Browsing by Author "Juang, Jonq"

Now showing 1 - 20 of 22

Analysis of a general SIS model with infective vectors on the complex networks
(2015-11-01)
Juang, Jonq
;
Liang, Yu-Hao
;
應用數學系
;
Department of Applied Mathematics
A general SIS model with infective vectors on complex networks is studied in this paper. In particular, the model considers the linear combination of three possible routes of disease propagation between infected and susceptible individuals as well as two possible transmission types which describe how the susceptible vectors attack the infected individuals. A new technique based on the basic reproduction matrix is introduced to obtain the following results. First, necessary and sufficient conditions are obtained for the global stability of the model through a unified approach. As a result, we are able to produce the exact basic reproduction number and the precise epidemic thresholds with respect to three spreading strengths, the curing strength or the immunization strength all at once. Second, the monotonicity of the basic reproduction number and the above mentioned epidemic thresholds with respect to all other parameters can be rigorously characterized. Finally, we are able to compare the effectiveness of various immunization strategies under the assumption that the number of persons getting vaccinated is the same for all strategies. In particular, we prove that in the scale-free networks, both targeted and acquaintance immunizations are more effective than uniform and active immunizations and that active immunization is the least effective strategy among those four. We are also able to determine how the vaccine should be used at minimum to control the outbreak of the disease. (C) 2015 Elsevier ay. All rights reserved.
Bifurcation and Chaos in Synchronous Manifold of a Forest Model
(2015-11-01)
Huang, Chun-Ming
;
Juang, Jonq
;
應用數學系
;
數學建模與科學計算所(含中心)
;
Department of Applied Mathematics
In previous papers [Isagi et al., 1997; Satake & Iwasa, 2000], a forest model was proposed. The authors demonstrated numerically that the mature forest could possibly exhibit annual reproduction (fixed point synchronization), periodic and chaotic synchronization as the energy depletion constant d is gradually increased. To understand such rich synchronization phenomena, we are led to study global dynamics of a piecewise smooth map f(d,beta) containing two parameters d and beta. Here d is the energy depletion quantity and beta is the coupling strength. In particular, we obtain the following results. First, we prove that f(d,0) has a chaotic dynamic in the sense of Devaney on an invariant set whenever d > 1, which improves a result of [Chang & Chen, 2011]. Second, we prove, via the Schwarzian derivative and a generalized result of [Singer, 1978], that f(d,beta) exhibits the period adding bifurcation. Specifically, we show that for any beta > 0, f(d,beta) has a unique global attracting fixed point whenever d <= 1/(beta+ 1)(beta+ 1/beta+ 2)beta (< 1) and that for any beta > 0, f(d,beta) has a unique attracting period k + 1 point whenever d is less than and near any positive integer k. Furthermore, the corresponding period k + 1 point instantly becomes unstable as d moves pass the integer k. Finally, we demonstrate numerically that there are chaotic dynamics whenever d is in between and away from two consecutive positive integers. We also observe the route to chaos as d increases from one positive integer to the next through finite period doubling.
Boundary influence on the entropy of a Lozi-type map
(2010-11-15)
Juang, Jonq
;
Chang, Yu-Chuan
;
應用數學系
;
數學建模與科學計算所(含中心)
;
Department of Applied Mathematics
Let T be a Henon-type map induced from a spatial discretization of a reaction-diffusion system. With the above-mentioned description of T. the following open problems were raised in [V.S. Afraimovich, S B. Hsu, Lectures on Chaotic Dynamical Systems, AMS International Press, 2003]. Is it true that, in general, h(T) = h(D)(T)= h(N)(T) = hl((1)),(l(2)) (T)? Here h(T) and hl((1),(2)) (T) (see Definitions 1.1 and 1.2) are, respectively, the spatial entropy of the system T and the spatial entropy of T with respect to the lines e(l) and E(2), and h(D)(T) and h(N)(T) are spatial entropy with respect to the Dirichlet and Neuman boundary conditions. If it is not true, then which parameters of the lines l((1)), i = 1.2, are responsible for the value of h(T)? What kind of bifurcations occurs if the lines l((1)) move? In this paper, we show that this is in general not always true. Among other things, we further give conditions for which the above problem holds true. (C) 2010 Elsevier Inc. All rights reserved.
