Stable and Unstable Quasiperiodic Oscillations in Robot Dynamics

1993 ◽  
Author(s):  
G. Stépán ◽  
G. Haller
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Time Lag ◽  
Nonlinear Behavior ◽  
Degree Of Freedom ◽  
Robot Dynamics ◽  
Robot Arm ◽  
Delayed Systems ◽  
Nonlinear Dynamical ◽  
Quasiperiodic Oscillations

Abstract Delays in robot control may result in unexpectedly sophisticated nonlinear dynamical behavior. Experiments on force controlled robots frequently show periodic and quasiperiodic oscillations which cannot be explained without including the time lag and/or the sampling time of the system in our models. Delayed systems, even of low degree of freedom, can produce phenomena which are already well understood in the theory of nonlinear dynamical systems but hardly ever occur in simple mechanical models. To illustrate this, we analyze the delayed positioning of a single degree of freedom robot arm. The analytical results show typical nonlinear behavior in the system which may go through a codimension two Hopf bifurcation for an infinite set of parameter values, leading to the creation of two-tori in the phase space. These results give a qualitative explanation for the existence of self-excited quasiperiodic oscillations in the dynamics of force controlled robots.

Generating Fully Bounded Chaotic Attractors

2013 ◽  
pp. 148-153
Author(s):  
Zeraoulia Elhadj
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Nonlinear Dynamical ◽  
Dynamical Properties ◽  
The Given

Generating chaotic attractors from nonlinear dynamical systems is quite important because of their applicability in sciences and engineering. This paper considers a class of 2-D mappings displaying fully bounded chaotic attractors for all bifurcation parameters. It describes in detail the dynamical behavior of this map, along with some other dynamical phenomena. Also presented are some phase portraits and some dynamical properties of the given simple family of 2-D discrete mappings.

Is There a Relation Between Synchronization Stability and Bifurcation Type?

2020 ◽  
Vol 30 (08) ◽  
pp. 2050123
Author(s):  
Zahra Faghani ◽  
Zhen Wang ◽  
Fatemeh Parastesh ◽  
Sajad Jafari ◽  
Matjaž Perc
Keyword(s):  
Complex Networks ◽  
Nonlinear Dynamical Systems ◽  
General Relation ◽  
Dynamical Behavior ◽  
Stability Function ◽  
Nonlinear Dynamical ◽  
Practical Applications ◽  
Stability Functions ◽  
Stability And Bifurcation ◽  
The Stability

Synchronization in complex networks is an evergreen subject with many practical applications across the natural and social sciences. The stability of synchronization is thereby crucial for determining whether the dynamical behavior is stable or not. The master stability function is commonly used to that effect. In this paper, we study whether there is a relation between the stability of synchronization and the proximity to certain bifurcation types. We consider four different nonlinear dynamical systems, and we determine their master stability functions in dependence on key bifurcation parameters. We also calculate the corresponding bifurcation diagrams. By means of systematic comparisons, we show that, although there are some variations in the master stability functions in dependence on bifurcation proximity and type, there is in fact no general relation between synchronization stability and bifurcation type. This has important implication for the restrained generalizability of findings concerning synchronization in complex networks for one type of node dynamics to others.

Generating Fully Bounded Chaotic Attractors

2011 ◽  
Vol 2 (3) ◽  
pp. 36-42
Author(s):  
Zeraoulia Elhadj
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Nonlinear Dynamical ◽  
Dynamical Properties ◽  
The Given

Generating chaotic attractors from nonlinear dynamical systems is quite important because of their applicability in sciences and engineering. This paper considers a class of 2-D mappings displaying fully bounded chaotic attractors for all bifurcation parameters. It describes in detail the dynamical behavior of this map, along with some other dynamical phenomena. Also presented are some phase portraits and some dynamical properties of the given simple family of 2-D discrete mappings.

scholarly journals Computation of the topological entropy in chaotic biophysical bursting models for excitable cells

