Josephson結(jié)陣列中的超混沌控制

馮 玉 玲

(長(zhǎng)春理工大學(xué) 理學(xué)院,長(zhǎng)春 130022)

Josephson結(jié)廣泛應(yīng)用于制備工藝和超導(dǎo)材料等領(lǐng)域[1],文獻(xiàn)[2-3]在電阻電容分路的Josephson結(jié)(RCSJJ)模型[4-5]基礎(chǔ)上,提出了電阻電容電感分路的Josephson結(jié)(RCLSJJ)模型;文獻(xiàn)[6-11]研究了RCLSJJ模型的動(dòng)力學(xué)特性;文獻(xiàn)[12-13]利用Josephson結(jié)陣列獲得了較大的輸出功率;文獻(xiàn)[14]用電阻分路結(jié)模型研究了Josephson結(jié)陣列系統(tǒng)中的自同步；文獻(xiàn)[15]數(shù)值研究了由電阻分路結(jié)組成陣列中的時(shí)空混沌；文獻(xiàn)[16]研究了由電阻分路結(jié)組成陣列中的絕熱混沌；文獻(xiàn)[17]數(shù)值研究了電阻電容分路結(jié)組成的陣列.本文研究由RCLSJJ組成陣列中的超混沌控制,根據(jù)周期信號(hào)干擾法[18],通過在原來直流偏置電流中增加交變電流,實(shí)現(xiàn)該陣列系統(tǒng)中的超混沌控制.

圖1 Josephson結(jié)陣列結(jié)構(gòu)示意圖Fig.1 Schematic diagram of Josephson junction array

1 Josephson結(jié)陣列模型

本文研究的陣列包含2個(gè)RCLSJJ,如圖1所示,其中R為耦合電阻,Idc為直流電流源.根據(jù)Kirchhoff定律和Josephson方程,當(dāng)忽略噪聲時(shí),圖1所示陣列的動(dòng)力學(xué)方程[3]為

(1)

其中:n=1,2表示陣列中2個(gè)Josephson結(jié)的序號(hào);γn和Vn分別為第n個(gè)Josephson結(jié)的超導(dǎo)序參量相位差和結(jié)電壓；?為Planck常數(shù);e為電子電荷;ICn,RVn,Cn分別為第n個(gè)Josephson結(jié)的臨界電流、非線性電阻和結(jié)電容；Rsn,Ln,Isn分別是第n個(gè)Josephson結(jié)的分路電阻、分路電感和分路電流；IR為流過電阻R的電流.

本文設(shè)計(jì)的陣列中2個(gè)結(jié)對(duì)應(yīng)參數(shù)相等,即IC1=IC2=IC0,R1=R2=R0,C1=C2=C0,Rs1=Rs2=R0,L1=L2=L0.因此方程組(1)的無量綱形式為:

(2)

2 Josephson結(jié)陣列系統(tǒng)中的超混沌

圖2 陣列電壓υ1+υ2的時(shí)間序列(A)及對(duì)應(yīng)的功率譜(B)Fig.2 Time series (A) and corresponding power spectra (B) of array voltage υ1+υ2

由圖2(A)可見,電壓波形隨機(jī)變化,即非周期性變化；由圖2(B)可見,其功率譜為寬帶背景噪聲,因此陣列系統(tǒng)處于混沌狀態(tài).用Wolf方法[19]計(jì)算與圖2對(duì)應(yīng)的陣列Lyapunov指數(shù)譜,結(jié)果為(0.089 0,0.018 9,-0.007 1,-0.017 3,-0.682 8,-0.711 7),其中兩個(gè)是正的,表明陣列系統(tǒng)處于超混沌狀態(tài).

3 Josephson結(jié)陣列中超混沌控制的實(shí)現(xiàn)

3.1 超混沌控制方案

本文利用弱周期干擾法[18]將外部交變電流作為一個(gè)周期干擾信號(hào)加到圖1原來的直流偏置電流中,此時(shí)方程組(1)的第一個(gè)方程變?yōu)?/p>

其中:Idc為原來的直流偏置電流；Iac和ωt分別為后加作為周期干擾信號(hào)外部交變電流的幅值和角頻率,該干擾信號(hào)的歸一化強(qiáng)度和頻率分別為m=Iac/IC0和f=ωt/ωc.

因此被控制后陣列系統(tǒng)的歸一化動(dòng)力學(xué)方程為

(3)

3.2 超混沌控制的數(shù)值模擬和分析

本文分兩種情況研究陣列系統(tǒng)的超混沌控制.先以參數(shù)m作為控制參數(shù),取參數(shù)f=0.28,其他參數(shù)值與圖2參數(shù)值相同,數(shù)值求解方程組(3),所得陣列系統(tǒng)的Lyapunov指數(shù)λ1和λ2及電壓分岔圖如圖3所示.

