Dynamic Systems of a Volterra Quadratic Stochastic Operatorwith Variable Coefficients

Abstract In this work, we investigate the conditions under which a Volterra quadratic stochastic operator with variable coefficients can exhibit infinitely many periodic points. Our approach combines analytical methods from mathematical analysis with computational techniques. We begin by identifying conditions under which the operator admits a 3-periodic orbit, assuming the variation rule of its parameter is unknown but itself possesses a 3-periodic behavior. Subsequently, we analyze the set of periodic points when the parameter variation is modeled by a piecewise-linear function. By applying Sharkovsky’s theorem to the parameter’s dynamics, we demonstrate that the operator can indeed exhibit an infinite number of periodic points.

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