2006 ◽  
Vol 2006 ◽  
pp. 1-18 ◽  
Author(s):  
Jorge Duarte ◽  
Luís Silva ◽  
J. Sousa Ramos
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Excitable Cells ◽  
Physiological Models ◽  
Nonlinear Dynamical ◽  
Chaotic Bursting ◽  
Oscillatory State ◽  
One Dimensional Maps

One of the interesting complex behaviors in many cell membranes is bursting, in which a rapid oscillatory state alternates with phases of relative quiescence. Although there is an elegant interpretation of many experimental results in terms of nonlinear dynamical systems, the dynamics of bursting models is not completely described. In the present paper, we study the dynamical behavior of two specific three-variable models from the literature that replicate chaotic bursting. With results from symbolic dynamics, we characterize the topological entropy of one-dimensional maps that describe the salient dynamics on the attractors. The analysis of the variation of this important numerical invariant with the parameters of the systems allows us to quantify the complexity of the phenomenon and to distinguish different chaotic scenarios. This work provides an example of how our understanding of physiological models can be enhanced by the theory of dynamical systems.

Nonlinear Behavior Analysis of a Rotor Supported on Fluid-Film Bearings

2005 ◽  
Vol 128 (1) ◽  
pp. 35-40 ◽  
Author(s):  
Guangyan Shen ◽  
Zhonghui Xiao ◽  
Wen Zhang ◽  
Tiesheng Zheng
Keyword(s):  
Behavior Analysis ◽  
Dynamical Behavior ◽  
Nonlinear Behavior ◽  
Fluid Film ◽  
Diagnostic Tools ◽  
Nonlinear Phenomena ◽  
Frequency Spectra ◽  
Nonlinear Dynamical ◽  
Synchronous Motion ◽  
Period Motion

A fast and accurate model to calculate the fluid-film forces of a fluid-film bearing with the Reynolds boundary condition is presented in the paper by using the free boundary theory and the variational method. The model is applied to the nonlinear dynamical behavior analysis of a rigid rotor in the elliptical bearing support. Both balanced and unbalanced rotors are taken into consideration. Numerical simulations show that the balanced rotor undergoes a supercritical Hopf bifurcation as the rotor spin speed increases. The investigation of the unbalanced rotor indicates that the motion can be a synchronous motion, subharmonic motion, quasi-period motion, or chaotic motion at different rotor spin speeds. These nonlinear phenomena are investigated in detail. Poincaré maps, bifurcation diagram and frequency spectra are utilized as diagnostic tools.

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.

scholarly journals Adaptive Vector Nonsingular Terminal Sliding Mode Control for a Class of n-Order Nonlinear Dynamical Systems with Uncertainty

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Nannan Shi ◽  
Zhikuan Kang ◽  
Zhuo Zhao ◽  
Qiang Meng
Keyword(s):  
Dynamical Systems ◽  
Sliding Mode Control ◽  
Nonlinear Dynamical Systems ◽  
Sliding Mode ◽  
Degree Of Freedom ◽  
Terminal Sliding Mode Control ◽  
Terminal Sliding Mode ◽  
Nonlinear Dynamical ◽  
Nonsingular Terminal Sliding Mode ◽  
The Common

This paper proposed an adaptive vector nonsingular terminal sliding mode control (NTSMC) algorithm for the finite-time tracking control of a class of n-order nonlinear dynamical systems with uncertainty. The adaptive vector NTSMC incorporates a vector design idea and novel adaptive updating laws based on the commonly used NTSMC, which consider the common existence of the degree-of-freedom (DOF) directional differences and eliminate the chattering problem. The closed-loop stability of the n-order nonlinear dynamical systems under the adaptive vector NTSMC is demonstrated using Lyapunov direct method. Simulations performed on a two-degree-of-freedom (DOF) manipulator are provided to illustrate the effectiveness and advantages of the proposed adaptive vector NTSMC by comparing with the common NTSMC.

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