arXiv subjects

Hansol Park

Publications and source records attributed to Hansol Park.

At least 37 records · Page 2Linked to original sources

arXiv2022

Mean field Kuramoto models on graphs

One of a classical synchronization model is the Kuramoto model. We propose both first and second order Kuramoto dynamical models on graphs using discrete optimal transport dynamics. We analyze the synchronization behaviors for some examples of Kuramoto models on graphs. We also provide a generalized Hopf-Cole transformation for discrete optimal transport systems. Focus on the two points graph, we derive analytical formulas of the Kuramoto dynamics with various potential induced from entropy functionals. Several numerical examples for the Kuramoto model on general graphs are presented.

math.DS↗

arXiv2021

An algebraic approach for the weak coupling of multiple Lohe tensor models

We present a systematic algebraic approach for the weak coupling of Cauchy problems to multiple Lohe tensor models. For this, we identify an admissible Cauchy problem to the Lohe tensor (LT) model with a characteristic symbol consisting of four tuples in terms of a size vector, a natural frequency tensor, a coupling strength tensor and admissible initial configuration. In this way, the collection of all admissible Cauchy problems to the LT models is equivalent to the space of characteristic symbols. On the other hand, we introduce a binary operation, namely "{\it fusion operation}" as a binary operation between the characteristic symbols. It turns out that the fusion operation satisfies the associativity and admits the identity element in the space of characteristic symbols which naturally forms a monoid. By virtue of the fusion operation, the weakly coupled system of multiple LT models can be obtained by applying the fusion operation of multiple characteristic symbols corresponding to the LT models. As a concrete example, we consider a weak coupling of the swarm sphere model and the Lohe matrix model, and provide a sufficient framework leading to emergent dynamics to the proposed weakly coupled model.

math.DS↗

arXiv2021

Emergent behaviors of discrete Lohe aggregation flows

The Lohe sphere model and the Lohe matrix model are prototype continuous aggregation models on the unit sphere and the unitary group, respectively. These models have been extensively investigated in recent literature. In this paper, we propose several discrete counterparts for the continuous Lohe type aggregation models and study their emergent behaviors using the Lyapunov function method. For suitable discretization of the Lohe sphere model, we employ a scheme consisting of two steps. In the first step, we solve the first-order forward Euler scheme, and in the second step, we project the intermediate state onto the unit sphere. For this discrete model, we present a sufficient framework leading to the complete state aggregation in terms of system parameters and initial data. For the discretization of the Lohe matrix model, we use the Lie group integrator method, Lie-Trotter splitting method and Strang splitting method to propose three discrete models. For these models, we also provide several analytical frameworks leading to complete state aggregation and asymptotic state-locking.

math-ph↗

arXiv2021

Relaxation dynamics of SIR-flocks with random epidemic states

We study the collective dynamics of a multi-particle system with three epidemic states as an internal state. For the collective modeling of active particle system, we adopt modeling spirits from the swarmalator model and the SIR epidemic model for the temporal evolution of particles' position and internal states. Under suitable assumptions on system parameters and non-collision property of initial spatial configuration, we show that the proposed model does not admit finite-time collisions so that the standard Cauchy-Lipschitz theory can be applied for the global well-posedness. For the relaxation dynamics, we provide several sufficient frameworks leading to the relaxation dynamics of the proposed model. The proposed sufficient frameworks are formulated in terms of system parameters and initial configuration. Under such sufficient frameworks, we show that the state configuration relaxes to the fixed constant configuration via the exponentially perturbed gradient system and explicit dynamics of the SIR model. We present explicit lower and upper bounds for the minimal and maximal relative distances.

math.DS↗

arXiv2021

A velocity alignment model on quotient spaces of the Euclidean space

The Cucker-Smale(CS) model is a velocity alignment model, and this model also has been generalized on general manifolds. We modify the CS model on manifolds to get rid of a-priori condition on particles' positions and conditions on communication functions. Since the shortest geodesic is used to define an interaction between two particles, if there exist two or more than two shortest geodesics, then the system is not well-defined. In this paper, instead of using the shortest geodesic to define an interaction between two particles, we use all geodesics to define an interaction. From this assumption, we can relax the a-priori condition and conditions on communication functions. We also explain the relationship between the suggested model and previous models. Finally, we provide some emergent behaviors on some specific manifolds(e.g. flat torus, flat Möbius strip, and flat Klein bottle). From these results, we can discuss the effect of the topology of the domain.