Chaotic difference equations in two variables and their multidimensional perturbations
(2008-05-01)
Juang, Jonq
;
Li, Ming-Chia
;
Malkin, Mikhail
;
應用數學系
;
Department of Applied Mathematics
We consider difference equations Phi(lambda)(y(n), y(n+1), . . . , y(n+m)) = 0, n is an element of Z, of order m with parameter. close to that exceptional value lambda(0) for which the function Phi depends on two variables: Phi(lambda 0)(x(0), . . . , x(m)) = xi (x(N), x(N+L)) with 0 <= N, N + L <= m. It is also assumed that for the equation xi(x, y) = 0, there is a branch y = phi(x) with positive topological entropy h(top)(phi). Under these assumptions we prove that in the set of bi-infinite solutions of the difference equation with. in some neighbourhood of lambda(0), there is a closed ( in the product topology) invariant set to which the restriction of the shift map has topological entropy arbitrarily close to htop(phi)/vertical bar L vertical bar, and moreover, orbits of this invariant set depend continuously on. not only in the product topology but also in the uniform topology. We then apply this result to establish chaotic behaviour for Arneodo-Coullet-Tresser maps near degenerate ones, for quadratic volume preserving automorphisms of R(3) and for several lattice models including the generalized cellular neural networks (CNNs), the time discrete version of the CNNs and coupled Chua's circuit.
Cluster synchronization in networks of neurons with chemical synapses
(2014-03-01)
Juang, Jonq
;
Liang, Yu-Hao
;
應用數學系
;
數學建模與科學計算所(含中心)
;
Department of Applied Mathematics
In this work, we study the cluster synchronization of chemically coupled and generally formulated networks which are allowed to be nonidentical. The sufficient condition for the existence of stably synchronous clusters is derived. Specifically, we only need to check the stability of the origins of m decoupled linear systems. Here, m is the number of subpopulations. Examples of nonidentical networks such as Hindmarsh-Rose (HR) neurons with various choices of parameters in different subpopulations, or HR neurons in one subpopulation and FitzHugh-Nagumo neurons in the other subpopulation are provided. Explicit threshold for the coupling strength that guarantees the stably cluster synchronization can be obtained. (C) 2014 AIP Publishing LLC.
Coordinate transformation and matrix measure approach for synchronization of complex networks
(2009-09-01)
Juang, Jonq
;
Liang, Yu-Hao
;
應用數學系
;
Department of Applied Mathematics
Global synchronization in complex networks has attracted considerable interest in various fields. There are mainly two analytical approaches for studying such time-varying networks. The first approach is Lyapunov function-based methods. For such an approach, the connected-graph-stability (CGS) method arguably gives the best results. Nevertheless, CGS is limited to the networks with cooperative couplings. The matrix measure approach (MMA) proposed by Chen, although having a wider range of applications in the network topologies than that of CGS, works for smaller numbers of nodes in most network topologies. The approach also has a limitation with networks having partial-state coupling. Other than giving yet another MMA, we introduce a new and, in some cases, optimal coordinate transformation to study such networks. Our approach fixes all the drawbacks of CGS and MMA. In addition, by merely checking the structure of the vector field of the individual oscillator, we shall be able to determine if the system is globally synchronized. In summary, our results can be applied to rather general time-varying networks with a large number of nodes.
Diversity of hysteresis in a fully cooperative coinfection model
(2018-02-01)
Rodriguez, Jorge P.
