scholarly journals Parametric Resonance of Hopf Bifurcation

2005 ◽  
Author(s):  
Richard Rand ◽  
Albert Barcilon ◽  
Tina Morrison
Keyword(s):  
Hopf Bifurcation ◽  
Perturbation Methods ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Damping Parameter ◽  
Long Time ◽  
Negative Damping ◽  
Parametric Forcing

We investigate the dynamics of a system consisting of a simple, harmonic oscillator with small nonlinearity, small damping and small parametric forcing in the neighborhood of 2:1 resonance. We assume that the unforced system exhibits the birth of a stable limit cycle as the damping changes sign from positive to negative (a supercritical Hopf bifurcation). Using perturbation methods and numerical integration, we investigate the changes which occur in long-time behavior as the damping parameter is varied. We show that for large positive damping, the origin is stable, whereas for large negative damping a quasiperiodic behavior occurs. These two steady states are connected by a complicated series of bifurcations which occur as the damping is varied.

scholarly journals Reduction of Dynamics with Lie Group Analysis

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
M. Iwasa
Keyword(s):  
Dynamical System ◽  
Lie Group ◽  
Perturbation Methods ◽  
Group Analysis ◽  
Lie Symmetries ◽  
Time Behavior ◽  
Lie Group Analysis ◽  
Long Time Behavior ◽  
Singular Perturbation Methods ◽  
Long Time

This paper is mainly a review concerning singular perturbation methods by means of Lie group analysis which has been presented by the author. We make use of a particular type of approximate Lie symmetries in those methods in order to construct reduced systems which describe the long-time behavior of the original dynamical system. Those methods can be used in analyzing not only ordinary differential equations but also difference equations. Although this method has been mainly used in order to derive asymptotic behavior, when we can find exact Lie symmetries, we succeed in construction of exact solutions.

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Dynamics ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Initial Value ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

2021 ◽  
pp. 1-27
Author(s):  
Ahmad Makki ◽  
Alain Miranville ◽  
Madalina Petcu
Keyword(s):  
Fractal Dimension ◽  
Boundary Conditions ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Dynamic Boundary Conditions ◽  
Finite Dimensional ◽  
Dynamic Boundary ◽  
Long Time ◽  
Finite Fractal Dimension

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

scholarly journals Stochastic Models for a Chemostat and Long-Time Behavior

2013 ◽  
Vol 45 (03) ◽  
pp. 822-836 ◽  
Author(s):  
Pierre Collet ◽  
Servet Martínez ◽  
Sylvie Méléard ◽  
Jaime San Martín
Keyword(s):  
Population Size ◽  
Nutrient Concentration ◽  
Bacterial Population ◽  
Fixed Number ◽  
Death Process ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Nutrient Distribution ◽  
Long Time ◽  
Number Of Individuals

We introduce two stochastic chemostat models consisting of a coupled population-nutrient process reflecting the interaction between the nutrient and the bacteria in the chemostat with finite volume. The nutrient concentration evolves continuously but depends on the population size, while the population size is a birth-and-death process with coefficients depending on time through the nutrient concentration. The nutrient is shared by the bacteria and creates a regulation of the bacterial population size. The latter and the fluctuations due to the random births and deaths of individuals make the population go almost surely to extinction. Therefore, we are interested in the long-time behavior of the bacterial population conditioned to nonextinction. We prove the global existence of the process and its almost-sure extinction. The existence of quasistationary distributions is obtained based on a general fixed-point argument. Moreover, we prove the absolute continuity of the nutrient distribution when conditioned to a fixed number of individuals and the smoothness of the corresponding densities.

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