New criteria for globally exponential stability of delayed Cohen–Grossberg neural network

Chen, Shengshuang; Zhao, Wenyu; Xu, Yong

doi:10.1016/j.matcom.2008.07.002

Cited by 20 publications

(18 citation statements)

References 24 publications

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“…The main results of this paper are demonstrated using Cohen-Grossberg Neural (CGN) Networks, which are dynamical networks whose stability is often studied in the presence of constant-type and time-varying time-delays [19,20]. It is worth noting that the results of this paper justify the modeling of dynamical networks and switched systems without formally including delays in the model if it is known that the system is intrinsically stable.…”

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confidence: 91%

“…We also show that the asymptotic state of intrinsically stable switched systems is independent of the system's initial conditions (see Main Result 1). This allows us to show that the globally attracting state of any intrinsically stable network and any time-delayed version of the network are identical (see Proposition 1).The main results of this paper are demonstrated using Cohen-Grossberg Neural (CGN) Networks, which are dynamical networks whose stability is often studied in the presence of constant-type and time-varying time-delays [19,20]. It is worth noting that the results of this paper justify the modeling of dynamical networks and switched systems without formally including delays in the model if it is known that the system is intrinsically stable.…”

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confidence: 91%

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Intrinsic stability: stability of dynamical networks and switched systems with any type of time-delays

Reber

Webb

2020

Nonlinearity

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In real-world networks the interactions between network elements are inherently time-delayed. These timedelays can not only slow the network but can have a destabilizing effect on the network's dynamics leading to poor performance. The same is true in computational networks used for machine learning etc. where time-delays increase the network's memory but can degrade the network's ability to be trained. However, not all networks can be destabilized by time-delays. Previously, it has been shown that if a network or high-dimensional dynamical system is intrinsically stabile, which is a stronger form of the standard notion of global stability, then it maintains its stability when constant time-delays are introduced into the system. Here we show that intrinsically stable systems, including intrinsically stable networks and a broad class of switched systems, i.e. systems whose mapping is time-dependent, remain stable in the presence of any type of time-varying time-delays whether these delays are periodic, stochastic, or otherwise. We apply these results to a number of well-studied systems to demonstrate that the notion of intrinsic stability is both computationally inexpensive, relative to other methods, and can be used to improve on some of the best known stability results. We also show that the asymptotic state of an intrinsically stable switched system is exponentially independent of the system's initial conditions.Here we refer to the emergent behavior of these interacting elements, which is the changing state of these network elements, as the network's dynamics. The network's dynamics can be periodic, as is found in many biological networks [2], synchronizing, which is the desired condition for transmitting power over large distance in power grids [3], and stable or multistable dynamics such as is found in gene-regulatory networks [4], etc.In real-world networks the interactions between network elements are inherently time-delayed. This comes from the fact that network elements are spatially separated, that information and other quantities can only be processed and transmitted at finite speeds, and that these quantities can be slowed by network traffic [5]. These time-delays not only slow the network, leading to poor performance, but can have a destabilizing effect on the network's dynamics, which can lead to network failure [6,7]. The same is true in computational networks used for machine learning etc. where time-delays can be used to increase the network's ability to detect long term temporal dependencies [8] but at the cost of potentially degrading the ability to train the network.Not all networks can be destabilized by time-delays. In [9] the notion of intrinsic stability was introduced, which is a stronger form of the standard notion of stability, i.e. a system in which there is a globally attracting fixed point (see [10] for more details). If a network is intrinsically stable the authors showed that constant-type time-delays, which are delays that do not vary in time, have no effect on the network's stabi...