;
Liang, Yu-Hao
;
Huang, Yu-Jhe
;
Juang, Jonq
;
應用數學系
We propose a fully cooperative coinfection model in which singly infected individuals are more likely to acquire a second disease than susceptible ones and doubly infected individuals are also assumed to be more contagious than singly infected ones. The dynamics of such a fully cooperative coinfection model is investigated through the well-mixed approach. In particular, discontinuous outbreak transitions from the disease free state or the low prevalence state to the high prevalence state can be separately observed as a disease transmission rate crosses a threshold alpha(o) from the below when the epidemic is still in the early stages. Moreover, discontinuous eradications from the high prevalence state to the low prevalence or disease free state are also separately seen as the transmission rate reaches a threshold alpha(e)(< alpha(o)) from the above when the outbreak occurs. Such phenomena constitute three types of hysteresis, where only one type has been identified before. Complete characterization of these three types of hysteresis in terms of parameters measuring the uniformity of the model is both analytically and numerically provided. Published by AIP Publishing.
Eigenvalue problems and their application to the wavelet method of chaotic control
(2006-07-01)
Juang, Jonq
;
Li, Chin-Lung
;
應用數學系
;
Department of Applied Mathematics
Controlling chaos via wavelet transform was recently proposed by Wei, Zhan, and Lai [Phys. Rev. Lett. 89, 284103 (2002)]. It was reported there that by modifying a tiny fraction of the wavelet subspace of a coupling matrix, the transverse stability of the synchronous manifold of a coupled chaotic system could be dramatically enhanced. The stability of chaotic synchronization is actually controlled by the second largest eigenvalue lambda(1)(alpha,beta) of the (wavelet) transformed coupling matrix C(alpha,beta) for each alpha and beta. Here beta is a mixed boundary constant and alpha is a scalar factor. In particular, beta=1 (respectively, 0) gives the nearest neighbor coupling with periodic (respectively, Neumann) boundary conditions. The first, rigorous work to understand the eigenvalues of C(alpha,1) was provided by Shieh [J. Math. Phys. (to be published)]. The purpose of this paper is twofold. First, we apply a different approach to obtain the explicit formulas for the eigenvalues of C(alpha,1) and C(alpha,0). This, in turn, yields some new information concerning lambda(1)(alpha,1). Second, we shed some light on the question whether the wavelet method works for general coupling schemes. In particular, we show that the wavelet method is also good for the nearest neighbor coupling with Neumann boundary conditions.
FASTEST SYNCHRONIZED NETWORK AND SYNCHRONY ON THE JULIA SET OF COMPLEX-VALUED COUPLED MAP LATTICES
(2016-01-01)
Liang, Yu-Hao
;
Wu, Wan-Rou
;
Juang, Jonq
;
應用數學系
;
Department of Applied Mathematics
The purpose of this paper is to address synchronous chaos on the Julia set of complex-valued coupled map lattices (CCMLs). Our main results contain the following. First, we solve an inf min max problem for which its solution gives the fastest synchronized rate in a certain class of coupling matrices. Specifically, we show that for the class of real circulant matrices of size 4, the coupling weights, possible complex numbers, yielding the fastest synchronized rate can be exactly obtained. Second, for individual map of the form f(c)(z) = z(2) vertical bar c with vertical bar c vertical bar < 1/4 we show that the corresponding CCMLs acquires global synchrony on its Julia set with the number of the oscillators being 3 or 4 for the diffusive coupling. For c = 0 and 2, the corresponding CCMLs obtain local synchronization if and only if the number of oscillators is less than or equal to 5. Global synchronization for the individual map of the form g(c)(z) = z(3) + cz is also reported.
Global stability for epidemic models on multiplex networks
(2018-05-01)
Huang, Yu-Jhe
;
Juang, Jonq
;
Liang, Yu-Hao
;
Wang, Hsin-Yu
;
應用數學系
In this work, we consider an epidemic model in a two-layer network in which the dynamics of susceptible-infected-susceptible process in the physical layer coexists with that of a cyclic process of unaware-aware-unaware in the virtual layer. For such multiplex network, we shall define the basic reproduction number in the virtual layer, which is similar to the basic reproduction number defined in the physical layer. We show analytically that if and , then the disease and information free equilibrium is globally stable and if and , then the disease free and information saturated equilibrium is globally stable for all initial conditions except at the origin. In the case of , whether the disease dies out or not depends on the competition between how well the information is transmitted in the virtual layer and how contagious the disease is in the physical layer. In particular, it is numerically demonstrated that if the difference in and is greater than the product of , the deviation of from 1 and the relative infection rate for an aware susceptible individual, then the disease dies out. Otherwise, the disease breaks out.
