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Related Concept Videos

Stability01:28

Stability

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The time response of a linear time-invariant (LTI) system can be divided into transient and steady-state responses. The transient response represents the system's initial reaction to a change in input and diminishes to zero over time. In contrast, the steady-state response is the behavior that persists after the transient effects have faded.
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The transfer function is a fundamental concept representing the ratio of two polynomials. The numerator and denominator encapsulate the system's dynamics. The zeros and poles of this transfer function are critical in determining the system's behavior and stability.
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Stability is an important concept in oscillation. If an equilibrium point is stable, a slight disturbance of an object that is initially at the stable equilibrium point will cause the object to oscillate around that point. For an unstable equilibrium point, if the object is disturbed slightly, it will not return to the equilibrium point. There are three conditions for equilibrium points—stable, unstable, and half-stable. A half-stable equilibrium point is also unstable, but is named so...
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In classical mechanics, the two-body problem is one of the fundamental problems describing the motion of two interacting bodies under gravity or any other central force. When considering the motion of two bodies, one of the most important concepts is the reduced mass coordinates, a quantity that allows the two-body problem to be solved like a single-body problem. In these circumstances, it is assumed that a single body with reduced mass revolves around another body fixed in a position with an...
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Understanding the stability of equilibrium configurations is a fundamental part of mechanical engineering. In any system, there are three distinct types of equilibrium: stable, neutral, and unstable.
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Root Loci for Positive-Feedback Systems01:23

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The Hartley oscillator is a positive feedback system that sustains oscillations by feeding the output back to the input in phase, thereby reinforcing the signal. Positive feedback systems can be viewed as negative feedback systems with inverted feedback signals. In these systems, the root locus encompasses all points on the s-plane where the angle of the system transfer function equals 360 degrees.
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Related Experiment Video

Updated: Mar 10, 2026

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
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Isostable reduction of periodic orbits.

Dan Wilson1, Jeff Moehlis1

  • 1Department of Mechanical Engineering, University of California, Santa Barbara, California 93106, USA.

Physical Review. E
|December 15, 2016
PubMed
Summary

Classical phase reduction fails for some oscillators. A new method using isostable coordinates enhances phase reduction, overcoming limitations with Floquet multipliers and enabling analysis of unstable periodic orbits.

Area of Science:

  • Dynamical Systems and Nonlinear Science
  • Mathematical Physics

Background:

  • Phase reduction is a standard technique for analyzing limit-cycle oscillators.
  • Its applicability is limited by the magnitude of Floquet multipliers, restricting its use for systems with rapid approach to periodic orbits.

Purpose of the Study:

  • To develop an augmented phase reduction method that overcomes limitations of the classical approach.
  • To enable the study of oscillator dynamics near periodic orbits, including unstable ones.

Main Methods:

  • Definition of isostable coordinates for periodic orbits.
  • Augmentation of classical phase reduction using these new coordinates.

Main Results:

  • The proposed isostable reduction method removes the restriction on Floquet multiplier magnitudes.

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  • This framework facilitates the analysis of dynamics near periodic orbits where standard phase reduction is inadequate.
  • It provides a self-contained characterization of dynamics near unstable periodic orbits.

    • Conclusions:

    • Isostable reduction offers a more general and powerful tool for analyzing limit-cycle oscillators.
    • This method expands the applicability of phase reduction to a wider range of dynamical systems, including those with unstable periodic orbits.