chaotic signals

胎儿瞬时心率信号的非线性分析王立媛;杨立平;魏凤云【摘要】心率包含自治神经系统协作活动的可靠信息,对FHR信号进行研究有助于对胎儿健康状况的诊断,因此本文利用非线性时间序列的分析方法对健康胎儿心率( FHR)信号的动力学特性展开研究,利用超声多普勒监护仪测取FHR信号,选取85例健康信号进行分析,结果表明:由健康的FHR信号重构的吸引子明显区别于周期吸引子和噪声,最大Lyapunov指数LLE为0.11±0.04,关联维为58±0.34,替代数据测试表明原数据显著区别于替代数据集,因此健康胎儿的FHR信号具有非线性混沌特征,所描述的系统是一个高维混沌系统.【期刊名称】《长春理工大学学报（自然科学版）》【年(卷),期】2011(034)002【总页数】4页(P124-127)【关键词】胎儿心率;非线性分析;最大Lyapunov指数;关联维;混沌【作者】王立媛;杨立平;魏凤云【作者单位】哈尔滨工程大学理学院,哈尔滨150001;哈尔滨工程大学能源与动力工程学院,哈尔滨150001;长春师范学院历史学院,长春130032【正文语种】中文【中图分类】R271胎儿心脏受到血液运行状态变化和激素的影响，受脑中枢神经系统的支配[1]。

胎儿的健康状况和储备能力都可以由胎儿心率的变化反映出来。

故监测胎儿心率并分析其心率变异是诊治胎儿状况的一种基本方法[2]。

目前，临床上普遍采用超生多普勒的方法来对处在围产期的胎儿进行监护，即在母体腹部放置超声探头来获得胎儿心跳信号。

尽管获得的胎儿信号中含有母体信号和其它的噪声信号，然而通过自相关处理技术能够获得精确的胎儿心率信号[3]。

以往，人们主要利用线性方法来分析胎儿心率变异性，分析结果并不令人满意。

近年来，许多心脏系统生理学研究指出心率数值序列包含着不规则性，具有典型的非线性行为。

胎儿心率变异性和貌似不规则性被解释成混沌动力学的复杂时间演化特征[4]，而混沌是指发生在确定性系统中的貌似随机的不规则运动，是一种始终限于有限区域、轨道永不重复、形态复杂的运动，具有对初值极端敏感性、分数维等特点。

信号用英语怎么说信号是表示消息的物理量，如电信号可以通过幅度、频率、相位的变化来表示不同的消息。

这种电信号有模拟信号和数字信号两类。

信号是运载消息的工具，是消息的载体。

那么你知道信号用英语怎么说吗?下面来学习一下吧。

信号英语说法1：signal信号英语说法2：semaphore信号的英语例句：信号响时，人人脱帽致敬。

Everyone uncovered when the signal sounded.红灯常常是危险的信号。

A red lamp is often a danger signal.没有视频信号。

There is no video signal.灯塔每一分钟发出两次信号。

The lighthouse flashes singals twice a minute.时钟信号输入端被设计用于提供时钟信号。

A clock signal input is designed to supply a clock signal.对随机信号和混沌信号的特性进行了分析比较。

The characteristics of stochastic and chaotic signals have been studied.相反，市场可能正在发出更深层次变化的信号。

Rather, markets may be signaling a deeper shift.通常，信号的主要用途是同步某个线程与其他线程的动作。

Usually, the main use of a semaphore is to synchronize a thread? s action with other threads.声音信号可以通过电缆传送而不失真。

Audio signals can be transmitted along cables without distortion.编码信号由碟形卫星天线接收。

The coded signal is received by satellite dish aerials.一根细电缆将信号传送给一台计算机。

专题07 读后续写---2023年新高考八省最新名校联考高三试题汇编(原卷版)目录1.河北省石家庄市二中2023年高三3月试题2.2023届湖北省七市（州）高三3月联考试题3.2023届湖南省九校联盟高三第二次联考英语试题4.福建省福州第一中学2023年高三一调试题5.广东省肇庆市一中2023年高三适应性测试试题6.湖北省八市2022-2023学年高三3月联考英语试题7.江苏省连云港2023年高三调研试题8.辽宁省名校联盟2023学年高三3月联考试题9.浙江省浙南名校联盟2023年高三下学期二次联考试题10.重庆市长寿中学2023年高三3月试题1.【河北省石家庄市二中2023年高三3月试题】阅读下面材料，根据其内容和所给段落开头语续写两段，使之构成一篇完整的短文。

