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Related Concept Videos

Oscillations In An LC Circuit01:30

Oscillations In An LC Circuit

An idealized LC circuit of zero resistance can oscillate without any source of emf by shifting the energy stored in the circuit between the electric and magnetic fields. In such an LC circuit, if the capacitor contains a charge q before the switch is closed, then all the energy of the circuit is initially stored in the electric field of the capacitor. This energy is given by
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Linear time-invariant Systems01:23

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Synchronization in complex oscillator networks and smart grids.

Florian Dörfler1, Michael Chertkov, Francesco Bullo

  • 1Center for Control, Dynamical Systems, and Computation, University of California, Santa Barbara, CA 93106, USA. dorfler@engineering.ucsb.edu

Proceedings of the National Academy of Sciences of the United States of America
|January 16, 2013
PubMed
Summary
This summary is machine-generated.

We discovered a precise condition for synchronization in coupled oscillator networks. This finding applies to complex systems in physics, biology, and engineering, including smart grids.

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Area of Science:

  • Complex systems
  • Network science
  • Nonlinear dynamics

Background:

  • Synchronization is key in coupled oscillator networks, but the exact transition threshold remains unknown.
  • Existing models often use heterogeneous phase oscillators with diffusive, sinusoidal coupling.
  • Understanding synchronization is crucial across physics, biology, and engineering.

Purpose of the Study:

  • To derive a unique, concise, and closed-form condition for synchronization in nonlinear, nonequilibrium oscillator networks.
  • To provide an exact threshold for the transition from incoherence to synchrony.
  • To offer a condition applicable to diverse network topologies and parameters.

Main Methods:

  • Developed a novel, closed-form synchronization condition for dynamic networks.
  • Formulated the condition using network topology and parameters.
  • Introduced an equivalent, intuitive linear and static auxiliary system for analysis.

Main Results:

  • Presented an exact synchronization condition for fully nonlinear, nonequilibrium networks.
  • The condition is statistically correct for most networks and exact for specific topologies.
  • Demonstrated applicability to complex networks and smart grid scenarios.

Conclusions:

  • The derived condition offers significant improvements over existing methods.
  • Provides a unified framework for synchronization applicable to physics, biology, and engineering.
  • Highlights practical applications in areas like electrical power networks and smart grids.