Bifurcation and chaos in discrete-time BVP oscillator

Wang, Jinliang; Feng, Guang-Qing

doi:10.1016/j.ijnonlinmec.2009.04.004

Cited by 14 publications

(5 citation statements)

References 17 publications

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“…By using DV, RAGlib 5 (for obtaining sample points), and our program for checking the stability of steady states at each sample points, we were able to obtain, after about 1,228 s, the results shown in Table 5, where R 3 , R 4 , and R 5 are polynomials consisting of 20, 125, and 6 terms and of degree 12, 32, and 5 in s and h, respectively. We have also analyzed the stability and bifurcations of the discrete Bonhöffer-van der Pol model [46] and a discretized model of the Cdc2-cyclin B/Wee1 system [36]. Our experiments demonstrate the feasibility of algebraic methods for stability and bifurcation analysis of discrete biological models.…”

Section: Discrete Cinquin-demongeot Modelmentioning

confidence: 84%

Stability Analysis for Discrete Biological Models Using Algebraic Methods

Mou

Wei

et al. 2011

Math.Comput.Sci.

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This paper is concerned with stability analysis of biological networks modeled as discrete and finite dynamical systems. We show how to use algebraic methods based on quantifier elimination, real solution classification and discriminant varieties to detect steady states and to analyze their stability and bifurcations for discrete dynamical systems. For finite dynamical systems, methods based on Gröbner bases and triangular sets are applied to detect steady states. The feasibility of our approach is demonstrated by the analysis of stability and bifurcations of several discrete biological models using implementations of algebraic methods.

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Section: Discrete Cinquin-demongeot Modelmentioning

confidence: 84%

Stability Analysis for Discrete Biological Models Using Algebraic Methods

Mou

Wei

et al. 2011

Math.Comput.Sci.

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“…Below, we list several techniques used to determine whether a repelling fixed point is a snapback repeller. A few studies combining mathematical analysis with complex manual processing may be found in [17,40,45]. In addition, numerical computation techniques have been used to prove the existence of snapback repellers (see, e.g., [33,28]).…”

Section: Preliminariesmentioning

confidence: 99%

Stability and chaos of the duopoly model of Kopel: A study based on symbolic computations

Kongyan

Wei

et al. 2023

Preprint

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Since Kopel's duopoly model was proposed about three decades ago, there are almost no analytical results on the equilibria and their stability in the asymmetric case. The first objective of our study is to fill this gap. This paper analyzes the asymmetric duopoly model of Kopel analytically by using several tools based on symbolic computations. We discuss the possibility of the existence of multiple positive equilibria and establish necessary and sufficient conditions for a given number of positive equilibria to exist. The possible positions of the equilibria in Kopel's model are also explored. Furthermore, in the asymmetric model of Kopel, if the duopolists adopt the best response reactions or homogeneous adaptive expectations, we establish rigorous conditions for the local stability of equilibria for the first time. The occurrence of chaos in Kopel's model seems to be supported by observations through numerical simulations, which, however, is challenging to prove rigorously. The second objective is to prove the existence of snapback repellers in Kopel's map, which implies the existence of chaos in the sense of Li–Yorke according to Marotto's theorem.

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“…Below, we list several techniques used to determine whether a repelling fixed point is a snapback repeller. A few studies combining mathematical analysis with complex manual processing may be found in [17,38,43]. In addition, numerical computation techniques have been used to prove the existence of snapback repellers (see, e.g., [31,27]).…”

Section: Preliminariesmentioning

confidence: 99%