Numerical Solution of Finite Kuramoto Model with Time-Dependent Coupling Strength: Addressing Synchronization Events of Nature
The synchronization of an ensemble of oscillators is a phenomenon present in systems of different fields, ranging from social to physical and biological systems. This phenomenon is often described mathematically by the Kuramoto model, which assumes oscillators of fixed natural frequencies connected...
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MDPI AG
2022-10-01
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| author | Dharma Raj Khatiwada |
| author_facet | Dharma Raj Khatiwada |
| author_sort | Dharma Raj Khatiwada |
| collection | DOAJ |
| description | The synchronization of an ensemble of oscillators is a phenomenon present in systems of different fields, ranging from social to physical and biological systems. This phenomenon is often described mathematically by the Kuramoto model, which assumes oscillators of fixed natural frequencies connected by an equal and uniform coupling strength, with an analytical solution possible only for an infinite number of oscillators. However, most real-life synchronization systems consist of a finite number of oscillators and are often perturbed by external fields. This paper accommodates the perturbation using a time-dependent coupling strength <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> in the form of a sinusoidal function and a step function using 32 oscillators that serve as a representative of finite oscillators. The temporal evolution of order parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> and phases <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>θ</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, key indicators of synchronization, are compared between the uniform and time-dependent cases. The identical trends observed in the two cases give an important indication that the synchrony persists even under the influence of external factors, something very plausible in the context of real-life synchronization events. The occasional boosting of coupling strength is also enough to keep the assembly of oscillators in a synchronized state persistently. |
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| last_indexed | 2024-03-09T21:27:13Z |
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| spelling | doaj.art-13de88ec715b44f4851a9ea6ba6920bb2023-11-23T21:04:47ZengMDPI AGMathematics2227-73902022-10-011019363310.3390/math10193633Numerical Solution of Finite Kuramoto Model with Time-Dependent Coupling Strength: Addressing Synchronization Events of NatureDharma Raj Khatiwada0School of STEM, College of Agriculture, Community and Sciences, Kentucky State University, 400 E Main St., Frankfort, KY 40601, USAThe synchronization of an ensemble of oscillators is a phenomenon present in systems of different fields, ranging from social to physical and biological systems. This phenomenon is often described mathematically by the Kuramoto model, which assumes oscillators of fixed natural frequencies connected by an equal and uniform coupling strength, with an analytical solution possible only for an infinite number of oscillators. However, most real-life synchronization systems consist of a finite number of oscillators and are often perturbed by external fields. This paper accommodates the perturbation using a time-dependent coupling strength <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> in the form of a sinusoidal function and a step function using 32 oscillators that serve as a representative of finite oscillators. The temporal evolution of order parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> and phases <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>θ</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, key indicators of synchronization, are compared between the uniform and time-dependent cases. The identical trends observed in the two cases give an important indication that the synchrony persists even under the influence of external factors, something very plausible in the context of real-life synchronization events. The occasional boosting of coupling strength is also enough to keep the assembly of oscillators in a synchronized state persistently.https://www.mdpi.com/2227-7390/10/19/3633Kuramoto modelfinite oscillatorssynchronization |
| spellingShingle | Dharma Raj Khatiwada Numerical Solution of Finite Kuramoto Model with Time-Dependent Coupling Strength: Addressing Synchronization Events of Nature Mathematics Kuramoto model finite oscillators synchronization |
| title | Numerical Solution of Finite Kuramoto Model with Time-Dependent Coupling Strength: Addressing Synchronization Events of Nature |
| title_full | Numerical Solution of Finite Kuramoto Model with Time-Dependent Coupling Strength: Addressing Synchronization Events of Nature |
| title_fullStr | Numerical Solution of Finite Kuramoto Model with Time-Dependent Coupling Strength: Addressing Synchronization Events of Nature |
| title_full_unstemmed | Numerical Solution of Finite Kuramoto Model with Time-Dependent Coupling Strength: Addressing Synchronization Events of Nature |
| title_short | Numerical Solution of Finite Kuramoto Model with Time-Dependent Coupling Strength: Addressing Synchronization Events of Nature |
| title_sort | numerical solution of finite kuramoto model with time dependent coupling strength addressing synchronization events of nature |
| topic | Kuramoto model finite oscillators synchronization |
| url | https://www.mdpi.com/2227-7390/10/19/3633 |
| work_keys_str_mv | AT dharmarajkhatiwada numericalsolutionoffinitekuramotomodelwithtimedependentcouplingstrengthaddressingsynchronizationeventsofnature |