One day workshop on Mathematical Modelling of Swarming

28th December 2017, Department of Mathematics, Inha University

This workshop will be devoted to the mathematical modeling of swarming behaviors, such as aggregation of bacteria, flocking, of birds, herding of sheep, and synchronization of fireflies, etc. Among the numerous areas of applications, we will particularly focus on the mathematical modelling describing the synchronization phenomena of classical and quantum oscillators, and the herding phenomena in financial markets. Graduate students and postdocs are particularly encouraged to participate in. Please send an email to ypchoi at if you need further information.

Invited speakers:

Jinwook Jung(Seoul National University, Seoul, South Korea)
Dohyun Kim(Seoul National University, Seoul, South Korea)
Jaeseung Lee(Seoul National University, Seoul, South Korea)
Jinyeong Park(Universidad de Granada, Granada, Spain)
Seok-Bae Yun(Sungkyunkwan University, Suwon, South Korea)

Talk title/abstract and Schedule:

December 28, 2017 (15:00-16:00, 5E102)

A particle model for the herding phenomena induced by dynamic market signals

Seok-Bae Yun
Deparment of Mathematics, Sungkyunkwan University

Abstract: In this talk, we consider the herding phenomena in financial markets arising from the combined effect of (1) non-coordinated collective interactions between the market players and (2) concurrent reactions of market players to dynamic market signals. By interpreting the expected rate of return of an asset and the favorability on that asset as position and velocity in phase space, we construct an agent-based particle model for herding behavior in finance. We then define two types of herding functionals using this model, and show that they satisfy a Gronwall type estimate and a LaSalle type invariance property respectively, leading to the herding behavior of the market players. Various numerical tests are presented to numerically verify these results.

December 28, 2017 (16:00-16:20, 5E102)

Remarks on the stability and instability properties of the mean-field equations with diffusion

Jaeseung Lee
Deparment of Mathematics, Seoul National University

Abstract: We present stability and instability properties of two types of mean-field equation with diffusion: sphere Lohe model and Kuramoto-Sakaguchi-Fokker-Planck equation with frustration. First, we introduce the derivation of sphere Lohe model from particle Lohe model based on Sznitman's theory and give sufficient framework leading to the asymptotic stability of the incoherent state. For KS-FP equation with frustration, we study both stability and instability properties of the incoherent state. For the instability, we improve the previous result, i.e., we relax the constraint on the natural frequency distribution function.

December 28, 2017 (16:25-16:45, 5E102)

Emergent behaviors of the Schrödinger-Lohe model on cooperative-competitive networks

Dohyun Kim
Deparment of Mathematics, Seoul National University

Abstract: We present several sufficient frameworks leading to the emergent behaviors of the coupled Schrödinger-Lohe (S-L) model under the same one-body external potential on cooperative-competitive networks. The S-L model was first introduced as a possible phenomenological model exhibiting quantum synchronization and its emergent dynamics on all-to-all cooperative networks has been treated via two distinct approaches, Lyapunov functional approach and the finite-dimensional reduction based on pairwise correlations. In this talk, we introduce three types of the network structures(cooperative network, competitive network and cooperative-competitive network) and present their emergent behaviors which show different dynamic patterns according to the network structures.

December 28, 2017 (16:50-17:10, 5E102)

Remarks on the slow relaxation for the fractional Kuramoto model for synchronization

Jinwook Jung
Deparment of Mathematics, Seoul National University

Abstract: The collective behavior of an oscillatory system is ubiquitous in our nature, and one interesting issue in the dynamics of many-body oscillatory systems is the relaxation dynamics toward relative equilibria such as phase-locked states. For the Kuramoto model, relaxation dynamics occurs exponentially fast for generic initial data. However, some synchronization phenomena observed in our nature exhibits a slow sub-exponential relaxation. Thus, as one of possible attempts for such slow relaxation, second-order inertia term was added to the Kuramoto model in the previous literature so that the resulting second-order model can exhibit a slow relaxation dynamics for some range of inertia and coupling strength. In this paper, we present another Kuramoto type model exhibiting a slow algebraic relaxation. More precisely, our proposed model replaces the usual derivative by Caputo fractional derivative in the original Kuramoto model. For this new model, we present several sufficient frameworks for fractional complete synchronization and practical synchronization.

December 28, 2017 (17:20-18:20, 5E102)

Hebbian learning in the Kuramoto model with regular and singular weighted couplings

Jinyeong Park
Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada

Abstract: We study the synchronization of a generalized Kuramoto system in which the coupling weights are determined by the phase differences between oscillators. We employ the fast-learning regime in a Hebbian-like plasticity rule so that the interaction between oscillators is enhanced by the approach of phases. We present the dynamics of the system equipped with both regular and singular types of coupling weights. Especially, for singular weights, we inspect the well-posedness of the system for subcritical and critical cases. We also study the emergence of synchronization.

List of participants:

Gi-Chan Bae (Sungkyunkwan University, Suwon, South Korea)
Seung-Yeon Cho (Sungkyunkwan University, Suwon, South Korea)
Kwangseok Choe (Inha University, Incheon, South Korea)
Young-Pil Choi (Inha University, Incheon, South Korea)
Byung-Hoon Hwang (Sungkyunkwan University, Suwon, South Korea)
Yong-Gwan Ji (Inha University, Incheon, South Korea)
Junhong Jo (Inha University, Incheon, South Korea)
Jinwook Jung (Seoul National University, Seoul, South Korea)
Dohyun Kim (Seoul National University, Seoul, South Korea)
Seongtag Kim (Inha University, Incheon, South Korea)
Hyundae Lee (Inha University, Incheon, South Korea)
Jaeseung Lee (Seoul National University, Seoul, South Korea)
Myeong-Su Lee (Sungkyunkwan University, Suwon, South Korea)
Xiaofei Li (Inha University, Incheon, South Korea)
Chan Ho Min (Seoul National University, Seoul, South Korea)
Jinyeong Park (Universidad de Granada, Granada, Spain)
Sung-Jun Son (Sungkyunkwan University, Suwon, South Korea)
Seok-Bae Yun (Sungkyunkwan University, Suwon, South Korea)
Yinglong Zhang (Seoul National University, Seoul, South Korea)
Wonjun Chang (Inha University, Incheon, South Korea)


Young-Pil Choi (Inha University)

This scientific event is sponsored by the BRL(Basic Research Lab) grant from Institute of Applied Mathematics at Department of Mathematics, Inha University.