math-ph↗

arXiv2021

Emergent behaviors of the kinetic Lohe Hermitian sphere model

We study a global well-posedness of measure-valued solutions to the kinetic Lohe Hermitian sphere(LHS) model derived from the Lohe tensor(LT) model on the set of rank-1 complex tensors(i.e. complex vectors) with the same size and investigate emergent behaviors. The kinetic LHS model corresponds to a complex analogue of the kinetic LS model which has been extensively studied in the literature on the aggregation modeling of Lohe particles on the unit sphere in Euclidean space. In this paper, we provide several frameworks in terms of system parameters and initial data leading to the local and global well-posedness of measure-valued solutions. In particular, we show emergent behaviors of the kinetic LHS model with the same free flows by analyzing the temporal evolution of the order parameter.

math.AP↗

arXiv2021

Existence and emergent dynamics of quadratically separable states to the Lohe tensor model

A tensor is a multi-dimensional array of complex numbers, and the Lohe tensor model is an aggregation model on the space of tensors with the same rank and size. It incorporates previously well-studied aggregation models on the space of low-rank tensors such as the Kuramoto model, Lohe sphere and matrix models as special cases. Due to its structural complexities in cubic interactions for the Lohe tensor model, explicit construction of solutions with specific structures looks daunting. Recently, we obtained completely separable states by associating rank-1 tensors. In this paper, we further investigate another type of solutions, namely "{\it quadratically separable states}" consisting of tensor products of matrices and their component rank-2 tensors are solutions to the double matrix model whose emergent dynamics can be studied using the same methodology of the Lohe matrix model.

math-ph↗

arXiv2021

Well-posedness and asymptotic behaviour of an aggregation model with intrinsic interactions on sphere and other manifolds

We investigate a model for collective behaviour with intrinsic interactions on Riemannian manifolds. We establish the well-posedness of measure solutions (defined via mass transport) on sphere, as well as investigate the mean-field particle approximation. We study the long-time behaviour of solutions to the model on sphere, where the primary goal is to establish sufficient conditions for a consensus state to form asymptotically. Well-posedness of solutions and the formation of consensus are also investigated for other manifolds (e.g., a hypercylinder).

math.AP↗

arXiv2021

The Watanabe-Strogatz transform and constant of motion functionals for kinetic vector models

We present a kinetic version of the Watanabe-Strogatz(WS) transform for vector models in this paper. From the generalized WS-transform, we obtain the cross-ratio type constant of motion functionals for kinetic vector models under suitable conditions. We present the sufficient and necessary conditions for the existence of the suggested constant of motion functional. As an application of the constant of motion functional, we provide the instability of bipolar states of the kinetic swarm sphere model. We also provide the WS-transform and constant of motion functional for non-identical kinetic vector models.

math.AP↗

arXiv2021

Emergent behaviors of rotation matrix flocks

We derive an explicit form for the Cucker-Smale (CS) model on the special orthogonal group $\mathrm{SO}(3)$ by identifying closed form expressions for geometric quantities such as covariant derivative and parallel transport in exponential coordinates. We study the emergent dynamics of the model by using a Lyapunov functional approach and La Salle's invariance principle. Specifically, we show that velocity alignment emerges from some admissible class of initial data, under suitable assumptions on the communication weight function. We characterize the $ω$-limit set of the dynamical system and identify a dichotomy in the asymptotic behavior of solutions. Several numerical examples are provided to support the analytical results.

math.DS↗

arXiv2021

KS-pies: Kohn-Sham Inversion Toolkit

A Kohn-Sham (KS) inversion determines a KS potential and orbitals corresponding to a given electron density, a procedure that has applications in developing and evaluating functionals used in density functional theory. Despite the utility of KS inversions, application of these methods among the research community is disproportionately small. We implement the KS inversion methods of Zhao-Morrison-Parr and Wu-Yang in a framework that simplifies analysis and conversion of the resulting potential in real-space. Fully documented Python scripts integrate with PySCF, a popular electronic structure prediction software, and Fortran alternatives are provided for computational hot spots.

physics.comp-ph↗

arXiv2021

Generalization of the Winfree model to the high-dimensional sphere and its emergent dynamics

We present a high-dimensional Winfree model in this paper. The Winfree model is the first mathematical model for synchronization on the circle. We generalize this model to the high-dimensional sphere and we call it "the Winfree sphere model." We restricted the support of the influence function in the neighborhood of the attraction point with a small diameter to mimic the influence function as the Dirac-delta distribution. Restricting the support of the influence function allows several new conditions of the complete phase-locking states for the identical Winfree sphere model compare to previous results. We also provide the exponential $\ell^1$-stability and the existence of the equilibrium solution to obtain the complete oscillator death state of the Winfree sphere model.

math-ph↗

arXiv2021

Asymptotic interplay of states and adapted coupling gains in the Lohe hermitian sphere model