圖3 隨m變化的Lyapunov指數(shù)λ1和λ2(A)、陣列分岔圖(B)及陣列中2個(gè)單結(jié)的分岔圖(C),(D)Fig.3 Lyapunov exponents λ1 and λ2 versus m (A),bifurcation diagram of the array versus m (B) and the bifurcation diagrams of the first and second individual junctiona versus m (C),(D)

由圖3(A)可見,在0.17

為了展示控制超混沌的效果,根據(jù)圖3,分別取m=0.37,1.0,1.67,其他參數(shù)值和圖3參數(shù)值相同,數(shù)值求解方程組(3),得到陣列電壓的時(shí)間序列,如圖4所示,其中(A)～(C)分別表示該陣列處于4,5,6周期狀態(tài).由圖4可見,通過適當(dāng)調(diào)節(jié)干擾信號(hào)強(qiáng)度m值,即可將圖2所示的陣列超混沌態(tài)控制進(jìn)入具有不同周期數(shù)的不同周期狀態(tài).

圖4 參數(shù)m取不同值時(shí)陣列電壓的時(shí)間序列Fig.4 Time series of the array voltage with different m values

為了闡明圖4中3個(gè)周期狀態(tài)的振蕩性質(zhì),本文給出了與其對(duì)應(yīng)陣列中2個(gè)單結(jié)間的相位關(guān)系,如圖5(A)所示.由圖5(A)可見,3個(gè)周期狀態(tài)均有γ1(τ)=γ2(τ),即同相位狀態(tài)[20].圖5(B)～(D)分別給出了和圖4(A)～(C)對(duì)應(yīng)的2個(gè)單結(jié)及陣列電壓的時(shí)間序列.由圖5(B)～(D)可見,2個(gè)單結(jié)電壓振蕩波形與陣列電壓的振蕩波形同步變化,陣列電壓的幅值是2個(gè)單結(jié)電壓幅值的和.因此弱周期干擾法控制陣列中的超混沌可使其進(jìn)入同相位的周期狀態(tài),從而使陣列周期狀態(tài)的輸出功率增加.

圖5 2個(gè)單結(jié)的相位關(guān)系(A)及與圖4(A)～(C)對(duì)應(yīng)陣列和單結(jié)電壓的時(shí)間序列(B)～(D)Fig.5 Phase relationship (A) between two individual junctions time series (B)～(D) of the voltage of the array and indiviual junctions in the three states mentioned in Fig.4 (A)～(C)

再以干擾信號(hào)的頻率f作為控制參數(shù),取干擾信號(hào)的強(qiáng)度m=0.7,其他參數(shù)值與圖2參數(shù)值相同,數(shù)值求解方程組(3),得到陣列系統(tǒng)的Lyapunov指數(shù)λ1和λ2及分岔圖,如圖6所示.由圖6(A)可見,在0.150 1

圖6 隨f變化Lyapunov指數(shù)λ1 和λ2 (A)、陣列分岔圖(B)及2個(gè)單結(jié)的分岔圖(C),(D)Fig.6 Lyapunov exponents λ1 和λ2 versus f (A),bifurcation diagram of the array versus f (B), and the bifurcation diagrams (C),(D) of the first and second individual junctions versus f

為說明該陣列系統(tǒng)能被控制進(jìn)入具有不同周期數(shù)的同相位周期狀態(tài),根據(jù)圖6(B)～(D),分別取f=0.4,0.56,1.0,對(duì)應(yīng)陣列電壓的時(shí)間序列如圖7所示,其中圖7(A)～(C)分別表示該陣列系統(tǒng)處于4,3,2周期狀態(tài).由圖7可見,通過適當(dāng)選取干擾信號(hào)頻率f值,即可控制陣列系統(tǒng)中的超混沌態(tài),使其進(jìn)入具有不同周期數(shù)的不同周期狀態(tài).

圖7 陣列電壓在不同f值時(shí)不同周期狀態(tài)的時(shí)間序列Fig.7 Time series of the array voltage in different periodic states with diferent f values

為了討論圖7中3種周期狀態(tài)的振蕩性質(zhì),本文計(jì)算了與其對(duì)應(yīng)陣列中2個(gè)單結(jié)的相位關(guān)系,結(jié)果均為γ1(τ)=γ2(τ),即同相位狀態(tài)[20],因此圖7所示的3種周期狀態(tài)均為同相位周期狀態(tài).

