Some results and open problems concerning symmetries, first integrals and limit cycles of dynamical systems

1980 ◽  
Vol 29 (3) ◽  
pp. 73-81
Author(s):  
F. González-Gascón
Keyword(s):  
Dynamical Systems ◽  
Limit Cycles ◽  
First Integrals ◽  
Open Problems

New results and open problems concerning the obtention of first integrals and pseudosymmetries of dynamical systems

1980 ◽  
Vol 27 (12) ◽  
pp. 363-368 ◽  
Author(s):  
F. González-Gascón ◽  
E. Rodriguez-Camino
Keyword(s):  
Dynamical Systems ◽  
First Integrals ◽  
Open Problems

Higher order first integrals of autonomous dynamical systems

2021 ◽  
Vol 170 ◽  
pp. 104383
Author(s):  
Antonios Mitsopoulos ◽  
Michael Tsamparlis
Keyword(s):  
Dynamical Systems ◽  
Higher Order ◽  
First Integrals ◽  
Autonomous Dynamical Systems

scholarly journals Maximum number of limit cycles for certain piecewise linear dynamical systems

2015 ◽  
Vol 82 (3) ◽  
pp. 1159-1175 ◽  
Author(s):  
Jaume Llibre ◽  
Douglas D. Novaes ◽  
Marco A. Teixeira
Keyword(s):  
Dynamical Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Linear Dynamical Systems

scholarly journals Quantifying Noninvertibility in Discrete Dynamical Systems

10.37236/9475 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Colin Defant ◽  
James Propp
Keyword(s):  
Dynamical Systems ◽  
Real Number ◽  
Arbitrary Function ◽  
Discrete Dynamical Systems ◽  
Open Problems ◽  
Natural Measure ◽  
Stack Sorting ◽  
Finite Set

Given a finite set $X$ and a function $f:X\to X$, we define the \emph{degree of noninvertibility} of $f$ to be $\displaystyle\deg(f)=\frac{1}{|X|}\sum_{x\in X}|f^{-1}(f(x))|$. This is a natural measure of how far the function $f$ is from being bijective. We compute the degrees of noninvertibility of some specific discrete dynamical systems, including the Carolina solitaire map, iterates of the bubble sort map acting on permutations, bubble sort acting on multiset permutations, and a map that we call "nibble sort." We also obtain estimates for the degrees of noninvertibility of West's stack-sorting map and the Bulgarian solitaire map. We then turn our attention to arbitrary functions and their iterates. In order to compare the degree of noninvertibility of an arbitrary function $f:X\to X$ with that of its iterate $f^k$, we prove that \[\max_{\substack{f:X\to X\\ |X|=n}}\frac{\deg(f^k)}{\deg(f)^\gamma}=\Theta(n^{1-1/2^{k-1}})\] for every real number $\gamma\geq 2-1/2^{k-1}$. We end with several conjectures and open problems.  

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