On Periodic Motions in the First-Order Nonlinear Systems

Initial Conditions ◽

Periodic Motions ◽

Nonlinear Dynamical ◽

First Order

In this paper, analytical solutions of periodic motions in the first-order nonlinear dynamical system are discussed from the finite Fourier series expression. The first-order nonlinear dynamical system is transformed to the dynamical system of coefficients in the Fourier series. From investigation of such dynamical system of coefficients, the analytical solutions of periodic motions are obtained, and the corresponding stability and bifurcation of periodic motions will be determined. In fact, this method provides a frequency-response analysis of periodic motions in nonlinear dynamical systems, which is alike the Laplace transformation of periodic motions for nonlinear dynamical systems. The harmonic frequency-amplitude curves are obtained for different-order harmonic terms in the Fourier series. Through such frequency-amplitude curves, the nonlinear characteristics of periodic motions in the first-order nonlinear system can be determined. From analytical solutions, the initial conditions are obtained for numerical simulations. From such initial conditions, numerical simulations are completed in comparison of the analytical solutions of periodic motions.

On a Global Sequential Scenario of Bifurcation Trees to Chaos in a First-Order, Periodically Excited, Time-Delayed System

10.1142/s0218127419501414 ◽

2019 ◽

Vol 29 (10) ◽

pp. 1950141

Author(s):

Siyuan Xing ◽

Albert C. J. Luo

Keyword(s):

Dynamical System ◽

Global Bifurcation ◽

Periodic Motions ◽

Bifurcation Tree ◽

Nonlinear Dynamical ◽

First Order ◽

Delayed System ◽

The Rich ◽

Stability And Bifurcations

In this paper, the global sequential scenario of bifurcation trees of periodic motions to chaos is studied for a first-order, time-delayed, nonlinear dynamical system with periodic excitation. The periodic motions of such a first-order time-delayed system is obtained semi-analytically, and the corresponding stability and bifurcations are determined by eigenvalue analysis. A global sequential scenario of bifurcation trees is given by [Formula: see text] where [Formula: see text] is a global bifurcation tree of an asymmetric period-[Formula: see text] motion to chaos, and [Formula: see text] is a global bifurcation tree of a symmetric period-[Formula: see text] motion to chaos. Each bifurcation tree of a specific periodic motion to chaos is presented in detail. Numerical simulations of periodic motions are performed from analytical predictions. From finite Fourier series, harmonic amplitudes and phases for periodic motions are obtained for frequency analysis. Through this study, the rich dynamics of the first-order, time-delayed, nonlinear dynamical system is presented.

NONLINEAR DYNAMICAL SYSTEM AND CHAOS SYNCHRONIZATION

10.1142/s0218127408021142 ◽

2008 ◽

Vol 18 (05) ◽

pp. 1531-1537 ◽

Cited By ~ 3

Author(s):

AYUB KHAN ◽

PREMPAL SINGH

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Chaos Synchronization ◽

Numerical Techniques ◽

Nonlinear Dynamical ◽

Chaotic Dynamical Systems ◽

Response System ◽

Identical Response

Chaos synchronization of nonlinear dynamical systems has been studied through theoretical and numerical techniques. For the synchronization of two identical nonlinear chaotic dynamical systems a theorem has been constructed based on the Lyapunov function, which requires a minimal knowledge of system's structure to synchronize with an identical response system. Numerical illustrations have been provided to verify the theorem.

