## 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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### References:

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