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Related Concept Videos

Stability01:28

Stability

186
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.
The stability of an LTI system is determined by the roots of its characteristic equation, known as poles. A system is stable if it produces a bounded...
186
Damped Oscillations01:07

Damped Oscillations

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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...
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Types of Damping01:20

Types of Damping

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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...
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Forced Oscillations01:06

Forced Oscillations

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When an oscillator is forced with a periodic driving force, the motion may seem chaotic. The motions of such oscillators are known as transients. After the transients die out, the oscillator reaches a steady state, where the motion is periodic, and the displacement is determined.
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Oscillations about an Equilibrium Position01:04

Oscillations about an Equilibrium Position

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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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Second Order systems II01:18

Second Order systems II

171
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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Stability of nonlinear Dirac solitons under the action of external potential.

Chaos (Woodbury, N.Y.)·2024
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Stability of parametrically driven, damped nonlinear Dirac solitons.

Bernardo Sánchez-Rey1, David Mellado-Alcedo2, Niurka R Quintero3

  • 1Departamento de Física Aplicada I, Escuela Politécnica Superior, Universidad de Sevilla, Virgen de África 7, 41011 Sevilla, Spain.

Chaos (Woodbury, N.Y.)
|August 22, 2025
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Summary
This summary is machine-generated.

This study investigates the stability of solutions to the nonlinear Dirac equation. Sufficient dissipation ensures the stability of certain solutions, with low-frequency solitons proving stable across all parameters.

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Area of Science:

  • Nonlinear dynamics
  • Quantum mechanics
  • Mathematical physics

Background:

  • The nonlinear Dirac equation describes various physical phenomena.
  • Understanding the stability of its solutions is crucial for theoretical and applied research.
  • Previous analyses suggested specific stability properties, requiring further investigation.

Purpose of the Study:

  • To investigate the linear stability of two exact stationary solutions of the parametrically driven, damped nonlinear Dirac equation.
  • To determine conditions under which these solutions are stable or unstable.
  • To map the stability regions in the parameter space.

Main Methods:

  • Linearization of the nonlinear Dirac equation around exact stationary solutions.
  • Resolution of the resulting eigenvalue problem to ascertain stability.
  • Extensive numerical simulations using a novel algorithm to corroborate analytical findings.

Main Results:

  • One stationary solution is proven to be always unstable, confirming prior variational method results.
  • Sufficient dissipation is shown to guarantee the stability of the second solution.
  • A stability curve is determined, separating stable and unstable parameter regions; low-frequency solitons are stable universally.

Conclusions:

  • The stability of nonlinear Dirac equation solutions is highly dependent on dissipation and driving frequency.
  • The study provides a comprehensive stability diagram for these solutions.
  • Numerical simulations validate the analytical stability predictions, highlighting the efficacy of the employed algorithm.