Unstable Limit Cycles and Singular Attractors in a Two-Dimensional Memristor-Based Dynamic System

Area of Science:

• Nonlinear Dynamics and Chaos Theory
• Electrical Engineering and Circuit Design
• Memristor Applications

Background:

• Dynamical systems exhibiting complex behaviors are crucial for understanding phenomena in physics and engineering.
• Memristors, as fundamental circuit elements, offer unique properties for designing advanced electronic systems.
• The analysis of unstable limit cycles and singular attractors is essential for characterizing system stability and predictability.

Purpose of the Study:

• To investigate the existence and characteristics of unstable limit cycles in a two-dimensional dynamical system incorporating a bistable memristor.
• To develop and validate a novel programmable scheme for the detection of unstable limit cycles based on the nested intervals theorem.
• To explore the emergence and coexistence of singular attractors within the memristor-based dynamical system.

Main Methods:

• Development of a two-dimensional dynamical system model featuring an inductor and a bistable bi-local active memristor.
• Implementation of a new programmable scheme inspired by the nested intervals theorem for identifying unstable limit cycles.
• Numerical simulations to verify the proposed scheme and analyze the system's dynamical behaviors, including bifurcations and attractor coexistence.

Main Results:

• Successful identification of unstable limit cycles and their evolution laws within subcritical Hopf bifurcation domains of the memristor system.
• Discovery of coexisting singular attractors in the twin local activity domains, distinct from the Hopf bifurcation regions.
• Observation of the coexistence of singular attractors with period-2 or period-3 attractors through numerical simulations.

Conclusions:

• The proposed programmable scheme is effective for finding unstable limit cycles in memristor-based dynamical systems.
• The studied system exhibits rich dynamics, including unstable limit cycles and coexisting singular attractors, particularly in its local activity domains.
• The findings contribute to a deeper understanding of complex dynamics in memristive circuits and potential applications in chaos-based systems.