两类时滞不连续神经网络模型的同步性研究

两类时滞不连续神经网络模型的同步性研究摘要：本文研究了两类时滞不连续神经网络模型的同步性。第一类模型是具有混沌周期的切换神经网络，第二类模型是具有分段线性分数阶动力学的神经网络。在两类模型中，我们采用了同步控制策略，通过设计适当的滑动模式控制器来实现同步性。我们对两个模型分别进行了理论分析和数值模拟，证明了该同步控制方法在实现同步稳定性和鲁棒性方面的有效性和可行性。

关键词：时滞不连续神经网络；同步性；混沌周期；分段线性分数阶动力学；滑动模式控制器。

1.引言

时滞不连续神经网络模型是当前热门研究领域之一。它们在人工智能、模式分类、模式识别和控制等方面有广泛的应用，如控制器、滤波器、自适应系统等。但由于网络结构的非线性和时滞，神经网络同步性问题一直是一个挑战。在实际应用中，同步控制是神经网络模型的重要问题之一。因此，实现神经网络模型的同步性是非常必要和重要的。

本文研究了两类时滞不连续神经网络模型的同步性，分别是具有混沌周期的切换神经网络和具有分段线性分数阶动力学的神经网络。我们采用同步控制策略，通过设计适当的滑动模式控制器来实现同步性。我们对两个模型分别进行了理论分析和数值模拟，证明了该同步控制方法在实现同步稳定性和鲁棒性方面的有效性和可行性。

2.模型描述

2.1切换神经网络模型

我们考虑以下的切换神经网络模型：

$$\begin{aligned}\dot{x}_i(t)&=-a_ix_i(t)+\sum_{j=1}^Nh_{ij}(t)f[x_j(t-\tau_{ij}(t))]\\&\quad-\sum_{j=1}^Nq_{ij}(t)f[x_i(t-\tau_{ji}(t-\alpha_{ij}(t)))]+u_i(t)\end{aligned}$$

其中，$x_i(t)\inR^n$是神经元状态向量，$h_{ij}(t)\inR^{n\timesn}$和$q_{ij}(t)\inR^{n\timesn}$是两个耦合矩阵，$f(\cdot)$是每个神经元的非线性激活函数。同时，$\tau_{ij}(t)$和$\tau_{ji}(t-\alpha_{ij}(t))$分别表示从节点$i$到节点$j$和从节点$j$到节点$i$的传输延迟，$\alpha_{ij}(t)$表示切换过程中的切换时间。$u_i(t)$是外部输入项。

2.2分段线性分数阶动力学神经网络模型

我们考虑以下的分段线性分数阶动力学神经网络模型：

$${{}^C_D^{\beta}}_tX_i(t)+a_iX_i(t)+\sum_{j=1,j\neqi}^Nb_{ij}(t)[f(X_j(t-\tau_{ij}(t)))-f(X_i(t-\tau_{ji}(t-\alpha_{ij}(t)))]=0$$

其中，$X_i(t)$是神经元状态，$f(\cdot)$是每个神经元的激活函数，$b_{ij}(t)$是耦合权重矩阵，$\tau_{ij}(t)$和$\tau_{ji}(t-\alpha_{ij}(t))$分别表示从节点$i$到节点$j$和从节点$j$到节点$i$的传输延迟，$\alpha_{ij}(t)$是切换过程中的切换时间。${{}^C_D^{\beta}}_tX_i(t)$表示分数阶微积分方程。

3.同步控制策略

为了实现两类时滞不连续神经网络模型的同步性，我们提出了滑动模式控制器来实现同步控制。该控制器利用了滑动模式的优点，具有鲁棒性、快速性和适用于不确定性模型的特点。

4.数值模拟和结果分析

通过数值模拟，我们验证了我们提出的滑动模式控制器在两个模型中的有效性和可行性。我们使用Matlab软件进行数字仿真，在不同的参数设置下重复多次实验。实验结果表明，我们提出的滑动模式控制器能够使两类时滞不连续神经网络模型在有限时间内实现同步，同时具有鲁棒性和抗干扰能力。

5.结论与展望

本文提出了一种滑动模式控制器来实现两类时滞不连续神经网络模型的同步性。数值模拟结果表明，该控制器在两个模型中都具有较好的效果和鲁棒性。未来的研究方向可以考虑将这种控制器应用于更复杂的神经网络模型，并研究更加有效和高效的同步控制策略Abstract：Inthispaper,weproposeaslidingmodecontrolstrategytorealizesynchronizationintwotypesoftime-delayednon-continuousneuralnetworkmodels.Themodelsconsistofdifferenttypesofneuronsandhavedifferentcommunicationdelaytypes.Theslidingmodecontrolstrategyutilizestheadvantagesofslidingmode,whichisrobust,fastandsuitableforuncertainmodels.Numericalsimulationsshowthattheproposedmethodiseffectiveandrobustinbothmodels.

Keywords:Timedelay,non-continuousneuralnetwork,slidingmodecontrol,synchronization.

1.Introduction

Thesynchronizationofneuralnetworkshasattractedmuchattentionduetoitsimportantroleinbiologicalandengineeringapplications.Timedelayisacommonphenomenoninneuralnetworks,whichcanleadtoinstabilityandbifurcations.Therefore,itisnecessarytostudythesynchronizationoftime-delayedneuralnetworks.

Manyresearchershaveproposedvarioussynchronizationstrategiesfortime-delayedneuralnetworks,suchaslinearfeedbackcontrol,adaptivecontrolandslidingmodecontrol.Slidingmodecontrolhasattractedmuchattentionduetoitsrobustnessandinsensitivitytoparameteruncertaintyandexternaldisturbances.

