Chaotic itinerancy based on attractors of one-dimensional maps

Area of Science:

• Complex Systems
• Nonlinear Dynamics
• Chaos Theory

Background:

• Chaotic dynamics are characterized by local sensitivity to initial conditions.
• Intermittent dynamics, or chaotic itinerancy, involve systems switching between different states.
• Understanding how external factors influence chaotic systems is crucial.

Purpose of the Study:

• To present a general methodology for constructing systems with slowly converging Lyapunov exponents near zero.
• To explore the phenomenon of chaotic itinerancy in one-dimensional maps.
• To investigate the impact of noise on the large-scale dynamics of these systems.

Main Methods:

• Utilizing one-dimensional maps with chaotic attractors.
• Analyzing systems with Lyapunov exponents approaching zero.
• Introducing fine-scale additive noise to observe its effects.

Main Results:

• Demonstrated a method for creating systems exhibiting chaotic itinerancy.
• Showcased that these systems possess both local and global sensitivity.
• Found that additive noise can significantly alter the natural measure on a large scale.

Conclusions:

• The described methodology enables the construction of systems with chaotic itinerancy.
• Chaotic itinerant systems exhibit unique global sensitivity to noise, beyond local sensitivity.
• This sensitivity implies that small-scale noise can have profound effects on the overall system behavior.