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The periodic flows predicted by the single-step implicit maps are discussed first, and the periodic flows predicted by the multistep implicit maps are also presented. Periodic flows in time-delay nonlinear dynamical systems are discussed by the single-step and multistep implicit maps. The time-delay nodes in discretization of time-delay nonlinear systems were treated by both an interpolation and a direct integration. Based on the discrete nodes of periodic flows in nonlinear dynamical systems with\/without time-delay, the discrete Fourier series responses of periodic flows are presented. To demonstrate the methodology, the bifurcation tree of period-1 motion to chaos in a Duffing oscillator is presented as a sampled problem. The method presented in this paper can be applied to nonlinear dynamical systems, which cannot be solved directly by analytical methods. <\/jats:p>","DOI":"10.1142\/s0218127415500443","type":"journal-article","created":{"date-parts":[[2015,4,2]],"date-time":"2015-04-02T12:22:48Z","timestamp":1427977368000},"page":"1550044","source":"Crossref","is-referenced-by-count":95,"title":["Periodic Flows to Chaos Based on Discrete Implicit Mappings of Continuous Nonlinear Systems"],"prefix":"10.1142","volume":"25","author":[{"given":"Albert C. 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