Stability of square oscillations in a delayed-feedback system

Area of Science:

• Nonlinear Dynamics
• Control Theory
• Chaos Theory

Background:

• Nonlinear systems with delayed feedback are crucial in various scientific and engineering fields.
• Understanding oscillation modes and their stability is essential for system design and prediction.
• Period-2 regimes and square oscillations present unique dynamic behaviors.

Purpose of the Study:

• To develop a semianalytical theory for assessing the stability of odd-harmonic square oscillation modes.
• To investigate the relationship between system stability and its approach to or departure from plateau states.
• To characterize the organization of stability domains concerning the delay parameter.

Main Methods:

• Development of a semianalytical theoretical framework.
• Analysis of system dynamics in the period-2 regime.
• Investigation of stability criteria related to plateau transitions.

Main Results:

• A novel theory for the stability of odd-harmonic square oscillations is established.
• System stability is directly linked to the manner in which the system approaches or leaves plateau regions.
• Stability domains exhibit a distinct organization within interrupted bands of delay values.

Conclusions:

• The proposed theory provides a new method for analyzing stability in complex nonlinear delayed-feedback systems.
• The findings highlight the critical role of plateau dynamics in determining system stability.
• The discovered organization of stability domains offers insights into controlling system behavior through delay parameter manipulation.