Global synchronization in lattices of coupled chaotic systems
(2007-09-01)
Juang, Jonq
;
Li, Chin-Lung
;
Liang, Yu-Hao
;
應用數學系
;
Department of Applied Mathematics
Based on the concept of matrix measures, we study global stability of synchronization in networks. Our results apply to quite general connectivity topology. In addition, a rigorous lower bound on the coupling strength for global synchronization of all oscillators is also obtained. Moreover, by merely checking the structure of the vector field of the single oscillator, we shall be able to determine if the system is globally synchronized.
Multistate and Multistage Synchronization of Hindmarsh-Rose Neurons With Excitatory Chemical and Electrical Synapses
(2012-06-01)
Jhou, Fang-Jhu
;
Juang, Jonq
;
Liang, Yu-Hao
;
應用數學系
;
數學建模與科學計算所(含中心)
The new phenomena of the multistate synchronization of Hindmarsh-Rose (HR) neurons with both excitatory chemical and electrical synapses over the complex network are analytically studied. The regions for coupling strengths to achieve local synchronization are explicitly obtained. Such regions are characterized by the second largest eigenvalue lambda(2) of the electrical connection matrix and the number k of chemical signals each neuron receives. The dynamics of the multistate synchronization includes the coexistence of stable regular bursting and periodic/steady-state behaviors. Our theory predicts that recurrent networks formed by a certain cell types in layers 4 and 6 in cat area 17 could lead to multistate synchronization. These are in contrast with coupled oscillator systems or coupled map lattices where only single-state synchronization is found. It should also be noted that if the parameters of HR neurons are chosen resulting in an irregular (chaotic) bursting, then the coexistence state would contain chaotic attractor. Our method employed here is quite general. For instance, it can be immediately applied to other coupled nervous systems such as FitzHugh-Nagumo and Morris-Lecar nervous systems. The analytical tools and concepts needed include coordinate transformations, matrix measures, monotone dynamics and time averaging estimates.
On the total variation, topological entropy and sensitivity for interval maps
(2008-05-15)
Juang, Jonq
;
Shieh, Shih-Feng
;
應用數學系
;
Department of Applied Mathematics
A concept related to total variation termed H-1 condition was recently proposed to characterize the chaotic behavior of an interval map f by Chen, Huang and Huang [G. Chen, T. Huang, Y. Huang, Chaotic behavior of interval maps and total variations of iterates, Internat. J. Bifur. Chaos 14 (2004) 2161-2186]. In this paper, we establish connections between H-1 condition, sensitivity and topological entropy for interval maps. First, we introduce a notion of restrictiveness of a piecewise-monotone continuous interval map. We then prove that H-1 condition of a piecewise-monotone continuous map implies the non-restrictiveness of the map. In addition, we also show that either H-1 condition or sensitivity then gives the positivity of the topological entropy of f. (C) 2007 Elsevier Inc. All rights reserved.
