Stabilizing long-period orbits via symbolic dynamics in simple limiter controllers.

Area of Science:

• Dynamical Systems and Control Theory
• Digital Circuit Design
• Symbolic Dynamics

Background:

• Symbolic dynamics offers a powerful framework for analyzing complex systems.
• Limiter controllers are crucial components in various control applications.
• One-dimensional unimodal maps exhibit rich dynamical behaviors.

Purpose of the Study:

• To develop an efficient method for determining control parameters of simple limiter controllers.
• To leverage symbolic dynamics for analyzing one-dimensional unimodal maps.
• To minimize digital circuit configurations for generating target sequences.

Main Methods:

• Application of addition- and subtraction-symbol rules for generating admissible periodic sequences.
• Analysis of the smallest base problem for the digital tent map.
• Utilizing limiter controllers to generate maximum-length shift-register sequences.

Main Results:

• An efficient approach for determining control parameters of limiter controllers is presented.
• The method successfully addresses the smallest base problem of the digital tent map.
• One-dimensional unimodal maps are shown to robustly generate maximum-length shift-register sequences.
• Analytical expressions for control parameters of arbitrary long Sarkovskii sequences are derived.

Conclusions:

• The proposed method provides an efficient way to determine control parameters for limiter controllers.
• This technique simplifies digital circuit design by minimizing configurations.
• One-dimensional unimodal maps can be reliably used for generating maximum-length sequences through limiter controllers.