Phase Space Transport in a Class of Multi-Degree of-Freedom Systems

1997 ◽  
Author(s):  
Shyh-Leh Chen ◽  
Steven W. Shaw
Keyword(s):  
Phase Space ◽  
Dynamic Systems ◽  
Nonlinear Dynamical Systems ◽  
Quantitative Information ◽  
Fluid Mixing ◽  
Dynamics Model ◽  
Analytical Estimate ◽  
Nonlinear Dynamical ◽  
Qualitative And Quantitative ◽  
Phase Space Transport

Abstract In this paper we describe some recent advances in the basic theory and applications of phase space transport in nonlinear dynamic systems. These methods offer both qualitative and quantitative information about the behavior of solutions near homoclinic and heteroclinic motions in nonlinear dynamical systems. Applications of these ideas are found in fluid mixing and the escape of solutions from potential energy wells under the action of disturbances, for example, in models of ship capsize. In this work the theory is extended to a certain class of higher-order systems in which several time scales are involved. In addition, a new analytical estimate is derived and used for the rate of transport in the case of two-dimensional Poincare maps. Extensive simulation results from a specific ship dynamics model are used to demonstrate and verify these results.

Linear and Nonlinear Dynamic Behaviour

2018 ◽  
Author(s):  
Ray Huffaker ◽  
Marco Bittelli ◽  
Rodolfo Rosa
Keyword(s):  
Dynamic Behavior ◽  
Dynamic Systems ◽  
Dynamic Behaviour ◽  
Nonlinear Dynamical Systems ◽  
Logistic Map ◽  
Nonlinear Dynamical ◽  
Deterministic Dynamic ◽  
Random Behavior ◽  
Single Solution ◽  
Equivalent Systems

In this chapter, we describe how highly erratic dynamic behavior can arise from a nonlinear logistic map, and how this apparently random behavior is governed by a surprising order. With this lesson in mind, we should not be overly surprised that highly erratic and random appearing observed data might also be generated by parsimonious deterministic dynamic systems. At a minimum, we contend that researchers should apply NLTS to test for this possibility. We also introduced tools to analyze dynamic behavior that form the foundation for NLTS. In particular, we have stressed the quite unexpected capability to achieve some form of predictability even with only one trajectory at hand. In subsequent chapters, we treat known nonlinear dynamical systems as unknown, and investigate how NLTS methods rely on a single solution (or multiple solutions) generated by them to reconstruct equivalent systems. This is a conventional approach in the literature for seeing how NLTS methods work since we know what needs to be reconstructed.

ANIMATION OF CHAOTIC MIXING BY A BACKWARD POINCARE CELL-MAP METHOD

2001 ◽  
Vol 11 (07) ◽  
pp. 1953-1960 ◽  
Author(s):  
LINXIANG WANG ◽  
YURUN FAN ◽  
YING CHEN
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Fluid Mixing ◽  
Chaotic Mixing ◽  
Mixing Process ◽  
Cell Mapping ◽  
Color Code ◽  
Nonlinear Dynamical ◽  
Curved Pipe ◽  
Phase Deformation ◽  
A Cell

A Backward Poincare cell-mapping (BPCM) method has been developed for animating chaotic fluid mixing. The chaotic mixing field considered is induced by periodically rotating the secondary flow of incompressible fluids in a curved pipe. The pipe's cross-section is transformed into a cell space where each cell is initially assigned with a color code and mapped by integrating the velocity field forward in time. The mixing process is thus animated efficiently with each cell being painted with its color on a computer screen. We propose the backward Poincare cell-mapping instead of direct Poincare cell-mapping as a useful tool for probing the chaotic fluid mixing and for animating the phase deformation of nonlinear dynamical systems.

scholarly journals A Review of Phase Space Topology Methods for Vibration-Based Fault Diagnostics in Nonlinear Systems

2019 ◽  
Vol 8 (3) ◽  
pp. 393-401 ◽  
Author(s):  
T. Haj Mohamad ◽  
Foad Nazari ◽  
C. Nataraj
Keyword(s):  
Phase Space ◽  
Nonlinear Systems ◽  
Dynamic Systems ◽  
Dynamical Behavior ◽  
Fault Diagnostics ◽  
Nonlinear Dynamical ◽  
Space Topology ◽  
Nonlinear Features ◽  
Phase Space Topology

Abstract Background In general, diagnostics can be defined as the procedure of mapping the information obtained in the measurement space to the presence and magnitude of faults in the fault space. These measurements, and especially their nonlinear features, have the potential to be exploited to detect changes in dynamics due to the faults. Purpose We have been developing some interesting techniques for fault diagnostics with gratifying results. Methods These techniques are fundamentally based on extracting appropriate features of nonlinear dynamical behavior of dynamic systems. In particular, this paper provides an overview of a technique we have developed called Phase Space Topology (PST), which has so far displayed remarkable effectiveness in unearthing faults in machinery. Applications to bearing, gear and crack diagnostics are briefly discussed.