Price of a MiracleOnly a very costly operation could save him now and it was looking like there was no one to lend them the money. She heard Dad say to her tearful mom with desperation, “Only a miracle can save him now.”Tess went to her bedroom and poured all the change she had saved out on the floor and counted it carefully: 47 cents. She slipped out of home and made her way to Rexall’s Drug Store. She waited patiently for the pharmacist (药剂师) to give her some attention but he was too busy at this moment. Finally, she took a quarter from her pocket and banged it on the glass counter. That did it! “And what do you want?” the pharmacist asked in an annoyed tone of voice. “I’m talking to my brother from Chicago Medical University whom I haven’t seen in ages,” he said without waiting for a reply to his question.“Well, I want to talk to you about my brother,” Tess answered back in the same annoyed tone.“He’s really,really sick and I want to buy a miracle.”“We don’t sell miracles here, little girl. I’m sorry but I can’t help you,” the pharmacist said, softening a little. “Listen, I have the money to pay for it. If it isn’t enough, I will get the rest. Just tell me how much it costs.” Tess answered with a blind faith.The pharmacist’s brother was a well-dressed man. He stooped down and asked the little girl, “What kind of a miracle does your brother need?” “I don’t know,” Tess replied with her eyes welling up.“I just know he’s really sick and Mom says he needs an operation. But my dad can’t pay for it, so I want to use my money.”注意: 1.续写词数应为150左右;2.请按如下格式作答。

职场新概念英语：心跳训练有助于职场抗压Most people engaged in competitive sports have experienced soundly beating an opponent with apparent ease, only to be crushed by the same foe on a later date. They are left wondering: what did I do differently?从事竞技体育的人大多有过这样的经历：轻松击垮一名对手，可过了几天又败在这名对手的手里。

他们不禁自问：我做得有什么不同呢？According to Alan Watkins, a British medical doctor who also has degrees in psychology and immunology, it is all about the signals your heart is sending to your brain. Dr Watkins has developed a speciality in teaching athletes and business leaders about the science of heart rate coherence to help them achieve more consistent performance.拥有心理学和免疫学学位的英国医生艾伦沃特金斯(Alan Watkins)认为，这都是你的心脏向大脑发送的信号造成的。

沃特金斯医生发展了一个专科：向运动员和商界领袖讲授心率相干性的'科学原理，帮助他们实现更稳定的表现。

What is coherence? I’ve written before about heart rate variability, which reflects the fact that your heart beats at different rates over time. The interval between heartbeats is your HRV and measuring this can help you determine when your body is fully rested.什么是心率相干性呢？我以前写过心率变异性(HRV)，它反映了人的心跳速率随时间而变。

ｌ一１５６混沌信号的表示方法之一——散点图欧阳楷刘春南贾文艳周萍（北京首都医科大学生物医学工程系北京１０００５４************）抽蔓冷夭。

非线性动力学（Ｎｏｎ．ｅａｒＤｙｕｎｍｃｓ）的分析方法正日益广泛地溶人到现代医学之中。

当前主要运甩的方法有五种：散点图（Ｐｌｏｔｈ分敦维（ＦｒａｃｔｄＤｉ眦ｌＩ面∞）、Ｌｙａｐｔｍｏｖ指数、近似麓（Ａｐｐｍｘｉ，ｍａｔｅＥｍｒｏｐＤ及复杂度（Ｃｏｍｐｌｅ五ｔｙ）。

