Year: 2026 | Month: June | Volume: 13 | Issue: 6 | Pages: 409-418
DOI: https://doi.org/10.52403/ijrr.20260640
Local Stability and Hopf Threshold in a Lienard-Type Oscillator with Exponentially Filtered Feedback
Yosephus Decupertino Sumanto1, Hafidh Khoerul Fata1
1Department of Mathematics, Faculty of Sciences and Mathematics, Universitas Diponegoro, Semarang, Indonesia.
Corresponding Author: Hafidh Khoerul Fata
ABSTRACT
This study investigates the local stability and Hopf threshold of a three-dimensional Lienard-type oscillator with exponentially filtered feedback. The model couples a nonlinear second-order oscillator with a first-order filter variable so that the feedback acts through a smoothed memory of the main state. The equilibrium structure is characterized explicitly, and linearization at the origin is used to derive the characteristic polynomial. By applying the Routh-Hurwitz criterion, explicit algebraic conditions for local asymptotic stability are obtained. The equality case of the final Routh-Hurwitz inequality yields a Hopf threshold that separates stable equilibrium dynamics from oscillatory behavior. Numerical illustrations are proposed to visualize the stability regions, Hopf threshold curve, convergence to equilibrium, and sustained oscillations after the loss of stability. The results provide a compact analytical framework for understanding how nonlinear damping, feedback strength, and memory rate affect the dynamics of filtered-feedback oscillators.
Keywords: Lienard-type oscillator, filtered feedback, exponential memory, Routh-Hurwitz criterion, Hopf threshold, continuous dynamical system.
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