ANALYTICAL DYNAMICS OF PERIOD-m FLOWS AND CHAOS IN NONLINEAR SYSTEMS

10.1142/s0218127412500939 ◽

2012 ◽

Vol 22 (04) ◽

pp. 1250093 ◽

Cited By ~ 53

Author(s):

ALBERT C. J. LUO ◽

JIANZHE HUANG

Keyword(s):

Dynamical Systems ◽

Analytical Solutions ◽

Duffing Oscillator ◽

Harmonic Balance Method ◽

Period Doubling ◽

Nonlinear Damping ◽

Periodic Motions ◽

Chaotic Motions ◽

Nonlinear Dynamical

In this paper, the analytical solutions for period-m flows and chaos in nonlinear dynamical systems are presented through the generalized harmonic balance method. The nonlinear damping, periodically forced, Duffing oscillator was investigated as an example to demonstrate the analytical solutions of periodic motions and chaos. Through this investigation, the mechanism for a period-m motion jumping to another period-n motion in numerical computation is found. In this problem, the Hopf bifurcation of periodic motions is equivalent to the period-doubling bifurcation via Poincare mappings of dynamical systems. The stable and unstable period-m motions can be obtained analytically. Even more, the stable and unstable chaotic motions can be achieved analytically. The methodology presented in this paper can be applied to other nonlinear vibration systems, which is independent of small parameters.

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

Optimal tracking control of nonlinear dynamical systems

10.1098/rspa.2008.0040 ◽

2008 ◽

Vol 464 (2097) ◽

pp. 2341-2363 ◽

Cited By ~ 56

Author(s):

Firdaus E Udwadia

Keyword(s):

Dynamical Systems ◽

Differential Equations ◽

Weighted Norm ◽

Analytical Dynamics ◽

Control Cost ◽

Nonlinear Dynamical ◽

First Order ◽

Time Optimal

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

On Algorithm of Integrability Classification of the Nonlinear Dynamical Systems via Computer Algebra Methods

Physico-mathematical modelling and informational technologies ◽

10.15407/fmmit2021.32.008 ◽

2021 ◽

pp. 7-12

Author(s):

Bohdan Fil ◽

Yaroslav Pelekh ◽

Myroslava Vovk ◽

Halyna Beregova ◽

Tatiana Magerovska ◽

...

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Conservation Laws ◽

Computer Algebra ◽

Nonlinear Dynamical ◽

Wolfram Mathematica ◽

De Vries

There is developed an algorithm to classify integrable nonlinear dynamical systems via Wolfram Mathematica. The hierarchy of conservation laws for the nonlinear dynamical system can be cal-culated by this algorithm. There are demonstrated some modifications of nonlinear Korteweg-de Vries equations integrated by inverse scatering method.

Aggregation of a class of large-scale, interconnected, nonlinear dynamical systems

Mathematical Problems in Engineering ◽

10.1155/s1024123x01001697 ◽

2001 ◽

Vol 7 (4) ◽

pp. 379-392 ◽

Cited By ~ 4

Author(s):

Swaroop Darbha ◽

K. R. Rajagopal

Keyword(s):

Dynamical System ◽

Differential Equation ◽

Dynamical Systems ◽

Large Scale ◽

Initial Conditions ◽

Hierarchical Systems ◽

Nonlinear Dynamical ◽

Analysis And Design ◽

Computational Perspective

In this paper, the authors consider the issue of the construction of a meaningful average for a collection of nonlinear dynamical systems. Such a collection of dynamical systems may or may not have well defined ensemble averages as the existence of ensemble averages is predicated on the specification of appropriate initial conditions. A meaningful “average” dynamical system can represent the macroscopic behavior of the collection of systems and allow us to infer the behavior of such systems on an average. They can also prove to be very attractive from a computational perspective. An advantage to the construction of the meaningful average is that it involves integrating a nonlinear differential equation, of the same order as that of any member in the collection. An average dynamical system can be used in the analysis and design of hierarchical systems, and will allow one to capture approximately the response of any member of the collection.

On Analytical Routes to Chaos in Nonlinear Systems

10.1142/s0218127414300134 ◽

2014 ◽

Vol 24 (04) ◽

pp. 1430013 ◽

Cited By ~ 12

Author(s):

Albert C. J. Luo

Keyword(s):

Dynamical Systems ◽

Analytical Solutions ◽

Analytical Methods ◽

Harmonic Balance ◽

Approximate Solutions ◽

Periodic Motions ◽

Approximate Methods ◽

Nonlinear Dynamical ◽

Analytical Bifurcation Trees

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.

