[Paper Review] Active Control of the Parametric Resonance in the Modified Rayleigh-Duffing Oscillator

C. H. Miwadinou, A. V. Monwanou+1 more|arXiv (Cornell University)|Mar 3, 2013

Nonlinear Dynamics and Pattern Formation12 references3 citations

TL;DR

This paper proposes an active control strategy using time-delayed feedback to suppress parametric resonance in a modified Rayleigh-Duffing oscillator. By applying the method of averaging and Routh-Hurwitz stability analysis, the authors demonstrate that appropriate selection of time-delay and feedback gain can significantly reduce vibration amplitude and broaden the stable region of non-trivial steady-state solutions, effectively controlling both Hopf and saddle-node bifurcations.

ABSTRACT

The present paper examines the active control of parametric resonance in modified Rayleigh-Duffing oscillator. We used the method of averaging to obtain steady-state solutions. We have found the critical value of the parametrical amplitude which indicates the boundary layer where the control is efficient in reducing the amplitude vibration. We have also found the effects of excitation parameters and time-delay on dynamical of this system with the principal parametric resonance. We have obtained for this oscillator the Hopf bifurcation and saddle-node bifurcation for certains values of parametric parameters and time-delay. We have studied the influence of parameter $k_2$ which is one of the parameters which modify the ordinary Rayleigh-Duffing oscillator. We have discussed the appropriate choice of the time-delay and control gain. We finally studied the stability of fixed point and it is found that the appropriate choice of the time-delay can broaden the stable region of the non-trivial steady-state solutions which will enhance the control efficiency. Numerical simulations are performed in order to confirm analytical results.

Motivation & Objective

• To investigate active control of parametric resonance in a modified Rayleigh-Duffing oscillator with time-delayed feedback.
• To determine the critical parametric amplitude threshold beyond which control becomes effective.
• To analyze the effects of excitation parameters, time-delay, and feedback gain on system stability and bifurcation behavior.
• To identify optimal time-delay and control gain values that enhance stability and reduce vibration amplitude.
• To study the influence of the parameter $k_2$ on the control performance and bifurcation structure.

Proposed method

• Application of the method of averaging to derive first-order approximate steady-state solutions of the modified Rayleigh-Duffing oscillator with time-delayed position and velocity feedback.
• Derivation of amplitude and phase equations from the original second-order nonlinear ODE to analyze periodic responses.
• Use of the Routh-Hurwitz criterion to assess the stability of non-trivial steady-state solutions by analyzing the eigenvalues of the Jacobian matrix.
• Construction of bifurcation diagrams and frequency-response curves to visualize the effects of parametric excitation and time-delay on system dynamics.
• Numerical simulations to validate analytical results, particularly the stability domains and amplitude reduction under control.
• Incorporation of time-delayed feedback terms $-b x(t- au)$ and $-c \\.dot{x}(t- au)$ into the oscillator equation to model active control.

Experimental results

Research questions

• RQ1 How does time-delayed feedback affect the amplitude of parametric resonance in the modified Rayleigh-Duffing oscillator?
• RQ2 What are the critical values of parametric excitation amplitude and detuning parameter where control becomes effective?
• RQ3 How do the feedback gain and time-delay influence the stability of non-trivial steady-state solutions?
• RQ4 What role does the parameter $k_2$ play in modifying the system’s bifurcation structure and control efficiency?
• RQ5 Under what conditions does the system exhibit Hopf or saddle-node bifurcations under active control?

Key findings

• The critical value of the parametric amplitude was identified as the threshold beyond which active control effectively reduces vibration amplitude.
• The stability domain of non-trivial steady-state solutions is broadened when the time-delay control gain for position ($b$) exceeds that for velocity ($c$).
• Appropriate selection of time-delay and feedback gain can suppress both Hopf and saddle-node bifurcations, enhancing system stability.
• The parameter $k_2$ significantly influences the shape and extent of the stability region and the amplitude of parametric resonance.
• Numerical simulations confirmed that the amplitude of vibration at primary resonance is reduced under optimal time-delay and feedback gain settings.
• The Routh-Hurwitz criterion revealed that stability requires $T > 0$ and $D > 0$, with explicit conditions derived for $T$ and $D$ in terms of system parameters and time-delay.