Related Experiment Video
Updated: Jun 30, 2026
07:56
Preparation of Liquid Crystal Networks for Macroscopic Oscillatory Motion Induced by Light
Published on: September 20, 2017
11.7K
Synchronization Transition of the Second-Order Kuramoto Model on Lattices.
1Centre for Energy Research, Institute of Technical Physics and Materials Science, P.O. Box 49, H-1525 Budapest, Hungary.
Entropy (Basel, Switzerland)
|January 21, 2023
Summary
The second-order Kuramoto equation shows hybrid phase transitions in 2D and 3D lattices. Unlike the first-order equation, its synchronization behavior is less understood, with distinct transitions for oscillator phases and frequencies.
Area of Science:
- Physics
- Complex Systems
- Nonlinear Dynamics
Background:
- The second-order Kuramoto equation models coupled oscillators with inertia, relevant to power grid synchronization.
- Its synchronization transition behavior is less understood compared to the first-order Kuramoto equation.
- For Gaussian self-frequencies, the transition is discontinuous, unlike the continuous transition in the first-order model.
Purpose of the Study:
- Investigate the synchronization transition of the second-order Kuramoto equation on large 2D and 3D lattices.
- Provide numerical evidence for hybrid phase transitions.
- Determine critical exponents and lower critical dimensions for phase and frequency transitions.
Main Methods:
- Numerical simulations on large 2D and 3D lattices.
- Analysis of oscillator phases (θi) and frequency spread.
- Investigation of initial conditions: aligned vs. random phase distributions.
Main Results:
- Evidence of hybrid phase transitions: crossover for phases and real phase transition for frequency in 3D.
- Estimated lower critical dimensions: dlO=2 for frequencies and dlR=4 for phases.
- Frequency spread decay rates: ~t-d/2 for aligned initial states and ~t-1.8(1) for random initial states in 3D due to nonlinearities.
Conclusions:
- The second-order Kuramoto equation exhibits complex synchronization dynamics distinct from its first-order counterpart.
- Hybrid phase transitions and dimension-dependent behaviors are key features.
- Nonlinear effects significantly influence transition dynamics, especially with random initial conditions.
Related Concept Videos
01:22Second Derivatives and Laplace Operator
The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
Consider a scalar function. The curl of its...
Consider a scalar function. The curl of its...
01:17Second-Order Circuits
Integrating two fundamental energy storage elements in electrical circuits results in second-order circuits, encompassing RLC circuits and circuits with dual capacitors or inductors (RC and RL circuits). Second-order circuits are identified by second-order differential equations that link input and output signals.
Input signals typically originate from voltage or current sources, with the output often representing voltage across the capacitor and/or current through the inductor. For example, in...
Input signals typically originate from voltage or current sources, with the output often representing voltage across the capacitor and/or current through the inductor. For example, in...
01:20Second 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...
By reinterpreting the system, one can derive the closed-loop transfer function, which...
01:18Second 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...
If ζ...
01:26Transmission-Line Differential Equations
Transmission lines are essential components of electrical power systems. They are characterized by the distributed nature of resistance (R), inductance (L), and capacitance (C) per unit length. To analyze these lines, differential equations are employed to model the variations in voltage and current along the line.
Line Section Model
A circuit representing a line section of length Δx helps in understanding the transmission line parameters. The voltage V(x) and current i(x) are measured from the...
Line Section Model
A circuit representing a line section of length Δx helps in understanding the transmission line parameters. The voltage V(x) and current i(x) are measured from the...
01:12Bewley Lattice Diagram
The Bewley lattice diagram, developed by L. V. Bewley, effectively organizes the reflections occurring during transmission-line transients. It visually represents how voltage waves propagate and reflect within a transmission line, making it easier to understand the complex interactions that occur.