A Wavelet Based Procedure to Study Vibrations and Stability

1999 ◽  
Author(s):  
S. Pernot ◽  
C. H. Lamarque
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Periodic Solutions ◽  
Time Varying ◽  
Differential Systems ◽  
Varying Coefficients ◽  
Linear Differential Systems ◽  
Time Varying Coefficients ◽  
Linear Differential

Abstract A Wavelet-Galerkin procedure is introduced in order to obtain periodic solutions of multidegrees-of-freedom dynamical systems with periodic time-varying coefficients. The procedure is then used to study the vibrations of parametrically excited mechanical systems. As problems of stability analysis of nonlinear systems are often reduced after linearization to problems involving linear differential systems with time-varying coefficients, we demonstrate the method provides efficient practical computations of Floquet exponents and consequently allows to give estimators for stability/instability levels. A few academic examples illustrate the relevance of the method.

Uniform global asymptotic stability for half-linear differential systems with time-varying coefficients

2011 ◽  
Vol 141 (5) ◽  
pp. 1083-1101 ◽  
Author(s):  
Masakazu Onitsuka ◽  
Jitsuro Sugie
Keyword(s):  
Linear System ◽  
Zero Solution ◽  
Integrable Case ◽  
Time Varying ◽  
Globally Asymptotically Stable ◽  
Varying Coefficients ◽  
Linear Differential Systems ◽  
Time Varying Coefficients ◽  
Linear Differential ◽  
Image Position

The present paper deals with the following system:where p and p* are positive numbers satisfying 1/p + 1/p* = 1, and ϕq(z) = |z|q−2z for q = p or q = p*. This system is referred to as a half-linear system. We herein establish conditions on time-varying coefficients e(t), f(t), g(t) and h(t) for the zero solution to be uniformly globally asymptotically stable. If (e(t), f(t)) ≡ (h(t), g(t)), then the half-linear system is integrable. We consider two cases: the integrable case (e(t), f(t)) ≡ (h(t), g(t)) and the non-integrable case (e(t), f(t)) ≢ (h(t), g(t)). Finally, some simple examples are presented to illustrate our results.

scholarly journals Limit Cycles Bifurcating from the Periodic Orbits of a Discontinuous Piecewise Linear Differentiable Center with Two Zones

2015 ◽  
Vol 25 (11) ◽  
pp. 1550144 ◽  
Author(s):  
Jaume Llibre ◽  
Douglas D. Novaes ◽  
Marco A. Teixeira
Keyword(s):  
Equilibrium Point ◽  
Periodic Solutions ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Piecewise Linear ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Straight Line ◽  
Linear Differential

We study a class of discontinuous piecewise linear differential systems with two zones separated by the straight line x = 0. In x > 0, we have a linear saddle with its equilibrium point living in x > 0, and in x < 0 we have a linear differential center. Let p be the equilibrium point of this linear center, when p lives in x < 0, we say that it is real, and when p lives in x > 0 we say that it is virtual. We assume that this discontinuous piecewise linear differential system formed by the center and the saddle has a center q surrounded by periodic orbits ending in a homoclinic orbit of the saddle, independent if p is real, virtual or p is in x = 0. Note that q = p if p is real or p is in x = 0. We perturb these three classes of systems, according to the position of p, inside the class of all discontinuous piecewise linear differential systems with two zones separated by x = 0. Let N be the maximum number of limit cycles which can bifurcate from the periodic solutions of the center q with these perturbations. Our main results show that N = 2 when p is on x = 0, and N ≥ 2 when p is a real or virtual center. Furthermore, when p is a real center we found an example satisfying N ≥ 3.

scholarly journals Asymptotic Stabilization by State Feedback for a Class of Stochastic Nonlinear Systems with Time-Varying Coefficients

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Hui Wang ◽  
Wuquan Li ◽  
Xiuhong Wang
Keyword(s):  
Nonlinear Systems ◽  
State Feedback ◽  
Original System ◽  
Feedback Stabilization ◽  
Stochastic Nonlinear Systems ◽  
Time Varying ◽  
Globally Asymptotically Stable ◽  
Varying Coefficients ◽  
Time Varying Coefficients ◽  
Control Coefficients

This paper investigates the problem of state-feedback stabilization for a class of upper-triangular stochastic nonlinear systems with time-varying control coefficients. By introducing effective coordinates, the original system is transformed into an equivalent one with tunable gain. After that, by using the low gain homogeneous domination technique and choosing the low gain parameter skillfully, the closed-loop system can be proved to be globally asymptotically stable in probability. The efficiency of the state-feedback controller is demonstrated by a simulation example.

scholarly journals Positive Periodic Solutions for an Generalized SIS Epidemic Model with Time-Varying Coefficients and Delays

2018 ◽  
Vol 2018 ◽  
pp. 1-5
Author(s):  
Zhiwen Long
Keyword(s):  
Periodic Solutions ◽  
Epidemic Model ◽  
Time Varying ◽  
Positive Periodic Solutions ◽  
Varying Coefficients ◽  
Sis Epidemic Model ◽  
Lyapunov Function Method ◽  
Time Varying Coefficients ◽  
Theoretical Results ◽  
Sis Epidemic

This paper mainly explores a generalized SIS epidemic model with time-varying coefficients and delays. By employing Lyapunov function method and differential inequality approach, a sufficient criterion to guarantee the existence and exponential stability of positive periodic solutions for the addressed model is obtained, which complements the earlier publications. Particularly, an example and its numerical simulations are given to demonstrate our theoretical results.

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