scholarly journals Interpolation on the special orthogonal group with high-dimensional Kuramoto model

2021 ◽  
Vol 1208 (1) ◽  
pp. 012037
Author(s):  
Aladin Crnkić ◽  
Zinaid Kapić
Keyword(s):  
Orthogonal Group ◽  
Coupled Oscillators ◽  
Specific Problem ◽  
Rotation Group ◽  
Kuramoto Model ◽  
High Dimensional ◽  
Special Orthogonal Group ◽  
Large Populations ◽  
Object Rotation ◽  
The Kuramoto Model

Abstract The construction of smooth interpolation trajectories in different non-Euclidean spaces finds application in robotics, computer graphics, and many other engineering fields. This paper proposes a method for generating interpolation trajectories on the special orthogonal group SO(3), called the rotation group. Our method is based on a high-dimensional generalization of the Kuramoto model which is a well-known mathematical description of self-organization in large populations of coupled oscillators. We present the method through several simulations and visualize each simulation as trajectories on unit spheres S2. In addition, we applied our method to the specific problem of object rotation interpolation.

Epistemological Aspects of Simulation Models for Decision Support

2013 ◽  
Vol 5 (2) ◽  
pp. 55-77 ◽  
Author(s):  
Anthony H. Dekker
Keyword(s):  
Coupled Oscillators ◽  
True Belief ◽  
Simulation Models ◽  
Decision Makers ◽  
Kuramoto Model ◽  
Model Parameters ◽  
Theoretical Construct ◽  
Formal Definitions ◽  
Complex Models ◽  
The Kuramoto Model

In this paper, the author explores epistemological aspects of simulation with a particular focus on using simulations to provide recommendations to managers and other decision-makers. The author presents formal definitions of knowledge (as justified true belief) and of simulation. The author shows that a simple model, the Kuramoto model of coupled-oscillators, satisfies the simulation definition (and therefore generates knowledge) through a justified mapping from the real world. The author argues that, for more complex models, such a justified mapping requires three techniques: using an appropriate and justified theoretical construct; using appropriate and justified values for model parameters; and testing or other verification processes to ensure that the mapping is correctly defined. The author illustrates these three techniques with experiments and models from the literature, including the Long House Valley model of Axtell et al., the SAFTE model of sleep, and the Segregation model of Wilensky.

scholarly journals New forms of structure in ecosystems revealed with the Kuramoto model

2021 ◽  
Vol 8 (3) ◽  
Author(s):  
John Vandermeer ◽  
Zachary Hajian-Forooshani ◽  
Nicholas Medina ◽  
Ivette Perfecto
Keyword(s):  
Natural History ◽  
Complex Systems ◽  
Coupled Oscillators ◽  
Ecological Systems ◽  
Kuramoto Model ◽  
Chaotic Orbits ◽  
The Kuramoto Model ◽  
Convection Rolls ◽  
Analytical Tools ◽  
Insight Into

Ecological systems, as is often noted, are complex. Equally notable is the generalization that complex systems tend to be oscillatory, whether Huygens' simple patterns of pendulum entrainment or the twisted chaotic orbits of Lorenz’ convection rolls. The analytics of oscillators may thus provide insight into the structure of ecological systems. One of the most popular analytical tools for such study is the Kuramoto model of coupled oscillators. We apply this model as a stylized vision of the dynamics of a well-studied system of pests and their enemies, to ask whether its actual natural history is reflected in the dynamics of the qualitatively instantiated Kuramoto model. Emerging from the model is a series of synchrony groups generally corresponding to subnetworks of the natural system, with an overlying chimeric structure, depending on the strength of the inter-oscillator coupling. We conclude that the Kuramoto model presents a novel window through which interesting questions about the structure of ecological systems may emerge.

