摘要:通過(guò)設(shè)計(jì)有效的時(shí)滯反饋控制器和開(kāi)環(huán)控制器,研究帶耦合時(shí)滯分?jǐn)?shù)階社團(tuán)網(wǎng)絡(luò)的聚類(lèi)滯后同步問(wèn)題。網(wǎng)絡(luò)中不同社團(tuán)的節(jié)點(diǎn)動(dòng)力學(xué)是不同的,且與同步目標(biāo)的滯后性也是不同的?;贚yapunov函數(shù)方法和分?jǐn)?shù)階系統(tǒng)穩(wěn)定性理論,給出了網(wǎng)絡(luò)實(shí)現(xiàn)聚類(lèi)滯后同步的充分條件。最后,通過(guò)數(shù)值與仿真實(shí)例驗(yàn)證理論結(jié)果的正確性和有效性。
關(guān)鍵詞:耦合時(shí)滯;分?jǐn)?shù)階;社團(tuán)網(wǎng)絡(luò);聚類(lèi)滯后同步
中圖分類(lèi)號(hào):O231.5;N94文獻(xiàn)標(biāo)識(shí)碼:A
基金項(xiàng)目:國(guó)家自然科學(xué)基金(61963019);江西省杰出青年人才資助計(jì)劃(20171BCB23031)
第一作者:吳聰玲(1999-),女,江西吉安人,碩士,主要研究方向?yàn)閯?dòng)力系統(tǒng)與混沌控制。
通信作者:吳召艷(1979-),男,江蘇徐州人,博士,教授,主要研究方向?yàn)閺?fù)雜網(wǎng)絡(luò)的建模、分析與控制。
Cluster Lag Synchronization in Fractional-order Community Networks with Coupling Delay
WU Congling,WU Zhaoyan
(College of Mathematics and Statistics,Jiangxi Normal University,Nanchang 330022,China)
Abstract:Through designing effective delayed feedback and open-loop controllers, the cluster lag synchronization problem of fractional order community network is studied. For different communities of the network, both the node dynamics and the lags with synchronization goals are nonidentical. Based on the Lyapunov function method and the stability theory of fractional order system, the sufficient condition for achieving the cluster lag synchronization is derived. Finally, the correctness and effectiveness of the theoretical results are verified by numerical and simulation examples.
Keywords: coupling delay;fractional-order;community network;cluster lag synchronization
0 引言
在自然界和人類(lèi)社會(huì)中,復(fù)雜網(wǎng)絡(luò)無(wú)處不在,例如,社交網(wǎng)絡(luò)、電力網(wǎng)絡(luò)、流行病傳播網(wǎng)絡(luò)、生物網(wǎng)絡(luò)等[1-4]。在復(fù)雜網(wǎng)絡(luò)的各種動(dòng)態(tài)行為中,同步是最重要的集群行為之一,指所有的節(jié)點(diǎn)相互同步或達(dá)到期望的軌道[5],其在數(shù)字通信、安全通信和圖像加密[6-9]中都得到了廣泛應(yīng)用。從同步的類(lèi)型來(lái)看,同步分為聚類(lèi)同步、滯后同步、完全同步、投影同步、反同步等[10-14]。其中,聚類(lèi)同步廣泛存在于社團(tuán)網(wǎng)絡(luò)中。在社團(tuán)網(wǎng)絡(luò)中,節(jié)點(diǎn)被劃分為若干個(gè)社團(tuán),同一個(gè)社團(tuán)中的節(jié)點(diǎn)有著相同的動(dòng)力學(xué)行為且相互之間聯(lián)系比較緊密,而不同社團(tuán)的節(jié)點(diǎn)具有不同的動(dòng)力學(xué)行為且相互之間聯(lián)系較為稀疏。當(dāng)同一個(gè)社團(tuán)中的節(jié)點(diǎn)實(shí)現(xiàn)同步,不同社團(tuán)的節(jié)點(diǎn)不相互同步時(shí),就說(shuō)網(wǎng)絡(luò)實(shí)現(xiàn)了聚類(lèi)同步。
