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On an application of convexity to discrete systems. (English) Zbl 0588.93048
We prove the following result: Let A be a symmetric matrix, f be a gradient (or certain subgradient) of a convex function, and $$\{y_ i\}$$ be a sequence defined by $$y_{i+1}=f(Ay_ i)$$, $$y_ 0$$ arbitrary. Then the only possible periods of $$\{y_ i\}$$ are 1 or 2.

##### MSC:
 93C55 Discrete-time control/observation systems 26B25 Convexity of real functions of several variables, generalizations 39A12 Discrete version of topics in analysis 34C25 Periodic solutions to ordinary differential equations
##### Keywords:
convexity, discrete systems
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##### References:
 [1] E. Goles, Dynamics on positive automata networks, Theor. Comput. Sci., to appear. · Zbl 0585.68059 [2] () [3] Goles, E; Martinez, S, A short proof of the cyclic behaviour of multithreshold symmetric automata, Information and control, 51, (1981) · Zbl 0503.68038 [4] Goles, E; Olivos, J, Compertement itératif des fonctions à multiseuil, Information and control, 45, 300-313, (1980) · Zbl 0445.94047 [5] Goles, E; Olivos, J, Comportement périodique des fonctions à seuil binaries et applications, Discrete appl. math., 3, 93-105, (1981) · Zbl 0454.68042 [6] Pelant, J; Poljak, S, Extensions of cyclically monotone mappings, (), to appear · Zbl 0647.90071 [7] J. Pelant, S. Poljak and D. Turzík, Cyclically monotonous evaluation in social influence models, submitted to Math. Operations Research. [8] Poljak, S; Süra, M, On periodical behaviour in societies with symmetric influencies, Combinatorica, 3, 119-121, (1983) · Zbl 0561.90008 [9] Poljak, S; Turzík, D, On systems, periods and semipositive mappings, Comm. math. univ. carolinae, 25, 4, 597-614, (1984) · Zbl 0576.05059 [10] Poljak, S; Turzík, D, On pre-periods of discrete influence systems, Discrete appl. math., 13, 33-39, (1986) · Zbl 0611.93046 [11] Poljak, S; Turzík, D, Social influence models with ranking alternatives and local election rules, Math. soc. sci., 10, 189-198, (1985) · Zbl 0586.90005 [12] S. Poljak and D. Turzík, A topological proof of existence of a certain potential, Scientific Papers of VŠCHT, to appear. [13] Rockafellar, R.T, Characterization of the subdifferentials of convex functions, Pacific J. math., 17, 97-510, (1966) · Zbl 0145.15901 [14] Rockafellar, R.T, Convex analysis, (1970), Princeton University Press Princeton, NJ · Zbl 0229.90020
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