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Relaxation to Fixed Points in the Logistic and Cubic Maps: Analytical and Numerical Investigation
Departamento de Física, UNESP, Univ Estadual Paulista Av.24A, 1515, Rio Claro, SP 13506-900, Brazil
UNESP, Univ Estadual Paulista, Câmpus São João da Boa Vista, São João da Boa Vista, SP 13874-149, Brazil
The Abdus Salam, ICTP, Strada Costiera, 11, Trieste 34151, Italy
* Author to whom correspondence should be addressed.
Received: 14 August 2013; in revised form: 25 September 2013 / Accepted: 1 October 2013 / Published: 14 October 2013
Abstract: Convergence to a period one fixed point is investigated for both logistic and cubic maps. For the logistic map the relaxation to the fixed point is considered near a transcritical bifurcation while for the cubic map it is near a pitchfork bifurcation. We confirmed that the convergence to the fixed point in both logistic and cubic maps for a region close to the fixed point goes exponentially fast to the fixed point and with a relaxation time described by a power law of exponent -1. At the bifurcation point, the exponent is not universal and depends on the type of the bifurcation as well as on the nonlinearity of the map.
Keywords: relaxation to fixed points; dissipative mapping; complex system; cubic map; logistic map
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MDPI and ACS Style
de Oliveira, J.A.; Papesso, E.R.; Leonel, E.D. Relaxation to Fixed Points in the Logistic and Cubic Maps: Analytical and Numerical Investigation. Entropy 2013, 15, 4310-4318.
de Oliveira JA, Papesso ER, Leonel ED. Relaxation to Fixed Points in the Logistic and Cubic Maps: Analytical and Numerical Investigation. Entropy. 2013; 15(10):4310-4318.
de Oliveira, Juliano A.; Papesso, Edson R.; Leonel, Edson D. 2013. "Relaxation to Fixed Points in the Logistic and Cubic Maps: Analytical and Numerical Investigation." Entropy 15, no. 10: 4310-4318.