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confidence: 91%

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confidence: 91%

Intrinsic stability: stability of dynamical networks and switched systems with any type of time-delays

Reber

Webb

2020

Nonlinearity

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“…As a network's dynamics can be related to the network's spectrum we can in certain cases determine how specialization of a network will effect the dynamics on the network. The type of dynamics we consider here is global stability, which is an important property in a number of systems including neural networks [10,12,11,14,36], network epidemic models [38], and in the study of congestion on computer networks [2]. The main example(s) we consider here and apply our results to are recurrent neural networks, which model the electrical activity in the brain and which form the basis of certain algorithms in machine learning [33].…”

Section: Introductionmentioning

confidence: 99%

Spectral and Dynamic Consequences of Network Specialization

Bunimovich

Passey

Smith

et al. 2020

Int. J. Bifurcation Chaos

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One of the hallmarks of real networks is the ability to perform increasingly complex tasks as their topology evolves. To explain this, it has been observed that as a network grows certain subsets of the network begin to specialize the function(s) they perform. A recent model of network growth based on this notion of specialization has been able to reproduce some of the most well-known topological features found in real-world networks including right-skewed degree distributions, the small world property, modular as well as hierarchical topology, etc. Here we describe how specialization under this model also effects the spectral properties of a network. This allows us to give the conditions under which a network is able to maintain its dynamics as its topology evolves. Specifically, we show that if a network is intrinsically stable, which is a stronger version of the standard notion of global stability, then the network maintains this type of dynamics as the network evolves. This is one of the first steps toward unifying the rigorous study of the two types of dynamics exhibited by networks. These are the dynamics of a network, which is the topological evolution of the network’s structure, modeled here by the process of network specialization, and the dynamics on a network, which is the changing state of the network elements, where the type of dynamics we consider is global stability. The main examples we apply our results to are recurrent neural networks, which are the basis of certain types of machine learning algorithms.

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“…This is advantageous from a computational point of view since it is significantly simpler to investigate the stability of an undelayed network than a network with delays.With this theory in place we consider the class of dynamical networks known as Cohen-Grossberg neural networks [4] whose stability has received a considerable amount of attention. See for instance [5,8,11,14,15,16]. By applying our theory to such systems we are able to derive new criteria for the stability of both the delayed and undelayed versions of this class of networks.To further apply this theory we note that one of the major obstacle in determining the dynamic behavior of a network (or high-dimensional system) is that the information needed to do so is spread throughout the network components.…”

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confidence: 99%

“…(Stability of Time-Delayed Cohen-Grossberg Networks) Let (H, R nT ) be the time-delayed Cohen-Grossberg network given by(15) where φ i has Lipschitz constant L. If |1 − | + Lρ(|W |) < 1 then (H, R nT ) has a globally attracting fixed point.The point of theorem 4.5 is that although time-delayed Cohen-Grossberg networks are more complicated systems than Cohen-Grossberg networks without delays, the criteria for their stability is not.4.3.Proving Theorem 4.2 and Theorem 4.4. In order to prove theorems 4.2 and theorem 4.4 we will use the well known theorem of Perron and Frobenius and the following standard terminology.…”

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confidence: 99%

Restrictions and stability of time-delayed dynamical networks

Bunimovich

Webb

2013

Nonlinearity

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Abstract. This paper deals with the global stability of time-delayed dynamical networks. We show that for a time-delayed dynamical network with nondistributed delays the network and the corresponding non-delayed network are both either globally stable or unstable. We demonstrate that this may not be the case if the network's delays are distributed. The main tool in our analysis is a new procedure of dynamical network restrictions. This procedure is useful in that it allows for improved estimates of a dynamical network's global stability. Moreover, it is a computationally simpler and much more effective means of analyzing the stability of dynamical networks than the procedure of isospectral network expansions introduced in [3]. The effectiveness of our approach is illustrated by applications to various classes of Cohen-Grossberg neural networks.

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New criteria for globally exponential stability of delayed Cohen–Grossberg neural network

Cited by 20 publications

References 24 publications

Intrinsic stability: stability of dynamical networks and switched systems with any type of time-delays

Intrinsic stability: stability of dynamical networks and switched systems with any type of time-delays

Spectral and Dynamic Consequences of Network Specialization

Restrictions and stability of time-delayed dynamical networks

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