cheng, and Ravi P. Agarwal. "Rotational periodic solutions for fractional iterative systems." AIMS Mathematics 6, no. 10 (2021): 11233–45. http://dx.doi.org/10.3934/math.2021651.

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<abstract><p>In this paper, we devoted to deal with the rotational periodic problem of some fractional iterative systems in the sense of Caputo fractional derivative. Under one sided-Lipschtiz condition on nonlinear term, the existence and uniqueness of solution for a fractional iterative equation is proved by applying the Leray-Schauder fixed point theorem and topological degree theory. Furthermore, the well posedness for a nonlinear control system with iteration term and a multivalued disturbance is established by using set-valued theory. The existence of solutions for a iterative neural network system is demonstrated at the end.</p></abstract>

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29

Li, Li, and Chen Jia. "Preview repetitive control for polytopic nonlinear discrete-time systems." Science Progress 105, no. 2 (April 2022): 003685042210754. http://dx.doi.org/10.1177/00368504221075472.

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In this study, a preview repetitive control (PRC) strategy was developed for uncertain nonlinear discrete-time systems subjected to a previewable periodic reference signal. The proposed preview repetitive controller was designed such that the system output tracked a previewable periodic reference signal even with model uncertainties and nonlinear terms. An augmented two-dimensional (2D) model was constructed based on the 2D model approach and state augmented technique. Second, considering the state unmeasured and periodic tracking reference signal, a static output PRC law was designed using the linear matrix inequality (LMI) techniques. Finally, the effectiveness of the proposed controller was verified through two illustrative examples.

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30

Xiang, Xiaolin, and Tao Luo. "Existence of periodic solutions of nonlinear systems with nonlinear boundary conditions." Acta Mathematica Sinica 11, no. 4 (December 1995): 439–45. http://dx.doi.org/10.1007/bf02248755.

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31

Zheng, Bo. "Multiple Periodic Solutions to Nonlinear Discrete Hamiltonian Systems." Advances in Difference Equations 2007 (2007): 1–14. http://dx.doi.org/10.1155/2007/41830.

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32

Naito, Koichiro. "Periodic Stability of Nonlinear Flexible Systems with Damping." SIAM Journal on Control and Optimization 33, no. 6 (November 1995): 1778–800. http://dx.doi.org/10.1137/s0363012993253522.

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33

Boskresenskii, E. V. "Comparison method and periodic solutions of nonlinear systems." Ukrainian Mathematical Journal 43, no. 10 (October 1991): 1254–58. http://dx.doi.org/10.1007/bf01061810.

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34

Geng, Jiansheng, Jiangong You, and Zhiyan Zhao. "Localization in One-dimensional Quasi-periodic Nonlinear Systems." Geometric and Functional Analysis 24, no. 1 (January 28, 2014): 116–58. http://dx.doi.org/10.1007/s00039-014-0256-9.

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35

Levy, Uri, Ken Yang, Noam Matzliah, and Yaron Silberberg. "Universal correlations after thermalization in periodic nonlinear systems." Journal of Physics B: Atomic, Molecular and Optical Physics 51, no. 3 (January 8, 2018): 035401. http://dx.doi.org/10.1088/1361-6455/aa9a97.

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36

Luo, Albert C. J. "Periodic motions and chaos in nonlinear dynamical systems." European Physical Journal Special Topics 228, no. 9 (September 2019): 1745–46. http://dx.doi.org/10.1140/epjst/e2019-900142-2.

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37

Kim, Wan Se. "Existence of periodic solutions for nonlinear Lienard systems." International Journal of Mathematics and Mathematical Sciences 18, no. 2 (1995): 265–72. http://dx.doi.org/10.1155/s0161171295000329.

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38

Martynyuk, D. I., and V. A. Dankanich. "Periodic solutions of nonlinear autonomous systems with delay." Ukrainian Mathematical Journal 39, no. 1 (1987): 52–55. http://dx.doi.org/10.1007/bf01056424.

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39

Zhou, Zhengxin. "Reflecting functions and periodic solutions of nonlinear systems." Applied Mathematics Letters 17, no. 10 (2004): 1121–25. http://dx.doi.org/10.1016/j.aml.2003.12.004.

