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Coexisting attractors in periodically modulated logistic maps.

Thounaojam Umeshkanta Singh¹, Amitabha Nandi, Ram Ramaswamy

¹School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110 067, India.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|July 23, 2008

Summary

This summary is machine-generated.

Periodic modulation of the logistic map leads to multistability. Increasing modulation periods, especially Fibonacci numbers, reduces multistable regions, with quasiperiodic driving creating strange nonchaotic attractors.

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Area of Science:

Dynamical systems
Nonlinear dynamics
Chaos theory

Background:

The logistic map is a fundamental model in chaos theory.
Multistability, where multiple attractors coexist, is a key phenomenon in nonlinear systems.
Understanding the influence of parameter modulation on system dynamics is crucial.

Purpose of the Study:

To investigate the impact of periodic parameter modulation on the logistic map.
To analyze the emergence and characteristics of multistability under modulated driving.
To explore the transition from periodic to quasiperiodic driving and its effect on attractors.

Main Methods:

Extension of kneading theory for one-dimensional maps.
Phase space analysis to identify regions of bistability.
Systematic variation of modulation period, including Fibonacci sequences and sinusoidal modulation.

Main Results:

For low modulation periods, bistability (two coexisting attractors) is observed and analyzed using extended kneading theory.
Increasing modulation periods, particularly Fibonacci numbers, leads to a decrease in the measure of multistable regions.
Quasiperiodic driving, achieved through dichotomous or sinusoidal modulation, results in the disappearance of multistability and the formation of strange nonchaotic attractors.

Conclusions:

The study provides a detailed analysis of multistability in the periodically modulated logistic map.
The findings demonstrate a reduction in multistability with increasing modulation periods.
The transition to quasiperiodic driving leads to the creation of strange nonchaotic attractors, highlighting a significant change in system behavior.