Browsing by Author "Cheng, Chang-Yuan"

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Coexistence and extinction for two competing species in patchy environments
(2019-01-01)
Cheng, Chang-Yuan
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Lin, Kuang-Hui
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Shih, Chih-Wen
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應用數學系
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Department of Applied Mathematics
A system of two competing species u and v that diffuse over a two-patch environment is investigated. When u-species has smaller birth rate in the first patch and larger birth rate in the second patch than v-species, and the average birth rate for u-species is larger than or equal to v-species, it was shown in a previous publication that two species coexist in a slow diffusion environment, whereas u-species drives v-species into extinction in a fast diffusion environment. In this paper, we analyze global dynamics and bifurcations for the same model with identical order of birth rates, but with opposite order of average birth rates, i.e., the average birth rate of u-species is less than that of v-species. We observe richer dynamics with two scenarios, depending on the relative difference between the variation in the birth rates of v-species on two patches and the variation in the average birth rates of two species. When the variation in average birth rates is relatively large, there is no stability switch for the semitrivial equilibria. On the other hand, such a stability switch takes place when the variation in average birth rates is relatively mild. In both cases, v-species, with larger average birth rate, prevails in a fast diffusion environment, whereas in a slow diffusion environment, the two species can coexist or u-species that has the greatest birth rate among both species and patches will persist and drive v-species to extinction.
Complete Stability in Multistable Delayed Neural Networks
(2009-03-01)
Cheng, Chang-Yuan
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Shih, Chih-Wen
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應用數學系
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Department of Applied Mathematics
We investigate the complete stability for multistable delayed neural networks. A new formulation modified from the previous studies on multistable networks is developed to derive componentwise dynamical property. An iteration argument is then constructed to conclude that every solution of the network converges to a single equilibrium as time tends to infinity. The existence of 3(n) equilibria and 2(n) positively invariant sets for the n-neuron system remains valid under the new formulation. The theory is demonstrated by a numerical illustration.
Multistability and convergence in delayed neural networks
(2007-01-01)
Cheng, Chang-Yuan
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Lin, Kuang-Hui
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Shih, Chih-Wen
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應用數學系
;
Department of Applied Mathematics
We present the existence of 2(n) stable stationary solutions for a general n-dimensional delayed neural networks with several classes of activation functions. The theory is obtained through formulating parameter conditions motivated by a geometrical observation. Positively invariant regions for the flows generated by the system and basins of attraction for these stationary solutions are established. The theory is also extended to the existence of 2(n) limit cycles for the n-dimensional delayed neural networks with time-periodic inputs. It is further confirmed that quasiconvergence is generic for the networks through justifying the strongly order preserving property as the self-feedback time lags are small for the neurons with negative self-connection weights. Our theory on existence of multiple equilibria is then incorporated into this quasiconvergence for the network. Four numerical simulations are presented to illustrate our theory. (c) 2006 Elsevier B.V. All rights reserved.
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
;
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.