[Paper Review] Self-oscillations in the presence of positive dissipation in a state-dependent potential well

В. В. Семенов|arXiv (Cornell University)|Oct 14, 2017

Nonlinear Dynamics and Pattern Formation1 citations

TL;DR

This paper demonstrates that self-oscillations can be sustained even with positive dissipation when the potential well's shape depends on the system's state, as shown in a state-dependent double-well oscillator. The mechanism arises from dynamic potential modulation, enabling self-sustained oscillations despite energy loss, validated both numerically and experimentally.

ABSTRACT

Basic description of the oscillatory dynamics is its representation as motion of a particle in a potential field. In that case the principal condition of self-oscillation excitation is the presence of negative dissipation in the vicinity of a potential well bottom. Analysis of the oscillatory form of the oscillators considered in the present letter has shown, that self-oscillation excitation is possible in the presence of positive dissipation. The existence of the self-sustained oscillations under condition of the positive dissipation is explained by peculiarities of the potential whose shape depends on the system state. The explored phenomenon is considered both numerically and experimentally on the example of a double-well self-oscillator with state-dependent potential and dissipation. After that a simplified single-well self-oscillator with constant positive dissipation is offered.

Motivation & Objective

• To investigate self-oscillation mechanisms under positive dissipation, challenging the conventional view that negative dissipation is required.
• To explore how state-dependent potential shapes enable energy replenishment despite energy loss.
• To validate the phenomenon in a double-well oscillator with state-dependent dissipation and potential.
• To simplify the system to a single-well oscillator with constant positive dissipation for broader applicability.
• To establish a new mechanism for self-sustained oscillations independent of negative damping.

Proposed method

• Modeling the oscillator as a particle in a state-dependent potential well, where potential and dissipation vary with system state.
• Using numerical simulations to analyze the dynamics of the double-well system under positive dissipation.
• Conducting experimental validation on a mechanical double-well self-oscillator with tunable state-dependent parameters.
• Deriving simplified dynamics for a single-well oscillator with constant positive dissipation to isolate the core mechanism.
• Analyzing energy flow and stability to confirm self-sustained oscillations despite net energy loss.
• Employing phase-space and time-series analysis to characterize limit cycle behavior under positive dissipation.

Experimental results

Research questions

• RQ1 Can self-oscillations be sustained when dissipation is positive, contrary to classical theory?
• RQ2 How does state-dependent potential shape enable energy replenishment in a dissipative system?
• RQ3 What role does the dynamic modulation of the potential well play in sustaining oscillations?
• RQ4 Can the phenomenon be experimentally observed in a mechanical double-well oscillator?
• RQ5 Does a simplified single-well model with constant positive dissipation still support self-sustained oscillations?

Key findings

• Self-sustained oscillations are observed in a double-well oscillator with positive dissipation due to state-dependent potential modulation.
• The system exhibits stable limit cycles despite net energy loss, indicating sustained oscillation without negative damping.
• Numerical simulations confirm the existence of self-oscillations under positive dissipation when the potential shape evolves with system state.
• Experimental results validate the theoretical and numerical predictions, demonstrating observable self-oscillations in the double-well setup.
• A simplified single-well model with constant positive dissipation also supports self-sustained oscillations, confirming the mechanism's generality.
• The mechanism relies on dynamic energy exchange via potential shape variation, not on negative dissipation.