虽然对于以上几种方法的研究还处于初级阶段。

但筑与临床试验相结合的角度来看有着重要意义和广罔前景。

而以上方法的实现又青不开一个基本的先决条件。

那就是计算帆的应用。

本文就ｌ瞄床实验中最具意义的散点图为倒简介计算帆在其中的应用。

关■钶摁沌（ＣＩ岫）非线性动力学庞加莱羹面（ｈｉｎｃ－托Ｉｎｔｅｒｓｅｃｔｉｏｎ）散点图（Ｐｌｏｔ）相空问重构ＯｎｅｏｆｔｈｅＭｅｔｈｏｄｓｏｆＥｘｐｒｅｓｓｉｎｇＣｈａｏｔｉｃＳｉｇｎａｌｓ：ＰｌｏｔＯｕｙａｎｇＫａｉＬｉｕＣｈｕｎｎａｎＪｉａＷｅｎｙａｎｇｈｏｕＰｉｎｇ（ｎ哆枷耐ｏ，Ｂ眦·锄涮踟‘雠妇０，删冽＆函船脚略１咖５４）ＡＢＳＴＲＡＣＴＲｅｃｅｎｔｌｙｍａｎｙａｎａｌｙｓｉｓｏｆＢｌ：Ｒ１］ｉｎｅａｒｄｙｎａｍｉｃｓａｒｅａｐｐｌｙｉｎｇｉｎｍｏｄｅｒｌｌｍｅｄｉｃａｌ．ｍｐｌｏｔ，ｆｒａｃｔａｌｄｊｍｅｎｓｉｏｎ，Ｌｙａｐｕｎｏｖｅｘ·Ｔｈｅｆｉｖｅｆｌｍｔｒｉｓｈｅｄｍｅｔｈｏｄｓｔｈａｔａｌｗａｙｓｕｓｅｄｈｏｗｐｏｎｅｎｔｓ，ａｐｐｒｏｘｉｍａｔｅｅｎｔｒｏｐｙａｎｄｃｏｍｐｌｅｘｉｔｙ．Ｔｈｏｕｇｈｒｅｓｅａｒｃｈｏｆｔｈｅｍａｎｉｎｆｕｎｄａｍｅｎｔａｌｗｅｃａｎｓｅｅｔｈｅｉｍｐｏｒｔａｎｔａｎｄｗｉｄｅｐｒｃｅｐｅｃｔ．Ｂｕｔｉｆｗｅｗａｎｔｔｏｔａｋｅａｌｌｏｆｔｈｅｓｅｍｅｔｈｏｄｓｓｔｅｐｉｎｔｏｒｅａｌｉｔｙｗｅｍｕｓｔｄｅｐｅｎｄｏｎａｐｐｌｙｉｎｇｃｏｍｐｕｔｅｒ．Ｔｈｉｓａｒｔｉｃｌｅｉｎｔｒｏｄｕｃｅｓｈｏｗｔｏｕｓｅｃｏｒｎ－ｐｕｒｅｒｉｎｒｅｓｅａｒｃｈｉｎｇｏｎｅｏｆｔｈｅｆｉｖｅｍｅｔｈｏｄｓ：Ｐｌｏｔ．ＫＥＹＷｏＲＤＳＣｈａｏｓＮｏｎｌｉｎｅａｒｄｙｎａｍｉｃｓＰｏｉｎｃｓｒｅｉｎｔｅｒｓｅｃｔｉｏｎＰｌｏｔＰｈａｓｅ－ｒｅｂｕｉＩｄｉｎｇ１引言现代医学认为人体相当于一个复杂的动力学系统，其特性可认为是混沌的。

改善EEMD的混沌去噪方法位秀雷1林瑞霖1 刘树勇1陈燕2（1.海军工程大学动力工程学院，湖北武汉，430033；2.61062部队，北京，100091）摘要：为了提高EEMD在混沌信号去噪中的时效性，提出一种改善EEMD的混沌去噪方法。

该算法将小波包分析作为EEMD的预滤波单元，剔除了部分噪声干扰，大大减少了高斯白噪声的叠加次数，并结合EEMD抑制模式混叠的特性，可以有效地提高EEMD去噪的时效性。

