Hopf Bifurcations and Limit Cycles in Evolutionary Network Dynamics
Hopf Bifurcations and Limit Cycles in Evolutionary Network Dynamics
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DOI：
10.1137/120878537
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发表时间：
2012-01-01
影响因子：
2.1
通讯作者：
Leonard, Naomi E.
中科院分区：
文献类型：
--
作者：
Pais, Darren;Caicedo-Nunez, Carlos H.;Leonard, Naomi E.
进化动力学中的复制者-变异者方程可以作为语言进化、社会网络中的行为动力学和网络多智能体系统中的决策动力学的模型。对这些动力学的稳定平衡的分析一直是文献中的一个焦点，其中通常假定适应度函数的对称性。我们研究了适应度的不对称性，证明了复制子-变子方程存在Hopf分支和极限环。在循环适应矩阵的情况下，我们证明了由动力学的多个不同的Hopf分支产生的稳定极限环存在的条件。在非循环的情况下，我们说明了动态的稳定极限环如何耦合到回报图中的嵌入有向环。极限环对应于语言进化中语法优势的振荡和社会网络中行为偏好的振荡；对于决策系统，极限环对应于整个群体决策的持续振荡。
The replicator-mutator equations from evolutionary dynamics serve as a model for the evolution of language, behavioral dynamics in social networks, and decision-making dynamics in networked multiagent systems. Analysis of the stable equilibria of these dynamics has been a focus in the literature, where symmetry in fitness functions is typically assumed. We explore asymmetry in fitness and show that the replicator-mutator equations exhibit Hopf bifurcations and limit cycles. We prove conditions for the existence of stable limit cycles arising from multiple distinct Hopf bifurcations of the dynamics in the case of circulant fitness matrices. In the noncirculant case we illustrate how stable limit cycles of the dynamics are coupled to embedded directed cycles in the payoff graph. The limit cycles correspond to oscillations of grammar dominance in language evolution and to oscillations in behavioral preferences in social networks; for decision-making systems, the limit cycles correspond to sustained oscillations in decisions across the group.