scholarly journals Distribution of Order Parameter for Kuramoto Model

2015 ◽  
Vol 3 (9) ◽  
pp. 52-68
Author(s):  
Hui Wu ◽  
Dongwook Kim
Keyword(s):  
Order Parameter ◽  
Initial Conditions ◽  
Oscillatory System ◽  
Kuramoto Model ◽  
Chemical System ◽  
Phase Oscillators ◽  
Numerical Result ◽  
Coupled Nonlinear Oscillators ◽  
Large Populations ◽  
The Kuramoto Model

The synchronization in large populations of interacting oscillators has been observed abundantly in nature, emergining in fields such as physical, biological and chemical system. For this reason, many scientists are seeking to understand the underlying mechansim of the generation of synchronous patterns in oscillatory system. The synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. The Kuramoto model can be used to understand the emergence of synchronization in nextworks of coupled, nonlinear oscillators. In particular, this model presents a phase transition from incoherence to synchronization. In this paper, we investigated the distribution of order parameter γ which describes the strength of synchrony of these oscillators. The larger the order parameter γ is, the more extent the oscillators are synchronized together. This order parameter γ is a critical parameter in the Kuramoto model. Kuramoto gave a initial estimate equation for the value of the order parameter by giving the value of the coupling constant. But our numerical results show that the distribution of the order parameter is slightly different from Kuramoto’s estimation. We gave an estimation for the distribution of order parameter for different values of initial conditions. We discussed how the numerical result will be distributed around Kuramoto’s analytical equation.

scholarly journals Viewing communities as coupled oscillators: elementary forms from Lotka and Volterra to Kuramoto

2020 ◽  
Author(s):  
Zachary Hajian-Forooshani ◽  
John Vandermeer
Keyword(s):  
Coupled Oscillators ◽  
Community Organization ◽  
Building Blocks ◽  
Kuramoto Model ◽  
Volterra Equations ◽  
Ecological Communities ◽  
Shared Resources ◽  
Resource Interactions ◽  
The Kuramoto Model ◽  
Consumer Resource

AbstractEcosystems and their embedded ecological communities are almost always by definition collections of oscillating populations. This is apparent given the qualitative reality that oscillations emerge from consumer-resource interactions, which are the simple building blocks for ecological communities. It is also likely always the case that oscillatory consumer-resource pairs will be connected to one another via trophic cross-feeding with shared resources or via competitive interactions among resources. Thus, one approach to understanding the dynamics of communities conceptualizes them as collections of oscillators coupled in various arrangements. Here we look to the pioneering work of Kuramoto on coupled oscillators and ask to what extent can his insights and approaches be translated to ecological systems. We explore all possible coupling arrangements of the simple case of three oscillator systems with both the Kuramoto model and with the classical Lotka-Volterra equations that are foundational to ecology. Our results show that the six-dimensional analogous Lotka-Volterra systems behave strikingly similarly to that of the corresponding Kuramoto systems across all possible coupling combinations. This qualitative similarity in the results between these two approaches suggests that a vast literature on coupled oscillators that has largely been ignored by ecologists may in fact be relevant in furthering our understanding of ecosystem and community organization.

scholarly journals The mathematics of asymptotic stability in the Kuramoto model

2018 ◽  
Vol 474 (2220) ◽  
pp. 20180467 ◽  
Author(s):  
Helge Dietert ◽  
Bastien Fernandez
Keyword(s):  
Asymptotic Stability ◽  
Coupled Oscillators ◽  
Continuum Limit ◽  
Heterogeneous Systems ◽  
Stationary Solutions ◽  
Kuramoto Model ◽  
The Kuramoto Model ◽  
The Continuum

Now a standard in Nonlinear Sciences, the Kuramoto model is the perfect example of the transition to synchrony in heterogeneous systems of coupled oscillators. While its basic phenomenology has been sketched in early works, the corresponding rigorous validation has long remained problematic and was achieved only recently. This paper reviews the mathematical results on asymptotic stability of stationary solutions in the continuum limit of the Kuramoto model, and provides insights into the principal arguments of proofs. This review is complemented with additional original results, various examples, and possible extensions to some variations of the model in the literature.

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