The Kuramoto model is a mathematical framework used to study synchronization phenomena in populations of interacting oscillators. It describes how individual oscillators, each with its own natural frequency, can synchronize their behavior through interactions. The model is particularly useful for analyzing synchronization in systems ranging from biological and chemical processes to social and technological systems. The model is based on phase oscillators, where each oscillator's phase is influenced by the phases of others. The key parameter in the model is the coupling strength, which determines how strongly oscillators influence each other. When the coupling strength is sufficiently large, the oscillators synchronize to a common frequency, leading to collective behavior. The model has been extended to include noise, time delays, and other factors, allowing for a broader range of applications. The analysis of the model includes both theoretical and numerical approaches, and it has been applied to various systems, including neural networks, Josephson junctions, and chemical oscillators. The model's ability to describe synchronization in diverse contexts makes it a fundamental tool in the study of collective behavior in complex systems.The Kuramoto model is a mathematical framework used to study synchronization phenomena in populations of interacting oscillators. It describes how individual oscillators, each with its own natural frequency, can synchronize their behavior through interactions. The model is particularly useful for analyzing synchronization in systems ranging from biological and chemical processes to social and technological systems. The model is based on phase oscillators, where each oscillator's phase is influenced by the phases of others. The key parameter in the model is the coupling strength, which determines how strongly oscillators influence each other. When the coupling strength is sufficiently large, the oscillators synchronize to a common frequency, leading to collective behavior. The model has been extended to include noise, time delays, and other factors, allowing for a broader range of applications. The analysis of the model includes both theoretical and numerical approaches, and it has been applied to various systems, including neural networks, Josephson junctions, and chemical oscillators. The model's ability to describe synchronization in diverse contexts makes it a fundamental tool in the study of collective behavior in complex systems.