在現(xiàn)實(shí)世界中,時(shí)滯是不可避免的。例如神經(jīng)元之間的信息交換、安全通信和動(dòng)力系統(tǒng)的電子實(shí)現(xiàn)等。一方面,復(fù)雜網(wǎng)絡(luò)的節(jié)點(diǎn)與節(jié)點(diǎn)之間在進(jìn)行信息交換時(shí)存在通信(耦合)時(shí)滯,另一方面,有時(shí)不希望兩個(gè)節(jié)點(diǎn)在同一時(shí)刻達(dá)到同步,而是一個(gè)節(jié)點(diǎn)達(dá)到某個(gè)狀態(tài)后,另一個(gè)節(jié)點(diǎn)在一段時(shí)間后也達(dá)到此狀態(tài),通常把這種帶有時(shí)滯的同步定義為滯后同步[15]。 文獻(xiàn)[16]研究了帶有通信(耦合)時(shí)滯的復(fù)雜動(dòng)力學(xué)網(wǎng)絡(luò)上的滯后同步。文獻(xiàn)[17]采用間歇控制實(shí)現(xiàn)變時(shí)滯混沌系統(tǒng)滯后同步。由于社團(tuán)網(wǎng)絡(luò)的特性,不同的社團(tuán)不僅實(shí)現(xiàn)的目標(biāo)可能不同,實(shí)現(xiàn)目標(biāo)的時(shí)間點(diǎn)也可能不同,即不同社團(tuán)實(shí)現(xiàn)不同的滯后同步,將其稱(chēng)為聚類(lèi)滯后同步。文獻(xiàn)[18]用間歇牽制控制方法研究了整數(shù)階社團(tuán)網(wǎng)絡(luò)的聚類(lèi)滯后同步問(wèn)題,但是帶有耦合時(shí)滯的分?jǐn)?shù)階社團(tuán)網(wǎng)絡(luò)的聚類(lèi)滯后同步卻少有研究。
分?jǐn)?shù)階微積分將經(jīng)典微積分的階數(shù)由整數(shù)拓展到了實(shí)數(shù)。分?jǐn)?shù)階微積分幾乎是和經(jīng)典微積分同時(shí)提出來(lái)的,但直到1984年Mandelbort提出客觀世界中有許多分?jǐn)?shù)維的現(xiàn)象,分?jǐn)?shù)階微積分才迎來(lái)了新的發(fā)展。由于分?jǐn)?shù)階微積分特有的記憶性,分?jǐn)?shù)階系統(tǒng)更能有效描述物理和工程中的遺傳和記憶的性質(zhì)。分?jǐn)?shù)階系統(tǒng)的實(shí)際應(yīng)用包括粘彈性、分?jǐn)?shù)階電感器、內(nèi)聚斷裂模型和量子力學(xué)等[19-22]。因此,越來(lái)越多的研究者開(kāi)始投入分?jǐn)?shù)階復(fù)雜網(wǎng)絡(luò)的同步研究。
5 結(jié)論
本文考慮了帶有耦合時(shí)滯的分?jǐn)?shù)階社團(tuán)網(wǎng)絡(luò)的聚類(lèi)滯后同步問(wèn)題,基于Lyapunov函數(shù)方法和分?jǐn)?shù)階微積分理論,給出了實(shí)現(xiàn)社團(tuán)網(wǎng)絡(luò)聚類(lèi)滯后同步的充分條件,并給出數(shù)值例子驗(yàn)證了理論結(jié)果的正確性。
本文中不同社團(tuán)實(shí)現(xiàn)滯后同步的時(shí)滯不同,但其分?jǐn)?shù)階階數(shù)和節(jié)點(diǎn)動(dòng)力學(xué)的維數(shù)相同。對(duì)于帶不同階數(shù)或維數(shù)的分?jǐn)?shù)階社團(tuán)網(wǎng)絡(luò)的同步問(wèn)題,如何設(shè)計(jì)有效的控制器是值得研究的一個(gè)問(wèn)題。
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