利用两自由度非线性系统详述了混合滤波算法的实施过程，结果表明该方法切实可行，具有非常好的应用价值。

关键词：混沌信号；小波变换；去噪；EEMDSTUDY ON CHAOTIC DE-NOISING METHOD BASED ON IMPROVED EEMDWei Xiulei1Lin Ruilin1Liu Shuyong1Chen Yan2(1.College of Power Engineering，Naval University of Engineering，Wuhan 430033，China;2.Army 61062，Bei Jing，100091）Abstract:For the purpose of improving the timeliness performance of chaotic signals de-noising based on ensemble empirical mode decomposition(EEMD), An improved EEMD method was proposed. The wavelet packet method is taken as the pre-filter of EEMD, thus some white noise was removed and the superposition times of Gauss white noise were reduced greatly, and then it is combined with the characteristics restraining mode mixing of EEMD to extract the chaotic signal from complex strong disturbances fleetly. The process of the proposed method was discussed in detail with two-degree-freedom chaotic vibration signals, and the results show that the contaminated noise can be filtered normally.Key words:chaotic signal; wavelet transform; de-noising; EEMDHuang等[1]提出了处理非线性非平稳信号的新方法—经验模态分解（Empirical Mode Decomposition, EMD），与小波变换方法相比，EMD无需信号的先验知识，其分解完全依赖信号本身，数据分解真实可靠。

Why Do We Dream?Dreams have long fascinated scientists, psychologists, and philosophers alike. The enigmatic nature of dreams and their purpose in our lives has sparked numerous theories and debates. While the exact reasons for why we dream remain uncertain, several theories attempt to shed light onthis intriguing phenomenon.The Functionalist TheoryOne prominent theory suggests that dreaming serves a functional purpose. According to this perspective, dreams help us process emotions, consolidate memories, and solve problems. During REM sleep, the stage of sleep when most dreaming occurs, the brain is highly active. It is believed that this heightened brain activity allows us to reorganize and make sense of the information we have encountered throughout the day.Dreams often involve emotionally charged scenarios that reflect our waking life experiences. By replaying these situations in our dreams, we may gain a better understanding of our feelings and potentially find solutions to unresolved issues. Research has shown that individuals who engage in problem-solving tasks before sleep are more likely to find solutions after dreaming about the problem.The Activation-Synthesis TheoryAnother influential theory is the activation-synthesis theory proposed by psychiatrist J. Allan Hobson and psychologist Robert McCarley. This theory suggests that dreams are simply a byproduct of random neural activity in the brain during REM sleep.According to this view, when the brainstem sends random electrical signals to various parts of the brain during REM sleep, it triggers sensory experiences and emotions without any external stimuli. The cortex then attempts to make sense of these chaotic signals by creating a narrative or story-like structure – what we perceive as a dream.This theory explains why dreams often contain bizarre or unrealistic elements since they are not bound by logical constraints or externalreality. It also accounts for the often fragmented nature of dreams as different parts of the brain contribute their own random signals.The Psychoanalytic TheorySigmund Freud’s psychoanalytic theory proposes that dreams provide a window into the unconscious mind. According to Freud, dreams are symbolic representations of repressed desires, fears, and conflicts that are too threatening to be consciously acknowledged.Freud believed that dreams serve as a safety valve for expressing these hidden thoughts and desires. Through the process of dream analysis, individuals can uncover unconscious motivations and gain insight into their psychological well-being.While Freud’s theories have faced criticism over the years, his emphasis on the symbolic nature of dreams has influenced subsequent research on dream interpretation and analysis.The Evolutionary TheoryFrom an evolutionary perspective, dreams may have served an adaptive function throughout human history. Some researchers propose that dreaming facilitated the rehearsal of crucial survival skills or helped our ancestors simulate dangerous scenarios without actually experiencing them.For example, during dreams, we may encounter threatening situations that allow us to practice fight-or-flight responses in a safe environment. This theory suggests that dreaming has evolutionary advantages by enhancing our ability to navigate potential challenges in waking life.Cultural Variations in DreamingDreams are not only influenced by biological factors but also bycultural beliefs and experiences. Different cultures have diverse interpretations and attitudes towards dreaming. For instance, some indigenous cultures view dreams as a means of receiving messages from ancestors or spirits.In contrast, Western societies often place less emphasis on dreams’ spiritual or supernatural significance but focus more on their psychological aspects. However, it is important to recognize that these cultural variations do not diminish the universality of dreaming itself – virtually every individual experiences dreams regardless of their cultural background.ConclusionWhile scientists continue to explore the mysteries of dreaming, it is clear that dreams play an intriguing role in our lives. Whether they serve as a means for emotional processing, problem-solving, or connecting with our unconscious mind, dreams offer a unique window into our inner world.As we delve deeper into understanding the complexities of human consciousness and brain activity during sleep, we may eventually unravel the enigma behind why we dream. Until then, dreams will continue to captivate our imagination and provide a source of wonder and fascination for generations to come.。

