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Bifurcation Chaos"],"published-print":{"date-parts":[[2025,3,30]]},"abstract":"<jats:p> Fractional chaotic maps have been recently proposed to describe long-memory effects. However, their bifurcation analysis is complicated. This paper considers a linearized system and introduces necessary and sufficient conditions of Neimark\u2013Sacker (N\u2013S) bifurcation by complex analysis method. Both critical values of fractional order and the system\u2019s parameters are determined when the fractional order maps undergo sudden changes. 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