Bifurcations Associated with a Double Zero and a Pair of Pure Imaginary Eigenvalues

1988 ◽  
Vol 48 (2) ◽  
pp. 229-261 ◽  
Author(s):  
Pei Yu ◽  
K. Huseyin
Keyword(s):  
Double Zero ◽  
Pure Imaginary Eigenvalues

BIFURCATION ANALYSIS OF A 2-DOF ROBOT MANIPULATOR DRIVEN BY CONSTANT TORQUES

1999 ◽  
Vol 09 (04) ◽  
pp. 617-627 ◽  
Author(s):  
FERNANDO VERDUZCO ◽  
JOAQUÍN ALVAREZ
Keyword(s):  
Robot Manipulator ◽  
Equilibrium Points ◽  
Double Zero ◽  
Zero Eigenvalue ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Codimension One ◽  
Form Theory ◽  
Pure Imaginary Eigenvalues ◽  
Two Degree Of Freedom

The bifurcations of a two-degree-of-freedom (2-DOF) robot manipulator with linear viscous damping and constant torques at joints are analyzed at three equilibrium points. We show that, provided some conditions on the parameters are satisfied, two of these equilibria have a Jacobian matrix with a double zero eigenvalue and a pair of pure imaginary eigenvalues, while the other one has a quadruple zero eigenvalue. We use the center manifold theorem and the normal form theory to show the presence of different kinds of local bifurcations, ranging from codimension one (Hopf and fold), three (cusp and degenerate zero-Hopf), and higher (double zero, double zero-Hopf, triple zero, quadruple zero and Hopf–Hopf).

scholarly journals Normal Form for High-Dimensional Nonlinear System and Its Application to a Viscoelastic Moving Belt

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
S. P. Chen ◽  
Y. H. Qian
Keyword(s):  
Normal Form ◽  
Nonlinear Dynamical Systems ◽  
High Dimensional ◽  
Computation Method ◽  
Double Zero ◽  
Nonlinear Dynamical ◽  
Novel Method ◽  
Explicit Formulae ◽  
The Stability ◽  
Pure Imaginary Eigenvalues

This paper is concerned with the computation of the normal form and its application to a viscoelastic moving belt. First, a new computation method is proposed for significantly refining the normal forms for high-dimensional nonlinear systems. The improved method is described in detail by analyzing the four-dimensional nonlinear dynamical systems whose Jacobian matrices evaluated at an equilibrium point contain three different cases, that are, (i) two pairs of pure imaginary eigenvalues, (ii) one nonsemisimple double zero and a pair of pure imaginary eigenvalues, and (iii) two nonsemisimple double zero eigenvalues. Then, three explicit formulae are derived, herein, which can be used to compute the coefficients of the normal form and the associated nonlinear transformation. Finally, employing the present method, we study the nonlinear oscillation of the viscoelastic moving belt under parametric excitations. The stability and bifurcation of the nonlinear vibration system are studied. Through the illustrative example, the feasibility and merit of this novel method are also demonstrated and discussed.

scholarly journals Global Dynamics of a Compressor Blade with Resonances

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Xiaoxia Bian ◽  
Fangqi Chen ◽  
Fengxian An
Keyword(s):  
Internal Resonance ◽  
Chaotic Dynamics ◽  
Primary Resonance ◽  
Compressor Blade ◽  
Invariant Sets ◽  
Global Dynamics ◽  
Resonance Case ◽  
Double Zero ◽  
Chaotic Motions ◽  
Pure Imaginary Eigenvalues

The global bifurcations and chaotic dynamics of a thin-walled compressor blade for the resonant case of 2 : 1 internal resonance and primary resonance are investigated. With the aid of the normal theory, the desired form associated with a double zero and a pair of pure imaginary eigenvalues for the global perturbation method is obtained. Based on the simpler form, the method developed by Kovacic and Wiggins is used to find the existence of a Shilnikov-type homoclinic orbit. The results obtained here indicate that the orbit homoclinic to certain invariant sets for the resonance case which may lead to chaos in the sense of Smale horseshoes for the system. The chaotic motions of the rotating compressor blade are also found by using numerical simulation.

L'option double zéro et le sort de l'Europe

1988 ◽  
Vol 53 (1) ◽  
pp. 73-78
Author(s):  
Youri Davydov
Keyword(s):  
Double Zero

Double zéro

Commentaire ◽  
1989 ◽  
Vol Numéro 45 (1) ◽  
pp. 221-223
Author(s):  
Roger Stéphane
Keyword(s):  
Double Zero

Invariant Neutral Subspaces for Symmetric and Skew Real Matrix Pairs

1994 ◽  
Vol 46 (3) ◽  
pp. 602-618 ◽  
Author(s):  
P. Lancaster ◽  
L. Rodman
Keyword(s):  
Lower Bounds ◽  
Low Rank ◽  
Real Matrix ◽  
Pure Imaginary Eigenvalues

AbstractReal matrix pairs (A,H) satisfying det H ≠ 0, HT = εH, and HA - ηATH, where ε, η take the values +1 or —1, are considered. It is shown that maximal A-invariant H-neutral subspaces have the same dimension (depending on ε and η), called the order of neutrality of the pair (A, H). The order of neutrality of definitizable pairs is investigated. In particular, this concept is used to obtain lower bounds for the number of pure imaginary eigenvalues of low rank perturbations of definitizable pairs when (ε,η) = (1, - 1 ) and when (ε,η) = (—1,—1).

scholarly journals Stability and Bifurcation for a Simply Supported Functionally Graded Material Plate with One-to-One Internal Resonance

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Dongmei Zhang ◽  
Fangqi Chen
Keyword(s):  
Numerical Methods ◽  
Numerical Solutions ◽  
Equilibrium Points ◽  
Time Integration ◽  
Functionally Graded ◽  
Integration Scheme ◽  
Double Zero ◽  
Internal Resonances ◽  
Simply Supported ◽  
Stability And Bifurcation

Stability and bifurcation behaviors for a model of simply supported functionally graded materials rectangular plate subjected to the transversal and in-plane excitations are studied by means of combination of analytical and numerical methods. The resonant case considered here is 1 : 1 internal resonances and primary parametric resonance. Two types of degenerated equilibrium points are studied in detail, which are characterized by a double zero and two negative eigenvalues, and a double zero and a pair of pure imaginary eigenvalues. For each case, the stability regions of the initial equilibrium solution and the critical bifurcation curves are obtained in terms of the system parameters which may lead to Hopf bifurcation and 2D torus. With both analytical and numerical methods, bifurcation behaviors on damping parameters and detuning parameters are studied, respectively. A time integration scheme is used to find the numerical solutions for these bifurcation cases, and numerical results agree with the analytic predictions.

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