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Second Order systems II

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In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.

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Updated: Sep 29, 2025

Rapid Repetition Rate Fluctuation Measurement of Soliton Crystals in a Microresonator

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Published on: December 15, 2021

Time crystal dynamics in a weakly modulated stochastic time delayed system.

Jordi Tiana-Alsina¹, Cristina Masoller²

¹Department de Física Aplicada, Facultat de Fisica, Universitat de Barcelona, Marti i Franques 1, 08028, Barcelona, Spain.

Scientific Reports

|March 23, 2022

Summary

This summary is machine-generated.

Researchers demonstrated time-crystal-like behavior in a semiconductor laser with delayed feedback. This system exhibits robust, regular oscillations locked to a modulation signal, even with noise.

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Area of Science:

Nonlinear Dynamics and Optics
Quantum Many-Body Physics

Background:

Time crystals exhibit robust, regular oscillations in periodically driven systems, with frequencies distinct from the driving signal.
Time-delayed systems (TDSs) share similarities with spatially-extended systems, offering a potential platform for exploring complex dynamics.
Semiconductor lasers with delayed feedback are known for irregular spiking behavior near their lasing threshold.

Purpose of the Study:

To experimentally demonstrate time-crystal-like behavior in a stochastic, weakly modulated time-delayed system.
To investigate the influence of delayed feedback and external modulation on the regularity of laser output spikes.
To identify characteristic features of time-crystal behavior, such as frequency locking and phase stability, in a non-quantum system.

Main Methods:

Utilized a semiconductor laser operating near threshold with optical delayed feedback.
Applied weak periodic modulation (pulsed and sinusoidal) to the laser's driving current.
Analyzed the output intensity spikes for regularity, phase locking, and robustness against noise over thousands of modulation cycles.

Main Results:

Observed long-range regularity in the timing of laser output spikes, locking to the applied modulation signal.
Demonstrated that the interaction between delayed feedback and modulation generates phase-stable oscillations despite the presence of noise.
Identified harmonic and subharmonic locking with pulsed modulation, and exclusively subharmonic locking with sinusoidal modulation, a hallmark of time-crystal behavior.

Conclusions:

Time-crystal-like behavior can be experimentally realized in a classical, stochastic time-delayed system (semiconductor laser).
The interplay of delayed feedback and external modulation is crucial for inducing robust, synchronized oscillations.
This work extends the concept of time crystals to classical systems and offers new avenues for studying complex dynamics.