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40

Nesterenko, Vitali F. "Waves in strongly nonlinear discrete systems." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2127 (July 23, 2018): 20170130. http://dx.doi.org/10.1098/rsta.2017.0130.

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The paper presents the main steps in the development of the strongly nonlinear wave dynamics of discrete systems. The initial motivation was prompted by the challenges in the design of barriers to mitigate high-amplitude compression pulses caused by impact or explosion. But this area poses a fundamental mathematical and physical problem and should be considered as a natural step in developing strongly nonlinear wave dynamics. Strong nonlinearity results in a highly tunable behaviour and allows design of systems with properties ranging from a weakly nonlinear regime, similar to the classical case of the Fermi–Pasta–Ulam lattice, or to a non-classical case of sonic vacuum. Strongly nonlinear systems support periodic waves and one of the fascinating results was a discovery of a strongly nonlinear solitary wave in sonic vacuum (a limiting case of a periodic wave) with properties very different from the Korteweg de Vries solitary wave. Shock-like oscillating and monotonous stationary stress waves can also be supported if the system is dissipative. The paper discusses the main theoretical and experimental results, focusing on travelling waves and possible future developments in the area of strongly nonlinear metamaterials. This article is part of the theme issue ‘Nonlinear energy transfer in dynamical and acoustical systems’.

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41

Zhang, Yandong, and S. C. Sinha. "Development of a Feedback Linearization Technique for Parametrically Excited Nonlinear Systems via Normal Forms." Journal of Computational and Nonlinear Dynamics 2, no. 2 (December 8, 2006): 124–31. http://dx.doi.org/10.1115/1.2447190.

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The problem of designing controllers for nonlinear time periodic systems via feedback linearization is addressed. The idea is to find proper coordinate transformations and state feedback under which the original system can be (exactly or approximately) transformed into a linear time periodic control system. Then a controller can be designed to guarantee the stability of the system. Our approach is designed to achieve local control of nonlinear systems with periodic coefficients desired to be driven either to a periodic orbit or to a fixed point. The system equations are represented by a quasi-linear system containing nonlinear monomials with periodic coefficients. Using near identity transformations and normal form theory, the original close loop problem is approximately transformed into a linear time periodic system with unknown gains. Then by using a symbolic computation method, the Floquet multipliers are placed in the desired locations in order to determine the control gains. We also give the sufficient conditions under which the system is feedback linearizable up to the rth order.

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42

Qin, Dongdong, Xianhua Tang, and Qingfang Wu. "Ground states of nonlinear Schrödinger systems with periodic or non-periodic potentials." Communications on Pure & Applied Analysis 18, no. 3 (2019): 1261–80. http://dx.doi.org/10.3934/cpaa.2019061.

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43

Tang, Xiao-Yan, and Sen-Yue Lou. "Quasi-periodic and Non-periodic Waves in (2+1)-Dimensional Nonlinear Systems." Communications in Theoretical Physics 44, no. 4 (October 2005): 583–88. http://dx.doi.org/10.1088/6102/44/4/583.

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44

Jun, Jin. "Existence and uniqueness of periodic solutions for fourth-order nonlinear periodic systems." Applied Mathematics and Mechanics 11, no. 1 (January 1990): 89–97. http://dx.doi.org/10.1007/bf02014575.

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45

Nayfeh, A. H., and B. Balachandran. "Modal Interactions in Dynamical and Structural Systems." Applied Mechanics Reviews 42, no. 11S (November 1, 1989): S175—S201. http://dx.doi.org/10.1115/1.3152389.

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We review theoretical and experimental studies of the influence of modal interactions on the nonlinear response of harmonically excited structural and dynamical systems. In particular, we discuss the response of pendulums, ships, rings, shells, arches, beam structures, surface waves, and the similarities in the qualitative behavior of these systems. The systems are characterized by quadratic nonlinearities which may lead to two-to-one and combination autoparametric resonances. These resonances give rise to a coupling between the modes involved in the resonance leading to nonlinear periodic, quasi-periodic, and chaotic motions.