Modeling of information channel by using of pseudorandom signals of nonlinear dynamical system

Proceedings of the higher educational institutions ENERGY SECTOR PROBLEMS ◽

10.30724/1998-9903-2020-22-4-79-87 ◽

2020 ◽

Vol 22 (4) ◽

pp. 79-87

Author(s):

I. K. Nasyrov ◽

V. V. Andreev

Keyword(s):

Dynamical Systems ◽

Detection Algorithm ◽

Main Step ◽

Operating Characteristics ◽

Nonlinear Dynamical ◽

Pseudorandom Signals ◽

Chaotic Signals ◽

Correlation Properties

Pseudorandom signals of nonlinear dynamical systems are studied and the possibility of their application in information systems analyzed. Continuous and discrete dynamical systems are considered: Lorenz System, Bernoulli and Henon maps. Since the parameters of dynamical systems (DS) are included in the equations linearly, the principal possibility of the state linear control of a nonlinear DS is shown. The correlation properties comparative analysis of these DSs signals is carried out.. Analysis of correlation characteristics has shown that the use of chaotic signals in communication and radar systems can significantly increase their resolution over the range and taking into account the specific properties of chaotic signals, it allows them to be hidden. The representation of nonlinear dynamical systems equations in the form of stochastic differential equations allowed us to obtain an expression for the likelihood functional, with the help of which many problems of optimal signal reception are solved. It is shown that the main step in processing the received message, which provides the maximum likelihood functionals, is to calculate the correlation integrals between the components and the systems under consideration. This made it possible to base the detection algorithm on the correlation reception between signal components. A correlation detection receiver was synthesized and the operating characteristics of the receiver were found.

The Generalized Shooting Arc-Length Method for Periodic Solutions and the Corresponding Periods of Nonlinear Dynamical Systems

10.1115/imece2001/de-23212 ◽

2001 ◽

Author(s):

Dexin Li ◽

Jianxue Xu

Keyword(s):

Dynamical System ◽

Periodic Solution ◽

Periodic Orbit ◽

Optimization Method ◽

Arc Length ◽

Van Der Pol Equation ◽

Van Der Pol ◽

Nonlinear Dynamical

Abstract In this paper, a generalized shooting/arc-length method for determining periodic orbit and its period of nonlinear dynamical system is presented. At first, by changing the time scale the period value of periodic orbit of the nonlinear system is drawn into the governing equation of this system. Then, by using the period value as a parameter, the shooting/arc-length procedure is taken for seeking such a periodic solution and its period simultaneously. The value of increment changed in iteration procedure is selected by using optimization method. The procedure involves the detennining of periodic orbit and its period value of the system. Thereby, the periodic orbit and period value of the system can be sought out rapidly and precisely. At last, the validity of such method is verified by determining the periodic orbit and period value for van der pol equation and nonlinear rotor-bear system.

A New Method for Equilibria and Stability Analysis of Nonlinear Dynamical Systems

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.534.131 ◽

2014 ◽

Vol 534 ◽

pp. 131-136

Author(s):

Long Cao ◽

Yi Hua Cao

Keyword(s):

Stability Analysis ◽

Dynamical Systems ◽

Numerical Continuation ◽

Vehicle Model ◽

Nonlinear Dynamical ◽

Novel Method ◽

Linearized System ◽

Continuation Algorithm

A novel method based on numerical continuation algorithm for equilibria and stability analysis of nonlinear dynamical system is introduced and applied to an aircraft vehicle model. Dynamical systems are usually modeled with differential equations, while their equilibria and stability analysis are pure algebraic problems. The newly-proposed method in this paper provides a way to solve the equilibrium equation and the eigenvalues of the locally linearized system simultaneously, which avoids QR iterations and can save much time.