Perturbed block circulant matrices and their application to the wavelet method of chaotic control
(2006-12-01)
Juang, Jonq
;
Li, Chin-Lung
;
Chang, Jing-Wei
;
應用數學系
;
Department of Applied Mathematics
Controlling chaos via wavelet transform was proposed by Wei [Phys. Rev. Lett. 89, 284103.1-284103.4 (2002)]. It was reported there that by modifying a tiny fraction of the wavelet subspace of a coupling matrix, the transverse stability of the synchronous manifold of a coupled chaotic system could be dramatically enhanced. The stability of chaotic synchronization is actually controlled by the second largest eigenvalue lambda(2)(alpha,beta) of the (wavelet) transformed coupling matrix C(alpha,beta) for each alpha and beta. Here beta is a mixed boundary constant and alpha is a scalar factor. In particular, beta=1 (0) gives the nearest neighbor coupling with periodic (Neumann) boundary conditions. In this paper, we obtain two main results. First, the reduced eigenvalue problem for C(alpha,0) is completely solved. Some partial results for the reduced eigenvalue problem of C(alpha,beta) are also obtained. Second, we are then able to understand behavior of lambda(2)(alpha,0) and lambda(2)(alpha,1) for any wavelet dimension j is an element of N and block dimension n is an element of N. Our results complete and strengthen the work of Shieh [J. Math. Phys. 47, 082701.1-082701.10 (2006)] and Juang and Li [J. Math. Phys. 47, 072704.1-072704.16 (2006)]. (c) 2006 American Institute of Physics.
Piecewise linear maps, Liapunov exponents and entropy
(2008-02-01)
Juang, Jonq
;
Shieh, Shih-Feng
;
應用數學系
;
Department of Applied Mathematics
Let L-A = {f(A,x):x is a partition of [0, 1]} be a class of piecewise linear maps associated with a transition matrix A. In this paper, we prove that if f(A,x) is an element of L-A, then the Liapunov exponent lambda(x) of f(A,x) is equal to a measure theoretic entropy h(mA,x) of f(A,x), where m(A,x) is a Markov measure associated with A and x. The Liapunov exponent and the entropy are computable by solving an eigenvalue problem and can be explicitly calculated when the transition matrix A is symmetric. Moreover, we also show that max(x) lambda(x) = max(x)h(mA,x) = log(lambda(1)), where lambda(1)is the maximal eigenvalue of A. (c) 2007 Elsevier Inc. All rights reserved.
Snap-back repellers and chaotic traveling waves in one-dimensional cellular neural networks
(2007-06-01)
Chang, Ya-Wen
;
Juang, Jonq
;
Li, Chin-Lung
;
應用數學系
;
Department of Applied Mathematics
In 1998, Chen et al. [1998] found an error in Marotto's paper [1978]. It was pointed out by them that the existence of an expanding fixed point z of a map F in Br(z), the ball of radius r with center at z does not necessarily imply that F is expanding in Br( z). Subsequent efforts (see e. g. [Chen et al., 1998; Lin et al., 2002; Li & Chen, 2003]) in. xing the problems have some discrepancies since they only give conditions for which F is expanding "locally". In this paper, we give suficient conditions so that F is "globally" expanding. This, in turn, gives more satisfying definitions of a snap-back repeller. We then use those results to show the existence of chaotic backward traveling waves in a discrete time analogy of one-dimensional Cellular Neural Networks (CNNs). Some computer evidence of chaotic traveling waves is also given.
Stable Synchrony in Globally Coupled Integrate-and-Fire Oscillators
(2008)
Chang, Yu-Chuan
;
Juang, Jonq
;
應用數學系
;
Department of Applied Mathematics
A model of integrate-and-fire oscillators is studied. In the special case of identical oscillators, the model was first proposed and analyzed by Mirollo and Strogatz [SIAM J. Appl. Math., 50 (1990), pp. 1645-1662]. We assume, as in Mirollo and Strogatz's model, that each oscillator x(i) evolves according to a map f(i). Our main results are to demonstrate that the concavity structure of f(i) plays an important role in determining whether Peskin's second conjecture holds true. Specifically, the following statements are proved. First, the system of convex oscillators (i.e., f(i)'' < 0 for all i), in general, synchronizes when the oscillators are not quite identical. Second, the system of a certain class of concave oscillators (i.e., f(i)'' > 0 for all i) will not achieve synchrony for initial conditions in a set of positive measure when the oscillators are nearly identical. Third, the system of concave oscillators may achieve synchrony under certain sufficient conditions, provided that the oscillators are not quite nonidentical and that its concavity is small.
