scholarly journals The direct method of Lyapunov for nonlinear dynamical systems with fractional damping

2020 ◽  
Vol 102 (4) ◽  
pp. 2017-2037
Author(s):  
Matthias Hinze ◽  
André Schmidt ◽  
Remco I. Leine
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Direct Method ◽  
Infinite Delay ◽  
Functional Differential Equations ◽  
Nonlinear Damping ◽  
Degree Of Freedom ◽  
Lyapunov Functionals ◽  
State Variables ◽  
Fractional Damping

AbstractIn this paper, we introduce a generalization of Lyapunov’s direct method for dynamical systems with fractional damping. Hereto, we embed such systems within the fundamental theory of functional differential equations with infinite delay and use the associated stability concept and known theorems regarding Lyapunov functionals including a generalized invariance principle. The formulation of Lyapunov functionals in the case of fractional damping is derived from a mechanical interpretation of the fractional derivative in infinite state representation. The method is applied on a single degree-of-freedom oscillator first, and the developed Lyapunov functionals are subsequently generalized for the finite-dimensional case. This opens the way to a stability analysis of nonlinear (controlled) systems with fractional damping. An important result of the paper is the solution of a tracking control problem with fractional and nonlinear damping. For this problem, the classical concepts of convergence and incremental stability are generalized to systems with fractional-order derivatives of state variables. The application of the related method is illustrated on a fractionally damped two degree-of-freedom oscillator with regularized Coulomb friction and non-collocated control.

Asymptotic stability criteria for delay-differential equations

1988 ◽  
Vol 110 (1-2) ◽  
pp. 31-44
Author(s):  
L.C. Becker ◽  
T.A. Burton
Keyword(s):  
Differential Equations ◽  
Direct Method ◽  
Infinite Delay ◽  
Functional Differential Equations ◽  
State Variables ◽  
Stability Criteria ◽  
Bounded State ◽  
Functional Differential ◽  
Delay Differential ◽  
Liapunov's Direct Method

SynopsisThis paper is concerned with the problem of showing uniform stability and equiasymptotic stability of thezero solution of functional differential equations with either finite or infinite delay. The investigations are based on Liapunov's direct method and attention is focused on those equations whose right-hand sides are unbounded for bounded state variables.

A sequential importance sampling filter with a new proposal distribution for state and parameter estimation of nonlinear dynamical systems

2007 ◽  
Vol 464 (2089) ◽  
pp. 25-47 ◽  
Author(s):  
Shuva J Ghosh ◽  
C.S Manohar ◽  
D Roy
Keyword(s):  
Dynamical Systems ◽  
Parameter Identification ◽  
Importance Sampling ◽  
Nonlinear Dynamical Systems ◽  
Gaussian White Noise ◽  
Sequential Importance Sampling ◽  
State Variables ◽  
Nonlinear Dynamical ◽  
Taylor Expansions ◽  
Non Gaussian

The problem of estimating parameters of nonlinear dynamical systems based on incomplete noisy measurements is considered within the framework of Bayesian filtering using Monte Carlo simulations. The measurement noise and unmodelled dynamics are represented through additive and/or multiplicative Gaussian white noise processes. Truncated Ito–Taylor expansions are used to discretize these equations leading to discrete maps containing a set of multiple stochastic integrals. These integrals, in general, constitute a set of non-Gaussian random variables. The system parameters to be determined are declared as additional state variables. The parameter identification problem is solved through a new sequential importance sampling filter. This involves Ito–Taylor expansions of nonlinear terms in the measurement equation and the development of an ideal proposal density function while accounting for the non-Gaussian terms appearing in the governing equations. Numerical illustrations on parameter identification of a few nonlinear oscillators and a geometrically nonlinear Euler–Bernoulli beam reveal a remarkably improved performance of the proposed methods over one of the best known algorithms, i.e. the unscented particle filter.

Refining the Method of Averaging for a More Accurate Analytical Approximation to the Behaviors of a Damped Pendulum

2012 ◽  
Author(s):  
Clark C. McGehee ◽  
Si Mohamed Sah ◽  
Brian P. Mann
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Damping ◽  
Analytical Approximation ◽  
Method Of Averaging ◽  
Restoring Force ◽  
Nonlinear Dynamical ◽  
Wide Range ◽  
Approximation Errors ◽  
Kbm Method

KBM averaging is a widely used technique in the analysis of nonlinear dynamical systems. The KBM method allows complex systems to be approximated as perturbations of simple harmonic oscillator. In many cases, such as in otherwise linear systems with various forms nonlinear damping, the KBM method performs exceptionally well, with error proportional to the size of the perturbations. However, when the largest perturbation in the system arises from nonlinearities in the restoring force, the KBM method falls short, and the interesting effects of other nonlinear terms are drowned out by the approximation errors generated by the KBM method. By generalizing the notion of KBM averaging and approximating systems as perturbations the isoenergy contours of their corresponding Hamiltonian, a greater degree of accuracy can be obtained. We extend the work of several authors to show that not only is this method more accurate, but it is also simple to implement and generalizable to a wide range of nonlinear systems. As an illustrative example, the motion of a pendulum on a tilted platform is studied.

