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Chaotic and stable perturbed maps: 2-cycles and spatial models
1Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary, Alberta T2N 1N4, Canada. maelena@ucalgary.ca
Chaos (Woodbury, N.Y.)
|July 2, 2010
Summary
Adding a perturbation to chaotic models like the Ricker map can reverse period doubling, leading to a stable two-cycle. This behavior differs in spatial models, where cell dynamics may not mirror the overall map.
Area of Science:
- Dynamical systems
- Nonlinear dynamics
- Chaos theory
Background:
- The Ricker and logistic maps commonly exhibit a period-doubling route to chaos as the growth rate increases.
- This route is typically irreversible in standard models.
Purpose of the Study:
- To investigate the effect of constant positive perturbations on the period-doubling route to chaos in discrete dynamical systems.
- To explore whether similar behaviors are observed in discrete spatial models derived from scalar maps.
Main Methods:
- Analysis of the Ricker and logistic maps under constant positive perturbations.
- Examination of discrete spatial models where cell values depend on nearest neighbors.
- Comparison of dynamics in scalar maps versus their spatial generalizations.
Main Results:
- The Ricker model, unlike the logistic map, shows period doubling reversals and a return to a stable two-cycle upon perturbation.
- Several other maps exhibiting similar perturbation-induced reversals are identified.
- Spatial generalizations of these maps do not necessarily display a uniform 2-cycle behavior across all cells, even when the scalar map achieves a stable 2-cycle.
Conclusions:
- Constant positive perturbations can introduce novel dynamics, including reversals of chaos, in certain discrete maps.
- The transition to chaos and its reversal are sensitive to model structure, particularly in spatial extensions.
- Predicting the behavior of discrete spatial models based solely on their scalar map counterparts is not always feasible.
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