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Related Concept Videos

Damped Oscillations01:07

Damped Oscillations

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...
Types of Damping01:20

Types of Damping

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...
Forced Oscillations01:06

Forced Oscillations

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.
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...
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
Time-Domain Interpretation of PD Control01:07

Time-Domain Interpretation of PD Control

Proportional-Derivative (PD) control is a widely used control method in various engineering systems to enhance stability and performance. In a system with only proportional control, common issues include high maximum overshoot and oscillation, observed in both the error signal and its rate of change. This behavior can be divided into three distinct phases: initial overshoot, subsequent undershoot, and gradual stabilization.
Consider the example of control of motor torque. Initially, a positive...

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Reduced dynamics for delayed systems with harmonic or stochastic forcing.

Jérémie Lefebvre1, Axel Hutt, Victor G Leblanc

  • 1Ottawa Hospital Research Institute, 501 Smyth Road, Ottawa, Ontario, K1H 8L6, Canada. Jeremie.Lefebvre@unige.ch

Chaos (Woodbury, N.Y.)
|January 3, 2013
PubMed
Summary
This summary is machine-generated.

This study introduces a novel center manifold reduction scheme for nonlinear delay-differential equations (DDEs). The method simplifies complex systems, enabling analysis of bifurcations under external forcing.

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Area of Science:

  • Dynamical Systems and Nonlinear Science
  • Mathematical Physics
  • Differential Equations

Background:

  • Analyzing nonlinear delay-differential equations (DDEs) is challenging due to their infinite-dimensional nature.
  • External forcing complicates the dynamics of DDEs, requiring advanced analytical techniques.
  • Existing methods struggle to capture the essential dynamics of forced DDEs effectively.

Purpose of the Study:

  • To develop a simplified, finite-dimensional model for analyzing nonlinear DDEs with external forcing.
  • To derive a time-dependent order parameter equation that accurately reflects the dynamics of delayed systems.
  • To investigate the behavior of bifurcations under periodic and stochastic forcing.

Main Methods:

  • Development of a non-homogeneous center manifold (CM) reduction scheme.
  • Application of an ansatz separating the CM into autonomous and time-dependent components.
  • Analysis of transcritical bifurcations subjected to periodic and additive white noise forcing.

Main Results:

  • A finite-dimensional, time-dependent order parameter equation is derived, capturing key dynamics of DDEs.
  • The time-dependent CM satisfies a non-homogeneous partial differential equation.
  • Additive white noise was shown to shift the probability density function's mode in a transcritical bifurcation.

Conclusions:

  • The proposed CM reduction scheme effectively simplifies the analysis of forced nonlinear DDEs.
  • The method provides accurate insights into bifurcations and the effects of various forcing types.
  • This approach offers a powerful tool for understanding complex dynamical systems with delays.