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| On the Periodic Logistic Map |
| Cui Ping LI, Ming ZHAO |
| LMIB-School of Mathematics and Systems Science, Beihang University, Beijing 100191, P. R. China |
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Abstract In this paper, the famous logistic map is studied in a new point of view. We study the boundedness and the periodicity of non-autonomous logistic map#br#xn+1=rnxn(1-xn), n=0, 1,...,#br#where {rn} is a positive p-periodic sequence. The sufficient conditions are given to support the existence of asymptotically stable and unstable p-periodic orbits. This appears to be the first study of the map with variable parameter r.
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Received: 08 January 2016
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| Fund:Supported by NSF (Grant No. 61134005) and the Fundamental Research Funds for the Central Universities |
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| [1] |
Boyce, M. S., Daley, D. J.:Population tracking of fluctuating environments and natural selection for tracking ability. Amer. Nature, 115, 480-491(1980)
|
| [2] |
Coleman, B. D., Hsich, Y. H., Knowles, G. P.:On the optimal choice of r for a population in a periodic environment. Math. Biosci, 46, 71-85(1979)
|
| [3] |
Gaut, G. R. J., Goldring, K., Grogan, F., et al.:Difference equations with the Allee effect and the periodic sigmoid Beverton-Holt equation revisited. J. Biological Dynamics, 6, 1019-1033(2012)
|
| [4] |
Jia, Y.:Robust control with decoupling performance for steering and traction of 4WS vehicles under velocity-varying motion. IEEE Trans. Control Systems Technology, 8(3), 554-569(2000)
|
| [5] |
Jia, Y.:Alternative proofs for improved LMI representations for the analysis and the design of continuoustime systems with polytopic type uncertainty:A predictive approach. IEEE Trans. Autom. Control, 48(8), 1413-1416(2003)
|
| [6] |
Li, C. P.:Semigroup structures and properties of some kinds of mappings. Sci. Sin. Math., 47(1), 135-146(2017)
|
| [7] |
Robinson, C.:Dynamical Systems:satbility, symbolic dynamics, and chaos. 2nd edition, CRC Press LLC, Florida, 2000
|
| [8] |
Saber N. Elaydi, Robert J. Sacker:Global stability of periodic orbits of non-autonumous difference equations and population biology. J. Differential Equations, 208, 258-273(2005)
|
| [9] |
Saber N. Elaydi, Robert J. Sacker:Population models with Allee maps:a new model. J. Biological Dynamics, 4, 397-408(2010)
|
| [10] |
Yang, Y., Robert J. Sacker:Periodic unimodal Allee maps, the semigroup peoperty and the λ-Ricker map with Allee effect. Discrete Continuous Dynam. Systems-B, 19, 589-606(2014)
|
| [11] |
Zeidler, E.:Applied Functional Analysis. Appl. Math. Phys., Springer, New York, 1991
|
|
|
|
2018, Vol. 34 
