Bifurcation and Stability in Dissipative Media (Plasticity, Friction, Fracture)

1994 ◽  
Vol 47 (1) ◽  
pp. 1-31 ◽  
Author(s):  
Quoc Son Nguyen
Keyword(s):  
Fracture Mechanics ◽  
Bifurcation Analysis ◽  
Recent Literature ◽  
Mechanical Systems ◽  
Stability And Bifurcation Analysis ◽  
Dissipative Media ◽  
The Stability ◽  
Bifurcation And Stability ◽  
Static Behaviour ◽  
Unified Description

This paper addresses stability and bifurcation analysis for common systems of solids in the framework of plasticity, of friction and of fracture mechanics. Although physically different, usual time-independent laws adopted in these domains lead to a certain mathematical similarity concerning the quasi-static behaviour of materials and structures. These mechanical systems can be described practically in the same mathematical manner concerning their quasi-static evolution and in particular concerning the stability of their response. Our objective is to present within this framework a review of principal results of the recent literature on these subjects in relation with some energy-related considerations and with an unified description based upon energy and dissipation analysis.

SINGULARITY, SWITCHABILITY AND BIFURCATIONS IN A 2-DOF, PERIODICALLY FORCED, FRICTIONAL OSCILLATOR

2013 ◽  
Vol 23 (03) ◽  
pp. 1330009 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
MOZHDEH S. FARAJI MOSADMAN
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Analysis ◽  
Degrees Of Freedom ◽  
Eigenvalue Analysis ◽  
Analytical Dynamics ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the analytical dynamics for singularity, switchability, and bifurcations of a 2-DOF friction-induced oscillator is investigated. The analytical conditions of the domain flow switchability at the boundaries and edges are developed from the theory of discontinuous dynamical systems, and the switchability conditions of boundary flows from domain and edge flows are presented. From the singularity and switchability of flow to the boundary, grazing, sliding and edge bifurcations are obtained. For a better understanding of the motion complexity of such a frictional oscillator, switching sets and mappings are introduced, and mapping structures for periodic motions are adopted. Using an eigenvalue analysis, the stability and bifurcation analysis of periodic motions in the friction-induced system is carried out. Analytical predictions and parameter maps of periodic motions are performed. Illustrations of periodic motions and the analytical conditions are completed. The analytical conditions and methodology can be applied to the multi-degrees-of-freedom frictional oscillators in the same fashion.

Stability and Bifurcation Analysis of a Prey–Predator Model

2021 ◽  
Vol 31 (04) ◽  
pp. 2150059
Author(s):  
T. N. Mishra ◽  
B. Tiwari
Keyword(s):  
Dynamical System ◽  
Linear Stability ◽  
Bifurcation Analysis ◽  
Finsler Space ◽  
Second Order ◽  
Critical Values ◽  
Numerical Examples ◽  
Stability And Bifurcation Analysis ◽  
Predator Model ◽  
The Stability

The purpose of the present paper is to study the stability of a prey–predator model using KCC theory. The KCC theory is based on the assumption that the second-order dynamical system and geodesics equation, in associated Finsler space, are topologically equivalent. The stability (Jacobi stability) based on KCC theory and linear stability of the model are discussed in detail. Further, the effect of parameters on stability and the presence of chaos in the model are investigated. The critical values of bifurcation parameters are found and their effects on the model are investigated. The numerical examples of particular interest are compared to the results of Jacobi stability and linear stability and it is found that Jacobi stability on the basis of KCC theory is global than the linear stability.

Periodic Motions in the Periodically Forced Oscillator With Multiple Discontinuities

2006 ◽  
Author(s):  
Albert C. J. Luo ◽  
Mehul T. Patel
Keyword(s):  
Dynamical System ◽  
Bifurcation Analysis ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Mapping Structures ◽  
Forced Oscillator ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the stability and bifurcation of periodic motions in periodically forced oscillator with multiple discontinuities is investigated. The generic mappings are introduced for the analytical prediction of periodic motions. Owing to the multiple discontinuous boundaries, the mapping structures for periodic motions are very complicated, which causes more difficulty to obtain periodic motions in such a dynamical system. The analytical prediction of complex periodic motions is carried out and verified numerically, and the corresponding stability and bifurcation analysis are performed. Due to page limitation, grazing and stick motions and chaos in this system will be investigated further.

Analytical Periodic Motions of a Parametric Oscillator With Quadratic Nonlinearity

2014 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Fourier Series ◽  
Series Expansion ◽  
Bifurcation Analysis ◽  
Analytical Solutions ◽  
Parametric Oscillator ◽  
Quadratic Nonlinearity ◽  
Fourier Series Expansion ◽  
Periodic Motions ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the analytical solutions of periodic motions in a parametric oscillator are presented by the finite Fourier series expansion, and the stability and bifurcation analysis of periodic motions are performed. Numerical illustrations of periodic motions are presented through phase trajectories and analytical spectrum.

