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Solutions of equivariance for iterative differential equations. (English) Zbl 1068.34069

The authors study an iterative differential equation of the form \(x'(t)= G[x^{n_1}(t),\dots, x^{n_k}(t)]\) on an internal \(I\subset\mathbb{R}\), where \(G: I^k\to\mathbb{R}\) is continuous, \(n_j\in\mathbb{N}\) for \(j= 1,\dots, k\), and \(x^n\) denotes the \(n\)th iteration of \(x: I\to I\). If \(\Gamma\) is a Lie group acting on \(\mathbb{R}\), a solution \(x\) of the above equation is said to be \(\Gamma\)-equivariant if \(x(\gamma t)= \gamma x(t)\), for \(\gamma\in\Gamma\) and \(t\in I\). Under the assumption that \(\Gamma\) is topologically finitely generated, they find sufficient conditions on \(G\) under which there exist \(\Gamma\)-equivariant solutions.

MSC:

34K17 Transformation and reduction of functional-differential equations and systems, normal forms
34A26 Geometric methods in ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations
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