綜上,本文對(duì)于由2個(gè)RCLSJJ和耦合電阻構(gòu)成的陣列,在選擇參數(shù)范圍內(nèi)給出了其超混沌行為.根據(jù)弱周期信號(hào)干擾理論提出了控制該陣列系統(tǒng)的超混沌方案.數(shù)值結(jié)果表明,該方案可有效控制陣列系統(tǒng)中的超混沌,使其進(jìn)入穩(wěn)定的周期狀態(tài),通過適當(dāng)調(diào)節(jié)干擾信號(hào)的幅值和頻率即可將陣列控制在不同的周期狀態(tài),且均為同相位周期狀態(tài),從而增大了陣列周期振蕩的輸出功率.

[1] Benz S P,Burroughs C J.Coherent Emission from Two-Dimensional Josephson Junction Arrays [J].Appl Phys Lett,1991,58(19):2162-2164.

[2] Whan C B,Lobb C J,Forrester M G.Effect of Inductance in Externally Shunted Josephson Tunnel Junctions [J].J Appl Phys,1995,77(1):382-389.

[3] Whan C B,Lobb C J.Complex Dynamics Behavior inRCL-Shunted Josephson Tunnel Junctions [J].Phys Rev E,1996,53(1):405-413.

[4] Kautz R L.The ac Josephson Effect in Hysteretic Junctions:Range and Stability of Phase Lock [J].J Appl Phys,1981,52(5):3528-3541.

[5] Kautz R L.Chaos and Thermal Noise in therf-Biased Josephson Junction [J].J Appl Phys,1985,58(1):424-440.

[6] Feng Y L,Shen K.Controlling Chaos inRCL-Shunted Josephson Junction by Delayed Linear Feedback [J].Chin Phys B,2008,17(1):111-116.

[7] Feng Y L,Zhang X H,Jiang Z G,et al.Generalized Synchronization of Chaos inRCL-Shunted Josephson Junctions by Unidirectionally Coupling [J].Int J Mod Phys B,2010,24(29):5675-5682.

[8] Feng Y L,Shen K.Chaos Synchronization inRCL-Shunted Josephson Junctions via a Common Chaos Driving [J].Eur Phys J B,2008,61(1):105-110.

[9] FENG Yu-ling,SHEN Ke.Synchronization of Chaos in Resistive-Capacitive-Inductive Shunted Josephson Junctions [J].Chin Phys B,2008,17(2):550-556.

[10] FENG Yu-ling,ZHANG Xi-he,JIANG Zhi-gang,et al.Controlling Chaos inRCL-Shunted Josephson Junction by Delayed Feedback [J].Int J Mod Phys B,2009,23(18):3803-3809.

[11] FENG Yu-ling,SHEN Ke.Controlling Chaos inRCL-Shunted Josephson Junction and Studying Influence of Thermal Noise [J].Chin J Phys,2010,48(3):316-323.

[12] Klushin A M,He M,Yan S L,et al.Arrays of High-TcJosephson Junctions in Open Millimeter Wave Resonators [J].Appl Phys Lett,2006,89(23):232505.

[13] ZHOU Tie-ge,YAN Shao-lin,FANG Lan,et al.Inductance of Long Intrinsic Josephson Junction Arrays Composed of Misaligned Tl2Ba2CaCu2O8Thin Films [J].Chin Phys Lett,2006,23(7):1935-1938.

[14] Hassel J,Gronberg L,Helisto P,et al.Self-synchronization in Distributed Josephson Junction Arrays Studied Using Harmonic Analysis and Power Balance [J].Appl Phys Lett,2006,89(7):072503.

[15] Bhagavatula R,Ebner C,Jayaprakash C.Spatiotemporal Chaos in Josephson-Junction Arrays [J].Phys Rev B,1994,50(1):9376-9379.

[16] Chernikov A A,Schmidt G.Adiabatic Chaos in Josephson-Junction Arrays [J].Phys Rev E,1994,50(5):3436-3445.

[17] ZHOU Tie-ge,MAO Jing,LIU Ting-shu,et al.Phase Locking and Chaos in a Josephson Junction Array Shunted by a Common Resistance [J].Chin Phys Lett,2009,26(7):077401.

[18] 胡崗,蕭井華,鄭志剛.混沌控制 [M].上海：上?？萍冀逃霭嫔?2000:54-57.

[19] Wolf A,Swift J B,Swinney H L,et al.Determining Lyapunov Exponent from a Time Series [J].Physica D:Nonlinear Phenomena,1985,16(3):285-317.

[20] Hadley P,Beasley M R.Dynamical States and Stability of Linear Arrays of Josephson Junctions [J].Appl Phys Lett,1987,50(10):621-623.