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Linear Approximation in Frequency Domain

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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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Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators

Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators

Published on: May 30, 2014

Phase reduction theory for hybrid nonlinear oscillators.

Sho Shirasaka¹, Wataru Kurebayashi², Hiroya Nakao³

¹Graduate School of Information Science and Engineering, Tokyo Institute of Technology, O-okayama 2-12-1, Meguro, Tokyo 152-8552, Japan.

Physical Review. E

|February 18, 2017

Summary

We developed a new phase reduction theory for hybrid dynamical systems, enabling analysis and control of nonlinear oscillators. This advances understanding of synchronization in systems like bipedal locomotion.

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Area of Science:

Nonlinear Dynamics
Control Theory
Robotics

Background:

Hybrid dynamical systems model rhythmic phenomena via switching dynamics.
Phase reduction theory traditionally applies to smooth systems, limiting analysis of hybrid oscillators.
Many real-world rhythmic systems are hybrid, lacking smooth models.

Purpose of the Study:

To develop a generalized phase reduction theory for hybrid dynamical systems.
To enable analysis, control, and optimization of nonlinear oscillators with intractable smooth models.
To investigate synchronization properties of hybrid oscillators.

Main Methods:

Generalized phase reduction theory for weakly perturbed hybrid limit cycles.
Analysis of injection locking using periodic forcing.
Application to a physical model of bipedal locomotion.

Main Results:

Demonstrated a novel phase reduction theory for hybrid systems.
Revealed ultrafast and robust entrainment in hybrid oscillators.
Observed logarithmic scaling at the synchronization transition.

Conclusions:

The generalized theory facilitates synchronization analysis and control of hybrid oscillators.
The findings offer insights into synchronization dynamics in complex rhythmic systems.
Applicable to diverse fields including robotics and neuroscience.