Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Second Order systems II01:18

Second Order systems II

445
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.
445
Second Order systems I01:20

Second Order systems I

680
A servo system exemplifies a second-order system, featuring a proportional controller and load elements that ensure the output position aligns with the input position. The relationship between these components is described by a second-order differential equation. Applying the Laplace transform under zero initial conditions yields the transfer function, showing how inputs are converted to outputs in the system.
By reinterpreting the system, one can derive the closed-loop transfer function, which...
680
Difference Equation Solution using z-Transform01:24

Difference Equation Solution using z-Transform

677
The z-transform is a powerful tool for analyzing practical discrete-time systems, often represented by linear difference equations. Solving a higher-order difference equation requires knowledge of the input signal and the initial conditions up to one term less than the order of the equation.
The z-transform facilitates handling delayed signals by shifting the signal in the z-domain, which corresponds to delaying the signal in the time domain, and advancing signals by similarly shifting in the...
677
Types of Responses of Series RLC Circuits01:11

Types of Responses of Series RLC Circuits

2.3K
A second-order differential equation characterizes a source-free series RLC circuit, marking its distinct mathematical representation. The complete solution of this equation is a blend of two unique solutions, each linked to the circuit's roots expressed in terms of the damping factor and resonant frequency.
2.3K
Relation between Mathematical Equations and Block Diagrams01:20

Relation between Mathematical Equations and Block Diagrams

3.7K
In a spring-mass-damper system, the second-order differential equation describes the dynamic behavior of the system. When transformed into the Laplace domain under zero initial conditions, this equation can be effectively analyzed and manipulated. The transformation into the Laplace domain converts differential equations into algebraic equations, simplifying the process of isolating the output.
3.7K
Introduction to Differential Equations01:20

Introduction to Differential Equations

187
A differential equation is a mathematical expression that establishes a relationship between a function and its derivatives. These equations are fundamental in modeling dynamic systems across various fields of science and engineering. The order of a differential equation is defined by the highest order derivative present in the equation. A first-order differential equation includes only the first derivative, while a second-order differential equation includes up to the second derivative of the...
187

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Stabilization of third-order differential equation by delay distributed feedback control.

Journal of inequalities and applications·2019
Same author

Effect of treatment on the global dynamics of delayed pathological angiogenesis models.

Journal of theoretical biology·2014
Same author

About positivity of green's functions for nonlocal boundary value problems with impulsive delay equations.

TheScientificWorldJournal·2014
See all related articles

Related Experiment Video

Updated: Mar 8, 2026

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
06:44

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Published on: September 23, 2025

653

W-transform for exponential stability of second order delay differential equations without damping terms.

Alexander Domoshnitsky1, Abraham Maghakyan1, Leonid Berezansky2

  • 1Department of Mathematics, Ariel University, Ariel, Israel.

Journal of Inequalities and Applications
|February 7, 2017
PubMed
Summary

This study introduces a novel method to analyze the stability of differential equations lacking an explicit first derivative. The research demonstrates that delay differential equations can achieve exponential stability even when their ordinary differential equation counterparts do not.

Keywords:
W-methodexponential stabilitysecond order delay differential equations

More Related Videos

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator
06:45

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Published on: October 28, 2022

2.2K
Interactive and Visualized Online Experimentation System for Engineering Education and Research
08:35

Interactive and Visualized Online Experimentation System for Engineering Education and Research

Published on: November 24, 2021

3.0K

Related Experiment Videos

Last Updated: Mar 8, 2026

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
06:44

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Published on: September 23, 2025

653
Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator
06:45

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Published on: October 28, 2022

2.2K
Interactive and Visualized Online Experimentation System for Engineering Education and Research
08:35

Interactive and Visualized Online Experimentation System for Engineering Education and Research

Published on: November 24, 2021

3.0K

Area of Science:

  • Mathematics
  • Differential Equations
  • Control Theory

Background:

  • Stability analysis is crucial for understanding the behavior of dynamical systems.
  • Ordinary differential equations (ODEs) and delay differential equations (DDEs) are fundamental models in various scientific fields.
  • Investigating stability in DDEs presents unique challenges compared to ODEs.

Purpose of the Study:

  • To propose a new method for analyzing the stability of a specific class of differential equations.
  • To investigate the conditions under which delay differential equations exhibit exponential stability.
  • To compare the stability properties of a delay differential equation with its corresponding ordinary differential equation.

Main Methods:

  • Development of a novel analytical technique tailored for differential equations without explicit first derivatives.
  • Comparative analysis of stability criteria for both ordinary and delay differential equations.
  • Mathematical proofs demonstrating exponential stability for the delay equation.

Main Results:

  • The proposed method effectively analyzes the stability of the target equation class.
  • It was shown that the delay differential equation can be exponentially stable.
  • The corresponding ordinary differential equation was found to be not exponentially stable under the same conditions.

Conclusions:

  • The proposed method offers a valuable tool for stability analysis in differential equations with delayed terms.
  • Exponential stability can be achieved in delay differential equations even when the non-delayed version is unstable.
  • This finding has implications for designing and controlling systems with time delays.