Skip Navigation Links
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland


C. Steve Suh (editor)

Texas A&M University, USA


Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China


Extreming curves and the parameter space of a generalized Logistic mapping

Journal of Vcibration Testing and System Dynamics 2(2) (2018) 109--118 | DOI:10.5890/JVTSD.2018.06.002

Diogo Ricardo da Costa$^{1}$, Matheus Hansen$^{2}$, Edson D. Leonel$^{1}$, Rene O. Medrano-T$^{3}$

$^{1}$ Departamento de Física, UNESP - Univ Estadual Paulista, Av. 24A, 1515, Bela Vista, 13506-900, Rio Claro, SP - Brazil

$^{2}$ Instituto de Física da Universidade de S~ao Paulo, Rua do Matão, Travessa R 187, Cidade Universitária, 05314-970 São Paulo, SP - Brazil

$^{3}$ Departamento de Física, UNIFESP - Universidade Federal de São Paulo, Rua São Nicolau, 210, Centro, 09913-030, Diadema, SP, Brazil

Download Full Text PDF



A logistic map with parametric perturbation is studied. We confirm the model exhibits self-similar structures in the parameter space known as shrimps. The organization of such structures may be describe through extreme curves, giving exactly the position of each one of them in a complicated behavior in the parameter space. Boundary crisis are also discussed. They lead to an abrupt destruction of the chaotic attractor and how such destruction affects the dynamics of the system is discussed, particularly affecting the time before the orbit leaves to infinite. By a specific choice of the parameters we show the existence of strange non-chaotic attractors in the parameter space.


DRC acknowledges to PNPD/CAPES. MH thanks to CAPES for the nancial support. EDL thanks to CNPq (303707/2015-1), FUNDUNESP and FAPESP (2017/14414-2), Brazilian agencies. RMT thanks to FAPESP (2015/50122-0). This research was supported by resources supplied by the Center for Scientific Computing (NCC/GridUNESP) of the São Paulo State University (UNESP).


