Browsing by Author "Tseng, Jui-Pin"

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A General Approach to Synchronization of Coupled Cells
(2013)
Shih, Chih-Wen
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Tseng, Jui-Pin
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應用數學系
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Department of Applied Mathematics
This investigation presents a general framework to establish synchronization of coupled cells and coupled systems. Each individual subsystem is represented by nonlinear differential equations with or without internal or intracellular delay. A general coupling function is employed to depict the communication or interaction between subsystems or cells. Under this framework, the problem of establishing the synchronization for delayed coupled nonlinear systems is transformed to solving a corresponding linear system of algebraic equations. We start by considering a cell-to-cell system under symmetric coupling to present the main idea of the approach. The framework is then extended to the N-cell system under circulant coupling. Delay-dependent, delay-independent, and network-scale-dependent criteria for global synchronization will be established, respectively. The developed scheme can accommodate a wide range of coupled systems. We demonstrate the applications of the present approach to establish synchronization for a gene regulation model, a neuronal model, and some neural networks.
Convergent dynamics for multistable delayed neural networks
(2008-10-01)
Shih, Chih-Wen
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Tseng, Jui-Pin
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應用數學系
;
Department of Applied Mathematics
This investigation aims at developing a methodology to establish convergence of dynamics for delayed neural network systems with multiple stable equilibria. The present approach is general and can be applied to several network models. We take the Hopfield-type neural networks with both instantaneous and delayed feedbacks to illustrate the idea. We shall construct the complete dynamical scenario which comprises exactly 2(n) stable equilibria and exactly (3(n)-2(n)) unstable equilibria for the n-neuron network. In addition, it is shown that every solution of the system converges to one of the equilibria as time tends to infinity. The approach is based on employing the geometrical structure of the network system. Positively invariant sets and componentwise dynamical properties are derived under the geometrical configuration. An iteration scheme is subsequently designed to confirm the convergence of dynamics for the system. Two examples with numerical simulations are arranged to illustrate the present theory.
FROM APPROXIMATE SYNCHRONIZATION TO IDENTICAL SYNCHRONIZATION IN COUPLED SYSTEMS
(2020-09-01)
Shih, Chih-Wen
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Tseng, Jui-Pin
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應用數學系
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Department of Applied Mathematics
We establish a framework to investigate approximate synchronization of coupled systems under general coupling schemes. The units comprising the coupled systems may be nonidentical and the coupling functions are nonlinear with delays. Both delay-dependent and delay-independent criteria for approximate synchronization are derived, based on an approach termed sequential contracting. It is explored and elucidated that the synchronization error, the distance between the asymptotic state and the synchronous set, decreases with decreasing difference between subsystems, difference between the row sums of connection matrix, and difference of coupling time delays between different units. This error vanishes when these factors decay to zero, and approximate synchronization becomes identical synchronization for the coupled system comprising identical subsystems and connection matrix with identical row sums, and with identical coupling delays. The application of the present theory to nonlinearly coupled heterogeneous FitzHugh-Nagumo neurons is illustrated. We extend the analysis to study approximate synchronization and asymptotic synchronization for coupled Lorenz systems and show that for some coupling schemes, the synchronization error decreases as the coupling strength increases, whereas in another case, the error remains at a substantial level for large coupling strength.
Global consensus for discrete-time competitive systems
(2009-07-15)
Shih, Chih-Wen
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Tseng, Jui-Pin
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應用數學系
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Department of Applied Mathematics
Grossberg established a remarkable convergence theorem for a class of competitive systems without knowing and using Lyapunov function for the systems. We present the parallel investigations for the discrete-time version of the Grossberg's model. Through developing an extended component-competing analysis for the coupled system, without knowing a Lyapunov function and applying the LaSalle's invariance principle, the global pattern formation or the so-called global consensus for the system can be achieved. A numerical simulation is performed to illustrate the present theory. (C) 2008 Elsevier Ltd. All rights reserved.
GLOBAL SYNCHRONIZATION AND ASYMPTOTIC PHASES FOR A RING OF IDENTICAL CELLS WITH DELAYED COUPLING
(2011)
Shih, Chih-Wen
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Tseng, Jui-Pin
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應用數學系
;
Department of Applied Mathematics
We consider a neural network which consists of a ring of identical neurons coupled with their nearest neighbors and is subject to self-feedback delay and transmission delay. We present an iteration scheme to analyze the synchronization and asymptotic phases for the system. Delay-independent, delay-dependent, and scale-dependent criteria are formulated for the global synchronization and global convergence. Under this setting, the possible asymptotic dynamics include convergence to single equilibrium, multiple equilibria, and synchronous oscillation. The study aims at elucidating the effects from the scale of network, self-decay, self-feedback strength, coupling strength, and delay magnitudes upon synchrony, convergent dynamics, and oscillation of the network. The disparity between the contents of synchrony induced by distinct factors is investigated. Two different types of multistable dynamics are distinguished. Moreover, oscillation and desynchronization induced by delays are addressed. We answer two conjectures in the literature.
Multistability for Delayed Neural Networks via Sequential Contracting
(2015-12-01)
Cheng, Chang-Yuan
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Lin, Kuang-Hui
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Shih, Chih-Wen
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Tseng, Jui-Pin
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應用數學系
In this paper, we explore a variety of new multistability scenarios in the general delayed neural network system. Geometric structure embedded in equations is exploited and incorporated into the analysis to elucidate the underlying dynamics. Criteria derived from different geometric configurations lead to disparate numbers of equilibria. A new approach named sequential contracting is applied to conclude the global convergence to multiple equilibrium points of the system. The formulation accommodates both smooth sigmoidal and piecewise-linear activation functions. Several numerical examples illustrate the present analytic theory.
Synchronized oscillations in a mathematical model of segmentation in zebrafish
(2012-04-01)
Liao, Kang-Ling
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Shih, Chih-Wen
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Tseng, Jui-Pin
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應用數學系
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Department of Applied Mathematics
Somitogenesis is a process for the development of somites which are transient, segmental structures that lie along the anterior-posterior axis of vertebrate embryos. The pattern of somites is governed by the segmentation clock and its timing is controlled by the clock genes which undergo synchronous oscillation over adjacent cells in the posterior presomitic mesoderm (PSM). In this paper, we analyze a mathematical model which depicts the kinetics of the zebrafish segmentation clock genes subject to direct autorepression by their own products under time delay, and cell-to-cell interaction through Delta-Notch signalling. Our goal is to elucidate how synchronous oscillations are generated for the cells in the posterior PSM, and how oscillations are arrested for the cells in the anterior PSM. For this system of delayed equations, an iteration technique is employed to derive the global convergence to the synchronous equilibrium, which corresponds to the oscillation-arrested. By applying the delay Hopf bifurcation theory and the center manifold theorem, we derive the criteria for the existence of stable synchronous oscillations for the cells at the tail bud of the PSM. Our analysis provides the basic parameter ranges and delay magnitudes for stable synchronous, asynchronous oscillation and oscillation-arrested. We exhibit how synchronous oscillations are affected by the degradation rates and delays. Extended from the analytic theory, further numerical findings linked to the segmentation process are presented.