Global Nonlinear Dynamics Based on the Method of Complete Bifurcation Groups and Rare Attractors

2009 ◽  
Author(s):  
Mikhail V. Zakrzhevsky
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Analysis ◽  
Degrees Of Freedom ◽  
Nonlinear Dynamical Systems ◽  
Piecewise Linear ◽  
Global Bifurcation ◽  
Main Idea ◽  
Degree Of Freedom ◽  
Strongly Nonlinear ◽  
Driven System

The paper is devoted to the global bifurcation analysis of the models of strongly nonlinear forced or autonomous dynamical systems with one or several-degree-of-freedom by direct numerical and/or analytical methods. A new approach for the global bifurcation analysis for strongly nonlinear dynamical systems, based on the ideas of Poincare´, Birkhoff and Andronov, is proposed. The main idea of the approach is a concept of complete bifurcation groups and periodic branch continuation along stable and unstable solutions, named by the author as a method of complete bifurcation groups (MCBG). The article is illustrated using four archetypal forced dynamical systems with one degree-of-freedom. They are Duffing model with positional force f(x) = x + x3, Duffing double-well potential driven system, pendulum driven system and piecewise-linear (bilinear soft impact) driven dynamical system (Eq. 1–4). x+¨bx+˙x+x3=h1coswt(1)x+¨bx−˙x+x3=h1coswt(2)x+¨bx+˙a1sin(πx)=h1coswt(3)x+¨bx+˙f(x)=h1coswt,(4)f(x)=c1xifx≤d1,c2x−(c2−c1)d1ifx>d1 This paper is a continuation of the author’s previous one [53] with new results such as new bifurcation groups, rare attractors (RA) and protuberances. Some new results for dynamical systems with several degrees-of-freedom, based on the method of complete bifurcation groups may be found in [46–52].

Homotopy Analysis Method for Multi-Degree-of-Freedom Nonlinear Dynamical Systems

2010 ◽  
Author(s):  
W. Zhang ◽  
Y. H. Qian ◽  
M. H. Yao ◽  
S. K. Lai
Keyword(s):  
Dynamical Systems ◽  
Homotopy Analysis Method ◽  
Nonlinear Dynamical Systems ◽  
Analytical Approximation ◽  
Degree Of Freedom ◽  
Analysis Method ◽  
Nonlinear Dynamical ◽  
Homotopy Analysis ◽  
Practical Engineering ◽  
Proof Of Convergence

In reality, the behavior and nature of nonlinear dynamical systems are ubiquitous in many practical engineering problems. The mathematical models of such problems are often governed by a set of coupled second-order differential equations to form multi-degree-of-freedom (MDOF) nonlinear dynamical systems. It is extremely difficult to find the exact and analytical solutions in general. In this paper, the homotopy analysis method is presented to derive the analytical approximation solutions for MDOF dynamical systems. Four illustrative examples are used to show the validity and accuracy of the homotopy analysis and modified homotopy analysis methods in solving MDOF dynamical systems. Comparisons are conducted between the analytical approximation and exact solutions. The results demonstrate that the HAM is an effective and robust technique for linear and nonlinear MDOF dynamical systems. The proof of convergence theorems for the present method is elucidated as well.

Periodic Solutions for Coupled Van Der Pol Oscillators of Two-Degree-of-Freedom Solved by Homotopy Analysis Method

2009 ◽  
Author(s):  
Wei Zhang ◽  
Youhua Qian ◽  
Qian Wang
Keyword(s):  
Dynamical Systems ◽  
Analytical Solutions ◽  
Nonlinear Dynamical Systems ◽  
Degree Of Freedom ◽  
Analysis Method ◽  
Van Der Pol ◽  
Nonlinear Dynamical ◽  
Homotopy Analysis ◽  
Van Der Pol Oscillators ◽  
Two Degree Of Freedom

Innumerable engineering problems can be described by multi-degree-of-freedom (MDOF) nonlinear dynamical systems. The theoretical modelling of such systems is often governed by a set of coupled second-order differential equations. Albeit that it is extremely difficult to find their exact solutions, the research efforts are mainly concentrated on the approximate analytical solutions. The homotopy analysis method (HAM) is a useful analytic technique for solving nonlinear dynamical systems and the method is independent on the presence of small parameters in the governing equations. More importantly, unlike classical perturbation technique, it provides a simple way to ensure the convergence of solution series by means of an auxiliary parameter ħ. In this paper, the HAM is presented to establish the analytical approximate periodic solutions for two-degree-of-freedom coupled van der Pol oscillators. In addition, comparisons are conducted between the results obtained by the HAM and the numerical integration (i.e. Runge-Kutta) method. It is shown that the higher-order analytical solutions of the HAM agree well with the numerical integration solutions, even if time t progresses to a certain large domain in the time history responses.

Maximum entropy approach to the identification of stochastic reduced-order models of nonlinear dynamical systems

2010 ◽  
Vol 114 (1160) ◽  
pp. 637-650 ◽  
Author(s):  
M. Arnst ◽  
R. Ghanem ◽  
S. Masri
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Quantitative Description ◽  
Polynomial Expansion ◽  
Reduced Order Models ◽  
State Variables ◽  
Restoring Force ◽  
Nonlinear Dynamical ◽  
Entropy Maximization ◽  
Reduced Order

AbstractData-driven methodologies based on the restoring force method have been developed over the past few decades for building predictive reduced-order models (ROMs) of nonlinear dynamical systems. These methodologies involve fitting a polynomial expansion of the restoring force in the dominant state variables to observed states of the system. ROMs obtained in this way are usually prone to errors and uncertainties due to the approximate nature of the polynomial expansion and experimental limitations. We develop in this article a stochastic methodology that endows these errors and uncertainties with a probabilistic structure in order to obtain a quantitative description of the proximity between the ROM and the system that it purports to represent. Specifically, we propose an entropy maximization procedure for constructing a multi-variate probability distribution for the coefficients of power-series expansions of restoring forces. An illustration in stochastic aeroelastic stability analysis is provided to demonstrate the proposed framework.

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