Cobweb Diagram for Generalized Logistic Maps with z-Unimodality
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
This Demonstration shows the cobweb diagram for 
, which generalizes the well-known logistic map 
[1–8]. Here 
is the iteration number, 
is the 
iterate of 
starting from the initial value 
, 
is the main control parameter, and 
is the subcontrol parameter (which determines the unimodality, the degree of the local maximum of 
).
Contributed by: Ki-Jung Moon (December 2013)
Open content licensed under CC BY-NC-SA
Snapshots
Details
References
[1] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed., Boulder: Westview Press, 2003.
[2] K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems, New York: Springer-Verlag, 1997.
[3] S. H. Strogatz, Nonlinear Dynamics and Chaos, Reading, MA: Perseus Books Publishing, 1994.
[4] S. Wolfram, A New Kind of Science, Champaign, IL: Wolfram Media, Inc., 2002.
[5] M. J. Feigenbaum, "Quantitative Universality for a Class of Nonlinear Transformations," Journal of Statistical Physics, 19(1), 1978 pp. 25–52. doi:10.1007/BF01020332.
[6] M. J. Feigenbaum, "The Universal Metric Properties of Nonlinear Transformations," Journal of Statistical Physics, 21(6), 1979 pp. 669–706. doi:10.1007/BF01107909.
[7] K.-J. Moon and S. D. Choi, "Reducible Expansions and Related Sharp Crossovers in Feigenbaum's Renormalization Field," Chaos: An Interdisciplinary Journal of Nonlinear Science, 18(2), 2008, 023104. doi:10.1063/1.2902826.
[8] K.-J. Moon, "Erratum: Reducible Expansions and Related Sharp Crossovers in Feigenbaum's Renormalization Field," Chaos: An Interdisciplinary Journal of Nonlinear Science, 20, 2010, 049902. doi:10.1063/1.3530128.
Permanent Citation
"Cobweb Diagram for Generalized Logistic Maps with z-Unimodality"
http://demonstrations.wolfram.com/CobwebDiagramForGeneralizedLogisticMapsWithZUnimodality/
Wolfram Demonstrations Project
Published: December 13 2013
Trajectory-Scaling Functions for Generalized Logistic Maps with z-Unimodality
Ki-Jung Moon
Time-Series Analysis for Generalized Logistic Maps with z-Unimodality
Ki-Jung Moon
Finite Lyapunov Exponent for Generalized Logistic Maps with z-Unimodality
Ki-Jung Moon
Binary Coding Functions for Generalized Logistic Maps with z-Unimodality
Ki-Jung Moon
High-Precision Newton Algorithm for Generalized Logistic Maps with Unimodality z
Ki-Jung Moon
Bifurcation Diagram for a Generalized Logistic Map
Ki-Jung Moon
Iterates of Generalized Logistic Maps for Superstable Parameter Values
Ki-Jung Moon
Feigenbaum's Scaling Law for the Logistic Map
Ki-Jung Moon
Box-Counting Algorithm of the Hénon Map
Ki-Jung Moon
Time Series and Cobwebs for Arbitrary Recursive Maps on the Unit Interval
Vitaliy Kaurov
-
Binary Coding Functions for Generalized Logistic Maps with z-Unimodality
Ki-Jung Moon -
Bifurcation Diagram for a Generalized Logistic Map
Ki-Jung Moon -
Playing with the Hénon Map Starting with a Circle or a Square
Ki-Jung Moon -
Box-Counting Algorithm of the Hénon Map
Ki-Jung Moon -
Iterates of Generalized Logistic Maps for Superstable Parameter Values
Ki-Jung Moon -
Trajectory-Scaling Functions for Generalized Logistic Maps with z-Unimodality
Ki-Jung Moon -
Finite Lyapunov Exponent for Generalized Logistic Maps with z-Unimodality
Ki-Jung Moon -
Cobweb Diagram for Generalized Logistic Maps with z-Unimodality
Ki-Jung Moon -
Degenerate Critical Points and Catastrophes: Fold Catastrophe
Ki-Jung Moon -
Time-Series Analysis for Generalized Logistic Maps with z-Unimodality
Ki-Jung Moon -
Estimating the Feigenbaum Constant from a One-Parameter Scaling Law
Ki-Jung Moon -
Stack Diagram for 1D Box-Counting Steps
Ki-Jung Moon -
Two-Color Pixel Division Game for Generalized Logistic Maps with z-Unimodality
Ki-Jung Moon -
High-Precision Newton Algorithm for Generalized Logistic Maps with Unimodality z
Ki-Jung Moon -
Feigenbaum's Scaling Law for the Logistic Map
Ki-Jung Moon