Circularly distributed multipliers with deterministic moduli assessing the stability of quasiperiodic response

Area of Science:

• Dynamical Systems Theory
• Nonlinear Dynamics
• Applied Mathematics

Background:

• Analyzing the stability of periodic solutions in dynamical systems is well-established using Floquet multipliers.
• Quantitative stability analysis for quasiperiodic (QP) responses, a natural extension of periodic solutions, remains a significant challenge.
• Existing methods struggle to provide precise stability criteria for QP systems.

Purpose of the Study:

• To propose a novel and efficient approach for defining and obtaining effective multipliers for quasiperiodic (QP) stability analysis.
• To extend the concept of Floquet multipliers to quasiperiodic dynamical systems.
• To provide a quantitative method for assessing the stability of QP responses.

Main Methods:

• Transformation of the perturbed system with QP coefficients into an approximately constant coefficient system using auxiliary variables.
• Efficient computation of effective multipliers via eigenvalue analysis of the resulting constant coefficient matrix.
• Demonstration of the method's ability to recover Floquet multipliers for periodic solutions when QP response degenerates.

Main Results:

• Introduction of effective multipliers for QP stability analysis.
• Identification of circularly distributed multipliers with deterministic moduli.
• Establishment of a clear stability criterion: QP response is stable if all moduli are less than or equal to unity.

Conclusions:

• The proposed approach offers an efficient and quantitative method for assessing QP stability.
• The effective multipliers serve as a generalization of Floquet multipliers for periodic solutions.
• This work bridges a critical gap in the stability analysis of dynamical systems with quasiperiodic forcing.