scholarly journals Test Models for Statistical Inference: Two-Dimensional Reaction Systems Displaying Limit Cycle Bifurcations and Bistability

2017 ◽  
pp. 3-27 ◽  
Author(s):  
Tomislav Plesa ◽  
Tomáš Vejchodský ◽  
Radek Erban
Keyword(s):  
Statistical Inference ◽  
Limit Cycle ◽  
Two Dimensional ◽  
Reaction Systems ◽  
Test Models

scholarly journals Asymptotic Limit-cycle Analysis of Oscillating Chemical Reactions

2021 ◽  
Author(s):  
Alain Brizard ◽  
Samuel Berry
Keyword(s):  
Limit Cycle ◽  
Chemical Reactions ◽  
Relaxation Oscillation ◽  
Three Dimensional ◽  
Van Der Pol Oscillator ◽  
Asymptotic Limit ◽  
Two Dimensional ◽  
Van Der Pol ◽  
Parameter Values ◽  
Analytical Expressions

Abstract The asymptotic limit-cycle analysis of mathematical models for oscillating chemical reactions is presented. In this work, after a brief presentation of mathematical preliminaries applied to the biased Van der Pol oscillator, we consider a two-dimensional model of the Chlorine dioxide Iodine Malonic-Acid (CIMA) reactions and the three-dimensional and two-dimensional Oregonator models of the Belousov-Zhabotinsky (BZ) reactions. Explicit analytical expressions are given for the relaxation-oscillation periods of these chemical reactions that are accurate within 5% of their numerical values. In the two-dimensional CIMA and Oregonator models, we also derive critical parameter values leading to canard explosions and implosions in their associated limit cycles.

scholarly journals Dynamics of a Nonstandard Finite-Difference Scheme for a Limit Cycle Oscillator with Delayed Feedback

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Yuanyuan Wang ◽  
Xiaohua Ding
Keyword(s):  
Finite Difference ◽  
Limit Cycle ◽  
Euler Method ◽  
Delayed Feedback ◽  
Dimensional System ◽  
Two Dimensional ◽  
Limit Cycle Oscillator ◽  
The Stability ◽  
Nsfd Scheme ◽  
Nonstandard Finite Difference

We consider a complex autonomously driven single limit cycle oscillator with delayed feedback. The original model is translated to a two-dimensional system. Through a nonstandard finite-difference (NSFD) scheme we study the dynamics of this resulting system. The stability of the equilibrium of the model is investigated by analyzing the characteristic equation. In the two-dimensional discrete model, we find that there are stability switches on the time delay and Hopf bifurcation when the delay passes a sequence of critical values. Finally, computer simulations are performed to illustrate the theoretical results. And the results show that NSFD scheme is better than the Euler method.

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