We study emergent dynamics of the Lohe hermitian sphere (LHS) model with the same free flows under the dynamic interplay between state evolution and adaptive couplings. The LHS model is a complex counterpart of the Lohe sphere (LS) model on the unit sphere in Euclidean space, and when particles lie in the Euclidean unit sphere embedded in $\bbc^{d+1}$, it reduces to the Lohe sphere model. In the absence of interactions between states and coupling gains, emergent dynamics have been addressed in [22]. In this paper, we further extend earlier results in the aforementioned work to the setting in which the state and coupling gains are dynamically interrelated via two types of coupling laws, namely anti-Hebbian and Hebbian coupling laws. In each case, we present two sufficient frameworks leading to complete aggregation depending on the coupling laws, when the corresponding free flow is the same for all particles.

math-ph↗

arXiv2021

Emergent behaviors of homogeneous Lohe Hermitian sphere particles under time-delayed interactions

We study emergent behaviors of the Lohe hermitian sphere(LHS) model with a time-delay for a homogeneous ensemble. The LHS model is a complex counterpart of the Lohe sphere(LS) aggregation model on the unit sphere in Euclidean space, and it describes the aggregation of particles on the unit hermitian sphere in $\mathbb{C}^d$ with $d \geq 2$, Recently it has been introduced by two authors of this work as a special case of the Lohe tensor model [23]. When the coupling gain pair satisfies a specific linear relation, namely the Stuart-Landau(SL) coupling gain pair, it can be embedded into the LS model on $\mathbb{R}^{2d}$. In this work, we show that if the coupling gain pair is close to the SL coupling pair case, the dynamics of the LHS model exhibits an emergent aggregate phenomenon via the interplay between time-delayed interactions and nonlinear coupling between states. For this, we present several frameworks for complete aggregation and practical aggregation in terms of initial data and system parameters using the Lyapunov functional approach.

math-ph↗

arXiv2020

Emergent behaviors in group ring flocks

We present a first-order aggregation model on a group ring, and study its asymptotic dynamics. In a positive coupling strength regime, we show that the flow generated by the proposed model tends to an equilibrium manifold asymptotically. For this, we introduce a Lyapunov functional which is non-increasing along the flow, and using the temporal decay of the nonlinear functional and the LaSalle invariance principle, we show that the flow converges toward an equilibrium manifold asymptotically. We also show that the structure of an equilibrium manifold is strongly dependent on the structure of an underlying group.

math.DS↗

arXiv2020

Constants of motion for the finite-dimensional Lohe type models with frustration and applications to emergent dynamics

We present constants of motion for the finite-dimensional Lohe type aggregation models with frustration and we apply them to analyze the emergence of collective behaviors. The Lohe type models have been proposed as possible non-abelian and higher-dimensional generalizations of the Kuramoto model, which is a prototype phase model for synchronization. The aim of this paper is to study the emergent collective dynamics of these models under the effect of (interaction) frustration, which generalizes phase-shift frustrations in the Kuramoto model. To this end, we present constants of motion, i.e., conserved quantities along the flow generated by the models under consideration, and, from the perspective of the low-dimensional dynamics thus so obtained, derive several results concerning the emergent asymptotic patterns of the Kuramoto and Lohe sphere models.

math-ph↗

arXiv2020

Collective behaviors of the Lohe hermitian sphere model with inertia

We present a second-order extension of the first-order Lohe hermitian sphere(LHS) model and study its emergent asymptotic dynamics. Our proposed model incorporates an inertial effect as a second-order extension. The inertia term can generate an oscillatory behavior of particle trajectory in a small time interval(initial layer) which causes a technical difficulty for the application of monotonicity-based arguments. For emergent estimates, we employ two-point correlation function which is defined as an inner product between positions of particles. For a homogeneous ensemble with the same frequency matrix, we provide two sufficient frameworks in terms of system parameters and initial data to show that two-point correlation functions tend to the unity which is exactly the same as the complete aggregation. In contrast, for a heterogeneous ensemble with distinct frequency matrices, we provide a sufficient framework in terms of system parameters and initial data, which makes two-point correlation functions close to unity by increasing the principal coupling strength.

math-ph↗

arXiv2020

An intrinsic aggregation model on the special orthogonal group $\mathrm{SO}(3)$: well-posedness and collective behaviours

We investigate an aggregation model with intrinsic interactions on the special orthogonal group $SO(3)$. We consider a smooth interaction potential that depends on the squared intrinsic distance, and establish local and global existence of measure-valued solutions to the model via optimal mass transport techniques. We also study the long-time behaviours of such solutions, where we present sufficient conditions for the formation of asymptotic consensus. The analytical results are illustrated with numerical experiments that exhibit various asymptotic patterns.

math.AP↗