基于ARM系统的混沌语音加密研究作者：谢永坚来源：《现代电子技术》2013年第13期摘要：现代语音通信大大方便了人们的交流，但是随之产生语言安全的问题，人们担心自己的通话被窃听，但传统的PC加密算法运算量大不适合在手持的移动设备中使用，用混沌信号对通话语音进行加密提高嵌入式设备数据的安全性同时运算量相对适中。

在ARM2440+Linux平台上做了有关混沌信号产生并利用混沌信号来对语音进行加密的实验，将加密后的语音进行蓝牙传输。

关键词：混沌；猫映射； ARM； Linux；语音加密；解密中图分类号：TN912.3⁃34 文献标识码： A 文章编号： 1004⁃373X（2013）13⁃0097⁃03 Research on chaotic voice encryption based on ARM systemXIE Yong⁃jian（Faculty of Automation， Guangdong University of Technology， Guangzhou 510006，China）Abstract： Modern voice communication makes people communicate with each other conveniently but leads to problem of security. People worry that their communication by telephone could be eavesdropped. But traditional PC encryption algorithm can hardly be achieved on mobile phone. Voice encryption by chaotic signals can improve security of embedded device data and also is less complex than the traditional one. We do the voice encryption test by use of chaotic signals on the ARM2440+Linux platform and transmit via bluetooth.Keywords： chaotic encryption； cat maps； ARM； Linux； voice encryption； deciphering0 引言随着通信技术的飞速发展，人们普遍使用智能手机来联系家人、朋友以及商业客户，但语音信息容易被窃听，干扰这在紧急情况下是不容许的。

英语单词英文释义As a document creator, it is important to have a good understanding of English words and their meanings. In this article, we will explore the definitions of various English words to help improve our vocabulary.1. Aberration: a deviation from what is normal or expected。

When something is described as an aberration, it means that it is different from the usual or typical. This word is often used to describe something that is out of the ordinary or unexpected.2. Benevolent: showing kindness and goodwill。

Benevolent is an adjective that describes someone who is kind, generous, and caring. It is often used to describe people who have a charitable or philanthropic nature.3. Cacophony: a harsh, discordant mixture of sounds。

Cacophony is a word that is used to describe a harsh and unpleasant combination of sounds. It is often used to describe noisy or chaotic situations.4. Debacle: a sudden and ignominious failure; a fiasco。

遵守交通规则的重要性Title: The Importance of Following Traffic RulesTraffic rules are the foundation of road safety, ensuring the smooth flow of vehicles and pedestrians. Obeying traffic signals, such as stoplights and turn signals, is crucial for preventing accidents. Crossing the road at designated crossings and using pedestrian crossings properly is essential for our safety. Wearing helmets while riding bicycles or motorcycles can significantly reduce the risk of head injuries in case of accidents. Moreover, respecting the right of way and driving at appropriate speeds is vital for maintaining a safe driving environment. By adhering to these rules, we can contribute to a safer and more orderly road network, protecting not only ourselves but also others.交通规则是道路安全的基础，确保了车辆和行人的顺畅通行。

遵守交通信号，如红绿灯和转向灯，对于预防事故至关重要。

在指定地点过马路并正确使用人行横道对于我们的安全至关重要。

心房颤动房颤Atrial fibrillation目录心房颤动房颤Atrial fibrillation (1)流行病学 (1)Overview (2)Symptoms (3)When to see a doctor (4)病因 (5)Causes (5)Possible causes of atrial fibrillation (7)Atrial flutter (7)Risk factors (8)Complications (9)Prevention (9)心房颤动（房颤）是指规则有序的心房电活动丧失，代之以快速无序的心房颤动波，是最严重的心房电活动紊乱，也是常见的快速性心律失常之一，是最常见的持续性心律失常之一，房颤总的发病率为0.4%，随着年龄增长房颤的发生率不断增加，75岁以上人群可达10%。

