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Related Concept Videos

BIBO stability of continuous and discrete -time systems01:24

BIBO stability of continuous and discrete -time systems

System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.
To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system.
Second Order systems II01:18

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.
If  ζ...
Linearization and Approximation01:26

Linearization and Approximation

Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...
Divergence and Stokes' Theorems01:06

Divergence and Stokes' Theorems

The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write numerous physical laws...
Application of Linearization and Approximation01:29

Application of Linearization and Approximation

A drone flying through complex terrain often relies on more than one sensing method to estimate small changes in altitude. Along with direct measurements, air pressure provides a useful indirect indicator of vertical movement. Atmospheric pressure decreases as altitude increases, and this relationship is commonly described using an exponential model. Although accurate, converting pressure measurements into altitude values requires calculations that are too complex to perform repeatedly during...
Second Order systems I01:20

Second Order systems I

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...

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Related Experiment Video

Updated: May 25, 2026

Parametric Optimization Design Method for Friction Plates of Hydro-Viscous Clutches
10:58

Parametric Optimization Design Method for Friction Plates of Hydro-Viscous Clutches

Published on: July 22, 2025

Variational approximations to homoclinic snaking in continuous and discrete systems.

P C Matthews1, H Susanto

  • 1School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 7, 2012
PubMed
Summary

Localized structures arise from pinning mechanisms, often with exponentially small pinning regions. A variational method successfully captures this behavior in both continuous and discrete nonlinear systems, aligning with prior findings and simulations.

Related Experiment Videos

Last Updated: May 25, 2026

Parametric Optimization Design Method for Friction Plates of Hydro-Viscous Clutches
10:58

Parametric Optimization Design Method for Friction Plates of Hydro-Viscous Clutches

Published on: July 22, 2025

Area of Science:

  • Nonlinear Dynamics
  • Mathematical Physics
  • Pattern Formation

Background:

  • Localized structures are observed in diverse systems, often linked to pinning mechanisms.
  • A key challenge is analyzing the exponentially small width of pinning regions when length scales separate.
  • Standard asymptotic methods struggle to address this scale separation.

Purpose of the Study:

  • To develop and apply a variational method for analyzing localized structures with exponentially small pinning regions.
  • To validate the method against established techniques and numerical simulations.
  • To demonstrate the broad applicability of the variational approach.

Main Methods:

  • A variational method is employed to analyze the behavior of localized structures.
  • The method is applied to the quadratic-cubic Swift-Hohenberg equation.
  • The approach is also tested on a discrete system with cubic-quintic nonlinearity.

Main Results:

  • The variational method successfully reproduces the exponentially small width of pinning regions.
  • Results for the Swift-Hohenberg equation align with recent exponential asymptotics studies.
  • The method shows good agreement with numerical simulations for the discrete nonlinear system.

Conclusions:

  • The variational method provides an effective tool for studying localized structures with scale separation.
  • This approach offers an alternative to complex asymptotic techniques.
  • The findings highlight the robustness of the variational method across different system types.