Periodicity and Nonlinearity of Nonlinear Dynamical Systems Under Periodic Excitation

2009 ◽  
Author(s):  
Lu Han ◽  
Liming Dai ◽  
Huayong Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Dynamic Systems ◽  
Nonlinear Dynamic ◽  
Nonlinear Dynamical Systems ◽  
Research Work ◽  
Nonlinear Dynamic Systems ◽  
Periodic Excitation ◽  
Nonlinear Dynamical

Periodicity and nonlinearity of nonlinear dynamic systems subjected to regular external excitations are studied in this research work. Diagnoses of regular and chaotic responses of nonlinear dynamic systems are performed with the implementation of a newly developed Periodicity Ratio in combining with the application of Lyapunov Exponent. The properties of the nonlinear dynamics systems are classified into four categories: periodic, irregular-nonchaotic, quasiperiodic and chaotic, each corresponding to their Periodicity Ratios. Detailed descriptions about diagnosing the responses of the four categories are presented with utilization of the Periodicity Ratio.

scholarly journals Existence of Perpetual Points in Nonlinear Dynamical Systems and Its Applications

2015 ◽  
Vol 25 (02) ◽  
pp. 1530005 ◽  
Author(s):  
Awadhesh Prasad
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Critical Points ◽  
Nonlinear Dynamical Systems ◽  
Transient Dynamics ◽  
Coexisting Attractors ◽  
Nonlinear Dynamical ◽  
New Class ◽  
Perpetual Points ◽  
Bifurcation Behavior

A new class of critical points, termed as perpetual points, where acceleration becomes zero but the velocity remains nonzero, are observed in dynamical systems. The velocity at these points is either maximum or minimum or of inflection behavior. These points also show the bifurcation behavior as the parameters of the system vary. These perpetual points are useful for locating the hidden oscillating attractors as well as coexisting attractors. Results show that these points are important for a better understanding of transient dynamics in the phase space. The existence of these points confirms whether a system is dissipative or not. Various examples are presented, and the results are discussed analytically as well as numerically.

scholarly journals Semigroup applications everywhere

2020 ◽  
Vol 378 (2185) ◽  
pp. 20190610
Author(s):  
Rainer Nagel ◽  
Abdelaziz Rhandi
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Volterra Equations ◽  
Special Issue ◽  
Theme Issue ◽  
Nonlinear Dynamical ◽  
Qualitative And Quantitative ◽  
Abstract Evolution Equation ◽  
Quantitative Properties ◽  
Functional Differential

Most dynamical systems arise from partial differential equations (PDEs) that can be represented as an abstract evolution equation on a suitable state space complemented by an initial or final condition. Thus, the system can be written as a Cauchy problem on an abstract function space with appropriate topological structures. To study the qualitative and quantitative properties of the solutions, the theory of one-parameter operator semigroups is a most powerful tool. This approach has been used by many authors and applied to quite different fields, e.g. ordinary and PDEs, nonlinear dynamical systems, control theory, functional differential and Volterra equations, mathematical physics, mathematical biology, stochastic processes. The present special issue of Philosophical Transactions includes papers on semigroups and their applications. This article is part of the theme issue ‘Semigroup applications everywhere’.

CONTROLLING CHAOTIC SYSTEMS VIA PHASE SPACE RECONSTRUCTION TECHNIQUE

2003 ◽  
Vol 13 (02) ◽  
pp. 467-471 ◽  
Author(s):  
Y. J. CAO ◽  
P. X. ZHANG ◽  
S. J. CHENG
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Chaotic Systems ◽  
Nonlinear Dynamical Systems ◽  
Linear Part ◽  
Phase Space Reconstruction ◽  
Control Law ◽  
Reconstruction Technique ◽  
Nonlinear Dynamical ◽  
Novel Approach

A novel approach to control chaotic systems has been developed. The approach employs the technique of phase space reconstruction in nonlinear dynamical systems theory to construct a linear part in the reconstructed system and design a feedback control law. The effectiveness of the proposed approach has been demonstrated through its applications to two well-known chaotic systems: Lorenz chaos and Rössler chaos.

CLOSURES OF FRACTAL SETS IN NONLINEAR DYNAMICAL SYSTEMS WITH SWITCHED INPUTS

2001 ◽  
Vol 11 (08) ◽  
pp. 2205-2215 ◽  
Author(s):  
RYOICHI WADA ◽  
KAZUTOSHI GOHARA
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Dynamical Systems ◽  
Limit Cycle ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Nonlinear Dynamical ◽  
Fractal Sets ◽  
Fractal Set

This paper studies closures of fractal sets observed in nonlinear dynamical systems excited stochastically by switched inputs. The Duffing oscillator and the forced dumped pendulum are analyzed as examples. The dynamics of the system is characterized by a fractal set in the phase space. We can numerically construct a closure that encloses the fractal set. Furthermore, it is shown that the closure is a limit cycle attractor of a dynamical system defined by the switching manifold.

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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