Synchronous Chaos in Coupled Map Lattices with General Connectivity Topology
(2008)
Juang, Jonq
;
Liang, Yu-Hao
;
應用數學系
;
Department of Applied Mathematics
The purpose of the paper is to address the synchronous chaos in coupled map lattices with general connectivity topology. Our main results contain the following. First, the master stability functions also hold for general connectivity topology with coupling through a nonlinear function that needs to be exactly the individual chaotic map. Second, the synchronization curve, composed of pieces of transverse Lyapunov exponent curves, is constructed. Third, necessary and sufficient conditions on coupling strength for yielding the synchronous chaos of the system are given. Moreover, the coupling strength d(c) giving the fastest convergence rate of the initial values toward the synchronous state is explicitly obtained. It is also proved that such dc is independent of the choice of the individual map. Finally, our results here can be applied to address questions of wavelength bifurcations and size instability.
The impact of vaccine failure rate on epidemic dynamics in responsive networks
(2015-04-01)
Liang, Yu-Hao
;
Juang, Jonq
;
應用數學系
;
數學建模與科學計算所(含中心)
;
Department of Applied Mathematics
An SIS model based on the microscopic Markov-chain approximation is considered in this paper. It is assumed that the individual vaccination behavior depends on the contact awareness, local and global information of an epidemic. To better simulate the real situation, the vaccine failure rate is also taken into consideration. Our main conclusions are given in the following. First, we show that if the vaccine failure rate alpha is zero, then the epidemic eventually dies out regardless of what the network structure is or how large the effective spreading rate and the immunization response rates of an epidemic are. Second, we show that for any positive alpha, there exists a positive epidemic threshold depending on an adjusted network structure, which is only determined by the structure of the original network, the positive vaccine failure rate and the immunization response rate for contact awareness. Moreover, the epidemic threshold increases with respect to the strength of the immunization response rate for contact awareness. Finally, if the vaccine failure rate and the immunization response rate for contact awareness are positive, then there exists a critical vaccine failure rate alpha(c) > 0 so that the disease free equilibrium (DFE) is stable (resp., unstable) if alpha < alpha(c) (resp., alpha > alpha(c)). Numerical simulations to see the effectiveness of our theoretical results are also provided. (C) 2015 AIP Publishing LLC.
The impact of vaccine success and awareness on epidemic dynamics
(2016-11)
Juang, Jonq
;
Liang, Yu-Hao
;
應用數學系
;
數學建模與科學計算所(含中心)
;
Department of Applied Mathematics
The role of vaccine success is introduced into an epidemic spreading model consisting of three states: susceptible, infectious, and vaccinated. Moreover, the effect of three types, namely, contact, local, and global, of infection awareness and immunization awareness is also taken into consideration. The model generalizes those considered in Pastor-Satorras and Vespignani [Phys. Rev. E 63, 066117 (2001)], Pastor-Satorras and Vespignani [Phys. Rev. E 65, 036104 (2002)], Moreno et al. [Eur. Phys. J. B 26, 521-529 (2002)], Wu et al. [Chaos 22, 013101 (2012)], and Wu et al. [Chaos 24, 023108 (2014)]. Our main results contain the following. First, the epidemic threshold is explicitly obtained. In particular, we show that, for any initial conditions, the epidemic eventually dies out regardless of what other factors are whenever some type of immunization awareness is considered, and vaccination has a perfect success. Moreover, the threshold is independent of the global type of awareness. Second, we compare the effect of contact and local types of awareness on the epidemic thresholds between heterogeneous networks and homogeneous networks. Specifically, we find that the epidemic threshold for the homogeneous network can be lower than that of the heterogeneous network in an intermediate regime for intensity of contact infection awareness while it is higher otherwise. In summary, our results highlight the important and crucial roles of both vaccine success and contact infection awareness on epidemic dynamics. Published by AIP Publishing.