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Linear Approximation in Frequency Domain

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....

Types of Damping

Types of Damping

If the amount of damping in a system is gradually increased, the period and frequency start to become affected because damping opposes, and hence slows, the back and forth motion (the net force is smaller in both directions). If there is a very large amount of damping, the system does not even oscillate; instead, it slowly moves toward equilibrium. In brief, an overdamped system moves slowly towards equilibrium, whereas an underdamped system moves quickly to equilibrium but will oscillate about...

Damped Oscillations

Damped Oscillations

In the real world, oscillations seldom follow true simple harmonic motion. A system that continues its motion indefinitely without losing its amplitude is termed undamped. However, friction of some sort usually dampens the motion, so it fades away or needs more force to continue. For example, a guitar string stops oscillating a few seconds after being plucked. Similarly, one must continually push a swing to keep a child swinging on a playground.
Although friction and other non-conservative...

Second Order systems II

Second Order systems II

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.

Plotting and Calibrating the Root Locus

Plotting and Calibrating the Root Locus

Root loci often diverge as system poles shift from the real axis to the complex plane. Key points in this transition are the breakaway and break-in points, indicating where the root locus leaves and reenters the real axis. The branches of the root locus form an angle of 180/n degrees with the real axis, where n is the number of branches at a breakaway or break-in point.
The maximum gain occurs at the breakaway points between open-loop poles on the real axis, while the minimum gain is...

Linear Approximation in Time Domain

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...

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Automation of Mode Locking in a Nonlinear Polarization Rotation Fiber Laser through Output Polarization Measurements

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Nonlinear Localization of Dissipative Modulation Instability.

Alexander U Nielsen^1,2, Yiqing Xu^1,2, Caleb Todd^1,2

¹The Dodd-Walls Centre for Photonic and Quantum Technologies, New Zealand.

Physical Review Letters

|October 1, 2021

Summary

This summary is machine-generated.

We demonstrate the nonlinear localization of dissipative modulation instability (MI) in optical resonators, creating stable, controllable domains of chaos. These localized chaos domains can be switched on/off and moved, offering new light control possibilities.

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Area of Science:

Nonlinear optics
Optical resonators
Spatiotemporal dynamics

Background:

Modulation instability (MI) typically causes plane wave disintegration.
Dissipative systems exhibit unique nonlinear phenomena.

Purpose of the Study:

To demonstrate nonlinear localization of dissipative MI.
To investigate the formation and control of localized chaos domains in optical resonators.

Main Methods:

Experiments in a coherently driven nonlinear optical resonator.
Utilizing bistability and complex spatiotemporal nonlinear dynamics.
Controlling localized chaos domains via shallow modulation.

Main Results:

Formation of persisting domains of MI-driven spatiotemporal chaos.
Stable quasi-plane-wave background surrounding chaos domains.
Individually addressable (on/off) localized chaos domains.
Exploration of chaos domain transport behavior.

Conclusions:

Nonlinear localization of dissipative MI is achievable.
Bistability and nonlinear dynamics enable chaos domain pinning.
Localized chaos domains offer novel light pattern generation and control.