Effects of local mutations in quadratic iterations

Area of Science:

• Complex dynamics
• Dynamical systems theory
• Chaos theory

Background:

• Complex quadratic maps (fc(z)=z^2+c) are fundamental in chaos theory.
• Understanding system evolution under perturbations is crucial for dynamical systems.
• The prisoner set is a key tool for analyzing long-term behavior in iterated maps.

Purpose of the Study:

• To introduce and analyze "mutated" complex quadratic maps.
• To investigate the impact of localized perturbations on system dynamics and orbit behavior.
• To use the prisoner set as a quantitative measure of these effects.

Main Methods:

• Definition of a "mutated" map interpolating between an erroneous map and the original map.
• Analysis of temporal evolution and asymptotic behavior under mutated iterations.
• Utilizing the prisoner set's topology to quantify long-term system changes.

Main Results:

• Mutations significantly alter the temporal evolution and asymptotic behavior of complex quadratic maps.
• The position, timing, and size of mutations demonstrably affect the system's long-term dynamics.
• Changes in system evolution are effectively encoded in the topology of the prisoner set.

Conclusions:

• The "mutated" map framework provides a novel way to study perturbations in dynamical systems.
• The prisoner set is a robust tool for analyzing the impact of localized errors.
• This model offers a metaphor for understanding copying errors in biological replication.