  1. [1]  de Oliveira, J.A., Papesso, E.R., and Leonel, E.D. (2013), Relaxation to Fixed Points in the Logistic and Cubic Maps: Analytical and Numerical Investigation, Entropy, 15(10), 4310-4318.
  2. [2]  Gade, P.M. and Sahasrabudhe, G.G. (2013), Universal persistence exponent in transition to antiferromagnetic order in coupled logistic maps, Physical Review E, 87, 052905.
  3. [3]  Challenger, J.D., Fanelli, D., and McKane, A.J. (2013), Intrinsic noise and discrete-time processes, Physical Review E, 88, 040102.
  4. [4]  de Souza, S.L.T., Lima, A.A., Caldas, I.L., Medrano-T, R.O., and Guimarães-Filho, Z.O. (2012), Selfsimilarities of periodic structures for a discrete model of a two-gene system, Physics Letters A, 376(15), 1290-1294.
  5. [5]  Cerbus, R.T. and Goldburg, W.I. (2013), Information content of turbulence, Physical Review E, 88, 053012.
  6. [6]  Hamacher, K. (2012), Dynamical regimes due to technological change in a microeconomical model of production, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22(3), 033149.
  7. [7]  McCartney, M. (2011), Lyapunov exponents for multi-parameter tent and logistic maps, Chaos: An Interdisciplinary Journal of Nonlinear Science, 21(4), 043104.
  8. [8]  Philominathan, P., Santhiah, M., Mohamed, I.R., Murali, K., and Rajasekar, S. (2011), Chaotic dynamics of a simple parametrically driven dissipative circuit, International Journal of Bifurcation and Chaos, 21(07), 1927-1933.
  9. [9]  Wen, H., Guang-Hao, Z., Gong, Z., Jing-Qiao, Z., and Xian-Long, L. (2012), Stabilities and bifurcations of sine dynamic equations on time scale, Acta Physica Sinica, 61(17), 170505.
  10. [10]  Medrano-T, R.O. and Rocha, R. (2014) The negative side of chua's circuit parameter space: stability analysis, period-adding, basin of attraction metamorphoses, and experimental investigation, International Journal of Bifurcation and Chaos, 24(09), 1430025.
  11. [11]  Urquizú, M. and Correig, A.M. (2007), Fast relaxation transients in a kicked damped oscillator, Chaos, Solitons & Fractals, 33(4), 1292-1306.
  12. [12]  Ilhem, D. and Amel, K. (2006) One-dimensional and two-dimensional dynamics of cubic maps, Discrete Dynamics in Nature and Society, 2006.
  13. [13]  Li, T.-Y. and Yorke, J.A. (1975) Period Three Implies Chaos, The American Mathematical Monthly, 82(10), 985-992.
  14. [14]  daCosta, D.R., Hansen, M., Guarise, G., Medrano-T, R.O., and Leonel, E.D. (2016) The role of extreme orbits in the global organization of periodic regions in parameter space for one dimensional maps, Physics Letters A, 380(18), 1610-1614.
  15. [15]  Leonel, E.D., daSilva, J.K.L., and Kamphorst, S.O. (2002), Relaxation and transients in a time-dependent logistic map, International Journal of Bifurcation and Chaos, 12(07), 1667-1674.
  16. [16]  Leonel, E.D., daSilva, J.K.L., and Kamphorst, S.O. (2001), Transients in a time-dependent logistic map, Physica A: Statistical Mechanics and its Applications, 295(1), 280-284.
  17. [17]  Lichtemberg, A.J. and Lieberman, M.A. Regular and chaotic dynamics, (1992).
  18. [18]  Rocha, R., Andrucioli, G.L.D., and Medrano-T, R.O. (2010), Experimental characterization of nonlinear systems: a real-time evaluation of the analogous Chua's circuit behavior, Nonlinear Dynamics, 62(1-2), 237-251.
  19. [19]  Alligood, K.T., Sauer, T.D., and Yorke, J.A. Chaos: An Introduction to Dynamical Systems, 1996. (1997).
  20. [20]  May, R.M. (1976), Simple mathematical models with very complicated dynamics, Nature, 261(5560), 459.
  21. [21]  May, R.M. and Oster, G.F. (1976) Bifurcations and dynamic complexity in simple ecological models, The American Naturalist, 110(974), 573-599.
  22. [22]  Feigenbaum, M.J. (1978) Quantitative universality for a class of nonlinear transformations, Journal of statistical physics, 19(1), 25-52.
  23. [23]  Grebogi, C., Ott, E., and Yorke, J.A. (1982), Chaotic attractors in crisis, Physical Review Letters, 48(22), 1507.
  24. [24]  Grebogi, C., Ott, E., and Yorke, J.A. (1983) Crises, sudden changes in chaotic attractors, and transient chaos, Physica D: Nonlinear Phenomena, 7(1-3), 181-200.
  25. [25]  Persohn, K.J. and Povinelli, R.J. (2012), Analyzing logistic map pseudorandom number generators for periodicity induced by finite precision floating-point representation, Chaos, Solitons & Fractals, 45(3), 238-245.
  26. [26]  Maranhão, D.M. (2016), Ordered and isomorphic mapping of periodic structures in the parametrically forced logistic map, Physics Letters A, 380(40), 3238-3243.
  27. [27]  Baptista, M.S. and Caldas, I.L. (1996) Dynamics of the kicked logistic map, Chaos, Solitons & Fractals, 7(3), 325-336.
  28. [28]  Medeiros, E.S., Medrano-T, R.O., Caldas, I.L., and deSouza, S.L.T. (2013), Torsion-adding and asymptotic winding number for periodic window sequences, Physics Letters A, 377(8), 628-631.
  29. [29]  Gallas, J.A.C. (1993) Structure of the parameter space of the Hénon map, Physical Review Letters, 70, 2714-2717.
  30. [30]  Hirsch, M.W., Smale, S., and Devaney, R.L. (2012) Differential equations, dynamical systems, and an introduction to chaos, Academic press.
  31. [31]  Radwan, A.G. (2013), On some generalized discrete logistic maps, Journal of advanced research, 4(2), 163- 171.
  32. [32]  Ames, W.F. and Rogers, C. (1988), Nonlinear equations in the applied sciences.
  33. [33]  Grebogi, C., Ott, E., Pelikan, S., and Yorke, J.A. (1984), Strange attractors that are not chaotic, Physica D: Nonlinear Phenomena, 13(1-2), 261-268.
  34. [34]  daCosta, D.R., Medrano-T, R.O., and Leonel, E.D. (2017), Route to chaos and some properties in the boundary crisis of a generalized logistic mapping, Physica A: Statistical Mechanics and its Applications, 486, 674-680.
  35. [35]  Fiedler-Ferrara, N. and doPrado, C.C. (1994), Caos: uma introdução, Edgar Blucher.