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Periodic orbits in periodic discrete dynamics. (English) Zbl 1165.37311
Summary: We study the combinatorial structure of periodic orbits of nonautonomous difference equations \(x_{n+1}=f_n(x_n)\) in a periodically fluctuating environment. We define the \(\Gamma \)-set to be the set of minimal periods that are not multiples of the phase period. We show that when the functions \(f_n\) are rational functions, the \(\Gamma \)-set is a finite set. In particular, we investigate several mathematical models of single-species without age structure, and find that periodic oscillations are influenced by periodic environments to the extent that almost all periods are divisors or multiples of the phase period.

MSC:
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
39A11 Stability of difference equations (MSC2000)
39B12 Iteration theory, iterative and composite equations
26A18 Iteration of real functions in one variable
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