Search for Periodic Solutions of Highly Nonlinear Dynamical Systems

2018 ◽  
Vol 58 (3) ◽  
pp. 384-393
Author(s):  
L. F. Petrov
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Highly Nonlinear

A Probabilistic Theory of Nonlinear Dynamical Systems Based on the Cell State Space Concept

1982 ◽  
Vol 49 (4) ◽  
pp. 895-902 ◽  
Author(s):  
C. S. Hsu
Keyword(s):  
Dynamical Systems ◽  
State Space ◽  
Nonlinear Dynamical Systems ◽  
Global Analysis ◽  
Cell Structure ◽  
The State ◽  
Probabilistic Theory ◽  
Nonlinear Dynamical ◽  
Linear Ordinary Differential Equations ◽  
Highly Nonlinear

Developed in the paper is a probabilistic theory for nonlinear dynamical systems. The theory is based on discretizing the state space into a cell structure and using the cell probability functions to describe the state of a system. Although the dynamical system may be highly nonlinear the probabilistic formulation always leads to a set of linear ordinary differential equations. The evolution of the probability distribution among the cells can then be studied by applying the theory of Markov processes to this set of equations. It is believed that this development possibly offers a new approach to the global analysis of nonlinear systems.

Prediction for the Rub-Impact Phenomena in Rotor Systems

2001 ◽  
Author(s):  
Zhiqiang Wu ◽  
Yushu Chen
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Rotor Systems ◽  
Different Types ◽  
Impact Phenomena ◽  
Direct Guidance

Abstract By the method (Wu, 2001) developed by authors for singularity analyzing of the bifurcation of the periodic solutions in nonlinear dynamical systems with clearance, the bifurcation patterns of non-impact-rub response and a method for predicting rub-impact are given. It is shown that there are much more types of bifurcation patterns when the clearance constraint is take into account. Given their physical meanings of the parameters in practical rotor systems, the resonant periodic solutions of rotor systems consist of 11 different types of bifurcation patterns among of which the following four types are more likely to appear, (1) patterns without impact and jump, (2) jump patterns without impact, (3) impact pattern without jump and (4) patterns with impact and jump. Based on these results, parameter conditions for rub-impact phenomena are derived. These conditions can give more direct guidance to the design of rotor systems. The method proposed here can be used to predict rub-impact phenomena in more complicated rotor systems.

The Necessary Condition for the Existence of Periodic Solutions of Certain Three Dimensional Nonlinear Dynamical Systems

2012 ◽  
Vol 252 ◽  
pp. 40-43
Author(s):  
Ting Ting Quan ◽  
Jing Li ◽  
Min Sun
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Nonlinear Dynamical Systems ◽  
Three Dimensional ◽  
Basic Research ◽  
Necessary Condition ◽  
Nonlinear Dynamical ◽  
Closed Orbits ◽  
Existence Of Periodic Solutions ◽  
Important Significance

In this paper, we investigate a class of three dimensional nonlinear dynamical systems whose unperturbed systems have a family of periodic orbits. Firstly, we establish the moving Frenet Frame on these closed orbits. Secondly, the successor functions are defined by the orbits which go through the normal plane. Finally, by judging the existence of solutions of the equations obtained from the Successor functions, we obtain the necessary condition for the existence of periodic solutions of these three dimensional nonlinear dynamical systems. The result has important significance for the basic research of applied mechanics.

TRUMPET AND TRUMPET PLAYER: A HIGHLY NONLINEAR INTERACTION STUDIED IN THE FRAMEWORK OF NONLINEAR DYNAMICS

2001 ◽  
Vol 11 (07) ◽  
pp. 1801-1814
Author(s):  
CHRISTOPHE VERGEZ ◽  
XAVIER RODET
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamical Systems ◽  
Real Time ◽  
Physical Model ◽  
Nonlinear Interaction ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
General Behavior ◽  
Highly Nonlinear

For musical purposes, a real-time physical model of trumpet-like instruments was built. The model exhibits different behaviors which are studied in the framework of nonlinear dynamical systems. Periodic, quasi-periodic, chaotic and intermittency regimes are identified and studied in order to better understand the general behavior of the model, which is essential for musical use.

NONEXISTENCE OF PERIODIC SOLUTIONS IN A CLASS OF DYNAMICAL SYSTEMS WITH CYLINDRICAL PHASE SPACE

2005 ◽  
Vol 15 (04) ◽  
pp. 1423-1431 ◽  
Author(s):  
YING YANG ◽  
ZHISHENG DUAN ◽  
LIN HUANG
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Linear Matrix ◽  
Specific Kind ◽  
Nonlinear Dynamical ◽  
Cylindrical Phase ◽  
Cylindrical Phase Space

This paper investigates the nonexistence of a specific kind of periodic solutions in a class of nonlinear dynamical systems with cylindrical phase space. Those types of systems can be viewed as an interconnection of several simpler subsystems with the interconnecting structure specified by a permutation matrix. Frequency-domain conditions as well as linear matrix inequalities conditions for nonexistence of limit cycles of the second kind are established. The main results also define the frequency range on which cycles of the second kind of the system cannot exist. Based on this LMI approach, an estimate of the frequency of cycles of the second kind can be explicitly computed by solving a generalized eigenvalue minimization problem. Numerical results demonstrate the applicability and validity of the proposed method and show the effect of nonlinear interconnections on dynamical behavior of entire interconnected systems.

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