This paper elucidates the bifurcation mechanism of an equidistant economy in spatial economics. To this end, we derive the rules of secondary and further bifurcations as a major theoretical contribution of this paper. Then we combine them with pre-existing results of direct bifurcation of the symmetric group [Formula: see text] [Elmhirst, 2004]. Particular attention is devoted to the existence of invariant solutions which retain their spatial distributions when the value of the bifurcation parameter changes. Invariant patterns of an equidistant economy under the replicator dynamics are obtained. The mechanism of bifurcations from these patterns is elucidated. The stability of bifurcating branches is analyzed to demonstrate that most of them are unstable immediately after bifurcation. Numerical analysis of spatial economic models confirms that almost all bifurcating branches are unstable. Direct bifurcating curves connect the curves of invariant solutions, thereby creating a mesh-like network, which appears as threads of warp and weft. The theoretical bifurcation mechanism and numerical examples of networks advanced herein might be of great assistance in the study of spatial economics.

中文翻译：

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SN-不变等距经济中的分岔的打破和维持

本文阐述了空间经济学中等距经济的分岔机制。为此，我们推导出二次和进一步分岔的规则作为本文的主要理论贡献。然后我们将它们与对称群的直接分岔的预先存在的结果结合起来[公式：见正文] [Elmhirst，2004]。特别关注不变解的存在，当分岔参数的值发生变化时，这些解保持其空间分布。获得了复制动力学下等距经济的不变模式。阐明了这些模式的分叉机制。对分叉分支的稳定性进行了分析，以证明它们中的大多数在分叉后立即不稳定。空间经济模型的数值分析证实，几乎所有的分叉分支都是不稳定的。直接分岔曲线将不变解的曲线连接起来，从而形成一个网状网络，表现为经纱和纬纱。本文提出的网络理论分岔机制和数值例子可能对空间经济学的研究有很大帮助。