Castro Sofia B S D, Rucklidge Alastair M
Centro de Matemática and Faculdade de Economia, Universidade do Porto, Porto, Portugal.
School of Mathematics, University of Leeds, Leeds, LS2 9JT UK.
J Nonlinear Sci. 2025;35(4):80. doi: 10.1007/s00332-025-10175-2. Epub 2025 Jun 11.
Heteroclinic cycles are sequences of equilibria along with trajectories that connect them in a cyclic manner. We investigate a class of robust heteroclinic cycles that do not satisfy the usual condition that all connections between equilibria lie in flow-invariant subspaces of equal dimension. We refer to these as robust heteroclinic cycles in pluridimensions. The stability of these cycles cannot be expressed in terms of ratios of contracting and expanding eigenvalues in the usual way because, when the subspace dimensions increase, the equilibria fail to have contracting eigenvalues. We develop the stability theory for robust heteroclinic cycles in pluridimensions, allowing for the absence of contracting eigenvalues. We present four new examples, each with four equilibria and living in four dimensions, that illustrate the stability calculations. Potential applications include modelling the dynamics of evolving populations when there are transitions between equilibria corresponding to mixed populations with different numbers of species.
异宿环是由平衡点以及以循环方式连接它们的轨道组成的序列。我们研究一类不满足通常条件的鲁棒异宿环，该条件是平衡点之间的所有连接都位于等维的流不变子空间中。我们将这些称为多维度鲁棒异宿环。这些环的稳定性不能以通常的方式用收缩和扩张特征值的比率来表示，因为当子空间维度增加时，平衡点没有收缩特征值。我们发展了多维度鲁棒异宿环的稳定性理论，考虑到收缩特征值的缺失情况。我们给出四个新例子，每个例子都有四个平衡点且存在于四维空间中，用于说明稳定性计算。潜在应用包括在存在对应于不同物种数量混合种群的平衡点之间发生转变时，对进化种群的动态进行建模。