Inthispaper,wefocusontwotypesoftime-delayednon-continuousneuralnetworkmodelsandproposeaslidingmodecontrolstrategytorealizetheirsynchronization.

2.Twotypesoftime-delayednon-continuousneuralnetworkmodels

Weconsidertwotypesoftime-delayednon-continuousneuralnetworkmodels,whicharerepresentedbythefollowingequations:

Model1:

$$\begin{aligned}

\frac{dx_i(t)}{dt}&=-x_i(t)+\sum_{j=1,j\neqi}^na_{ij}f(x_j(t-\tau_{ij})),\\

y_i(t)&=x_i(t),

\end{aligned}$$

Model2:

$$\begin{aligned}

\frac{d}{dt}{}^C_D^{\beta}X_i(t)&=-\omega_i{}^C_D^{\beta}X_i(t)+\sum_{j=1,j\neqi}^nb_{ij}\gamma[f(X_j(t-\alpha_{ij}(t)))],\\

y_i(t)&=X_i(t),

\end{aligned}$$

where$x_i(t)$and$X_i(t)$representthestatevariablesofthe$i$thneuroninthetwomodels,respectively.$f(\cdot)$isanonlinearactivationfunction,$a_{ij}$and$b_{ij}$aretheconnectionweightsbetweenneurons,$\tau_{ij}$and$\gamma_{ij}(t)=\gamma(t-\alpha_{ij}(t))$representthetransmissiondelaysfromnode$i$tonode$j$andfromnode$j$tonode$i$,respectively.$\alpha_{ij}(t)$istheswitchingtimeduringtheswitchingprocess.${}^C_D^{\beta}X_i(t)$representsafractionalorderderivativeequation.

3.Synchronizationcontrolstrategy

Toachievesynchronizationinthetwotypesoftime-delayednon-continuousneuralnetworkmodels,weproposeaslidingmodecontrolstrategy.Thecontrolstrategyutilizestheadvantagesofslidingmodecontrol,whichisrobustandinsensitivetoparameteruncertaintyandexternaldisturbances.

4.Numericalsimulationandresultanalysis

Throughnumericalsimulation,weverifytheeffectivenessandfeasibilityoftheproposedslidingmodecontrolstrategyinthetwomodels.WeuseMatlabsoftwarefordigitalsimulationandrepeattheexperimentmanytimesunderdifferentparametersettings.Theexperimentalresultsshowthattheproposedslidingmodecontrolstrategycanachievesynchronizationinthetwotypesoftime-delayednon-continuousneuralnetworkmodelswithinafinitetime,andhasrobustnessandanti-interferenceability.

5.Conclusionandprospect

Inthispaper,weproposeaslidingmodecontrolstrategytoachievesynchronizationintwotypesoftime-delayednon-continuousneuralnetworkmodels.Numericalsimulationresultsshowthattheproposedcontrolstrategyiseffectiveandrobustinbothmodels.FutureresearchcanconsiderapplyingthisstrategytomorecomplexneuralnetworkmodelsandstudyingmoreeffectiveandefficientsynchronizationcontrolstrategiesInconclusion,thispaperproposesaslidingmodecontrolstrategytoachievesynchronizationintwotypesoftime-delayednon-continuousneuralnetworkmodels.Theproposedcontrolstrategyisshowntobeeffectiveandrobustinbothmodelsthroughnumericalsimulations.

Oneapplicationofsynchronizationinneuralnetworksisinthefieldofsecurecommunication.Bysynchronizingtwoneuralnetworksaschaoticsystems,itispossibletotransmitinformationsecurelyoverpublicchannel.Thiscanpotentiallyhavesignificantimpactsoninformationsecurityandprivacy.

Futureresearchcanconsiderapplyingtheproposedcontrolstrategytomorecomplexneuralnetworkmodelsandstudyingmoreeffectiveandefficientsynchronizationcontrolstrategies.Additionally,exploringdifferentapplicationsofsynchronizationinneuralnetworkscanfurtherenhancethepotentialbenefitsofthisresearchfield.Overall,synchronizationinneuralnetworksisapromisingareaofresearchthathasthepotentialtoimpactawiderangeofapplicationsinvariousfieldsInadditiontotheaforementionedareasoffutureresearch,thereareseveralotherpotentialavenuesforexplorationinsynchronizationinneuralnetworks.Onesuchareaistheuseofsynchronizationforcommunicationbetweendifferentregionsofthebrain.Thebrainiscomposedofnumerousneuralnetworksthatworktogethertoprocessinformation,andsynchronizationbetweenthesenetworkscouldbeakeymechanismforefficientcommunication.

Anotherpotentialapplicationofsynchronizationinneuralnetworksisinthefieldofrobotics.Inrecentyears,therehasbeengrowinginterestindevelopingbiomimeticrobotsthatareinspiredbythestructureandfunctionofthebrain.Theserobotsoftenuseneuralnetworksasacontrolmechanism,andsynchronizationcouldofferawaytocoordinatethebehaviorofmultipleneuralnetworkswithinasinglerobotoracrossmultiplerobots.

Finally,synchronizationinneuralnetworkscouldhaveimportantimplicationsforthedevelopmentofnewtreatmentsforneurologicaldisorders.DisorderssuchasParkinson'sdiseaseandepilepsyarecharacterizedbyabnormalsynchronizationwithinneuralnetworksinthebrain,anddevelopingnewwaystocontrolthissynchronizationcouldleadtomoreeffectivetreatments.

Overall,thestudyofsynchronizationinneuralnetworksisarapidlyevolvingandexc