Analytical Solutions for Periodic Motions in a Hardening Mathieu-Duffing Oscillator

2013 ◽  
Author(s):  
Albert C. J. Luo ◽  
Dennis M. O’Connor
Keyword(s):  
Fourier Series ◽  
Bifurcation Analysis ◽  
Analytical Solutions ◽  
Duffing Oscillator ◽  
Series Solutions ◽  
Periodic Motions ◽  
Period 2 ◽  
Approximate Analytical Solutions ◽  
Stability And Bifurcation Analysis ◽  
The Stability

Analytical solutions for period-m motions in a hardening Mathieu-Duffing oscillator are obtained using the finite Fourier series solutions, and the stability and bifurcation analysis of such periodic motions are completed. To verify the approximate analytical solutions of periodic motions, numerical simulations of the hardening Mathieu-Duffing oscillator are presented. Period-1 asymmetric and period-2 symmetric motions are illustrated.

Analytical Solutions of Period-1 Motions in a Periodically Forced van der Pol Oscillator

2012 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arash Baghaei Lakeh
Keyword(s):  
Bifurcation Analysis ◽  
Analytical Solutions ◽  
Eigenvalue Analysis ◽  
Van Der Pol Oscillator ◽  
Comprehensive Understanding ◽  
Van Der Pol ◽  
Stability And Bifurcation ◽  
Approximate Analytical Solutions ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the approximate analytical solutions of period-1 motion in the periodically forced van der Pol oscillator are obtained by the generalized harmonic balanced method. The stability and bifurcation analysis of the period-1 solutions is completed through the eigenvalue analysis, and numerical illustrations of periodic-1 solutions are given to verify the approximate motion. This investigation provides more accurate solutions of period-1 motions in the van der pol oscillator for a better and comprehensive understanding of motions in such an oscillator.

scholarly journals Bifurcation Analysis of a Coupled Kuramoto-Sivashinsky- and Ginzburg-Landau-Type Model

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Lei Shi
Keyword(s):  
Asymptotic Behavior ◽  
Bifurcation Analysis ◽  
Trivial Solution ◽  
Type Model ◽  
Bifurcation Parameter ◽  
Local Behavior ◽  
Nontrivial Solutions ◽  
Ginzburg Landau ◽  
The Stability ◽  
Bifurcation And Stability

We study the bifurcation and stability of trivial stationary solution(0,0)of coupled Kuramoto-Sivashinsky- and Ginzburg-Landau-type equations (KS-GL) on a bounded domain(0,L)with Neumann's boundary conditions. The asymptotic behavior of the trivial solution of the equations is considered. With the lengthLof the domain regarded as bifurcation parameter, branches of nontrivial solutions are shown by using the perturbation method. Moreover, local behavior of these branches is studied, and the stability of the bifurcated solutions is analyzed as well.

Stability and Bifurcation Analysis in a Delayed Predator-Prey Model with Stage-Structure and Harvesting

2010 ◽  
Vol 143-144 ◽  
pp. 1358-1363
Author(s):  
Zhi Chao Jiang ◽  
Ming Wei Nie
Keyword(s):  
Bifurcation Analysis ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Hopf Bifurcations ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Stability And Bifurcation Analysis ◽  
Characteristic Values ◽  
The Stability

In this paper, we investigate a delayed stage-structured predator-prey model with continuous harvesting on prey. Positivity and boundness of solutions and sufficient conditions of the stability of equilibria are obtained. Using and as bifurcation parameters, the existence of Hopf bifurcations at equilibria is established by analyzing the distribution of the characteristic values.

scholarly journals Stability and Bifurcation Analysis on a Predator–Prey System with the Weak Allee Effect

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 432 ◽  
Author(s):  
Jianming Zhang ◽  
Lijun Zhang ◽  
Yuzhen Bai
Keyword(s):  
Allee Effect ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Hopf Bifurcations ◽  
Stability Switches ◽  
Predator Prey ◽  
Prey System ◽  
Stability And Bifurcation Analysis ◽  
The Stability ◽  
Bifurcation And Stability

In this paper, the dynamics of a predator-prey system with the weak Allee effect is considered. The sufficient conditions for the existence of Hopf bifurcation and stability switches induced by delay are investigated. By using the theory of normal form and center manifold, an explicit expression, which can be applied to determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions, are obtained. Numerical simulations are performed to illustrate the theoretical analysis results.

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