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46

Huang, Jianzhe, and Albert C. J. Luo. "Periodic Motions and Bifurcation Trees in a Buckled, Nonlinear Jeffcott Rotor System." International Journal of Bifurcation and Chaos 25, no. 01 (January 2015): 1550002. http://dx.doi.org/10.1142/s0218127415500029.

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In this paper, analytical solutions for period-m motions in a buckled, nonlinear Jeffcott rotor system are obtained. This nonlinear Jeffcott rotor system with two-degrees of freedom is excited periodically from the rotor eccentricity. The analytical solutions of period-m solutions are developed, and the corresponding stability and bifurcation are also analyzed by eigenvalue analysis. Analytical bifurcation trees of period-1 motions to chaos are presented. The Hopf bifurcations of periodic motions cause not only the bifurcation tree but quasi-periodic motions. The quasi-periodic motion can be stable or unstable. Displacement orbits of periodic motions in the buckled, nonlinear Jeffcott rotor systems are illustrated, and harmonic amplitude spectrums are presented for harmonic effects on periodic motions of the nonlinear rotor. Coexisting periodic motions exist in such a buckled nonlinear Jeffcott rotor.

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S.B. Kiwne. "Periodic solutions of nonlinear finite difference systems with time delays." Malaya Journal of Matematik 1, no. 04 (October 1, 2013): 81–88. http://dx.doi.org/10.26637/mjm104/009.

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In this paper a coupled system of nonlinear finite difference equations corresponding to a class of periodic-parabolic systems with time delays and with nonlinear boundary conditions in a bounded domain is investigated. Using the method of upper-lower solutions two monotone sequences for the finite difference system are constructed. Existence of maximal and minimal periodic solutions of coupled system of finite difference equations with nonlinear boundary conditions is also discussed. The proof of existence theorem is based on the method of upper-lower solutions and its associated monotone iterations. It is shown that the sequence of iterations converges monotonically to unique solution of the nonlinear finite difference system with time delays under consideration.

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Kishimoto, Nobu. "Remarks on periodic Zakharov systems." Electronic Journal of Differential Equations 2022, no. 01-87 (March 18, 2022): 20. http://dx.doi.org/10.58997/ejde.2022.20.

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In this article, we consider the Cauchy problem associated with the Zakharov system on the torus. We obtain unconditional uniqueness of solutions in low regularity Sobolev spaces including the energy space in one and two dimensions. We also prove convergence of solutions in the energy space, as the ion sound speed tends to infinity, to the solution of a cubic nonlinear Schrodinger equation, for dimensions one and two. Our proof of unconditional uniqueness is based on the method of infinite iteration of the normal form reduction; actually, we simply show a certain set of multilinear estimates, which was proposed as a criterion for unconditional uniqueness in [13]. The convergence result is obtained by a similar argument to the non-periodic case [13], which uses conservation laws and unconditional uniqueness for the limit equation.

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Duan, Yameng, Wieslaw Krawcewicz, and Huafeng Xiao. "Periodic solutions in reversible systems in second order systems with distributed delays." AIMS Mathematics 9, no. 4 (2024): 8461–75. http://dx.doi.org/10.3934/math.2024411.

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<abstract><p>In this paper, we study the existence and multiplicity of periodic solutions to a class of second-order nonlinear differential equations with distributed delay. Under assumptions that the nonlinearity is odd, differentiable at zero and satisfies the Nagumo condition, by applying the equivariant degree method, we prove that the delay equation admits multiple periodic solutions. The results are then illustrated by an example.</p></abstract>

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50

Llibre, Jaume, Durval José Tonon, and Mariana Queiroz Velter. "Crossing Periodic Orbits via First Integrals." International Journal of Bifurcation and Chaos 30, no. 11 (September 15, 2020): 2050163. http://dx.doi.org/10.1142/s0218127420501631.

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We characterize the families of periodic orbits of two discontinuous piecewise differential systems in [Formula: see text] separated by a plane using their first integrals. One of these discontinuous piecewise differential systems is formed by linear differential systems, and the other by nonlinear differential systems.

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