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In this paper we shall show amazing behavior of some discrete maps usingCollatze like problems and some advanced theories in analytic number theory anddynamical system,we have investigated the driven cubic-quintic Duffing equationsuch that , We were able to predict the number of limit cycles around theequilibrium and to develop a theoretical approach to chaos suppression indamped driven systems using Collatze like problem sequences , some new resultsregarding behavior of that sequence are presented.

Amazing behavior and transition to chaos of some sequences using Collatz like problems and Quibic duffing