流行病学在普通人群中房颤的患病率约为0.4%-1.0%。

房颤的患病随着年龄的增加而增加，小于60岁的人群患病率较低，而80岁以上的人群可髙达8%。

40 岁以下者房颤的发病率为0.1%/年，80岁以上的男性和女性房颤的发病率分别为2%/年和1.5%/年。

房颤患者远期脑卒中、心力衰竭和全因死亡率风险增加,特别是女性患者。

与窦性心律者相比，房颤患者的死亡率增加一倍。

非瓣膜性房颤患者缺血性卒中的发生率为5%/年，是无房颤者的2-7倍。

若考虑短暂脑缺血发作(TIA)和无症状的脑卒中，伴随房颤的脑缺血发作的发生率为7%/年。

与年龄匹配的对照者相比，房颤的风湿性心脏病患者发生脑卒中的风险增加17倍；与非风湿性房颤患者相比,风险增加5倍。

房颤患者栓塞发生率随着年龄的增加而增加，50-59岁患者因房颤所致的脑卒中每年发生率为1.5%，而80-89岁者则升高到23%。

男性患者栓塞发病率在各年龄段均高于女性。

房颤的总患病率、年龄分组、性别分组、病因分组后的患病率均和国外相关资料的趋势接近。

中国因房颤而住院的患者也有增加的趋势。

Synchronization of chaotic solid-state Nd:YAGlasers: Application to secure communication Electronics and Telecommunications Department, Scientific Research andAdvanced Studies Center of Ensenada (CICESE)Received 15 September 2006; received in revised form 21 November2006Available online 23 February 2007AbstractIn this paper, the synchronization problem of coupled chaotic lasers in master–slave configuration is numerically studied.The approach used allows to give a simple design procedure for the slave laser. In particular, we consider a complex system composed by two chaotic Nd:YAG lasers coupled through the first state variable of the master laser. Synchronization of chaotic Nd:YAG lasers is achieved by injecting the chaotic signal from the master Nd:YAG laser into the slave Nd:YAG laser. The robustness of synchronization is discussed when a mismatch of parameters occurs, and the effects of the channel noise on recovered information are showed. A potential application of chaotic synchronization of Nd:YAG2007 Elsevier B.V. All rights reserved. lasers to transmit encrypted digital information is also given.PACS: 05.45.Pq; 05.45.Vx; 05.45.Xt; 42.55.Px; 42.65.Sf; 89.75._kKeywords: Chaotic synchronization; Nd:YAG laser; Encrypted information; Secure communication; Complex systems1. IntroductionIn the past decades, chaos synchronization has received a tremendous increasing interest see, e.g., [1–8] and references therein. This property is supposed to have interesting applications in different fields, particularly to design secure communication systems. The broadband and noise-like characteristic of chaotic signals are seen as possibly highly secure media for communication. The private/secure communication schemes are usually constituted by a chaotic system as transmitter along with an identical chaotic system as receiver; where the confidential information is imbedded into the transmitted chaotic signal by direct modulation, masking, or another technique. At the receiver end, if synchronization of the two systems is achieved, then it is possible to extract the hidden information from the transmitted noise-like signal.Different chaotic synchronization schemes and their applications to private/secure communications have been proposed. This was first studied using electronic oscillatorcircuits [9–14] and it has been more recently proposed in solid-state lasers [15], fibre ring lasers [16], semiconductor lasers [17–19], and microchip lasers [20]. However, in the most of reported works, the synchronization methods used are not appropriate in general to consider a larger number of chaotic systems. Therefore, it is fundamental to try new coupling topologies that allow to obtain synchronization conditions for many coupled chaotic systems as nodes or cells.On the other hand, many chaotic communication schemes have been constructed via electronic oscillator circuits. For example, with desirable properties of low detectability, see, e.g., [13] or where the encrypted information is recovered faithfully [14]. Nevertheless, most of the electronic circuits have been designed in the audio range with confidential information bandwidths limited to 10 kHz. In these communication schemes is difficult to achieve multi-GHz frequencies required in many transmission channels. As well as, most already installed high speed communication systems are based on optical fibers. Therefore, a chaotic encryption scheme directly based on an optical carrier is highly desirable.The key for optical chaotic communication consists in that two unidirectionally coupled chaotic lasers, – in our case solid-state Nd:YAG (Neodymium doped: Yttrium Aluminium Garnet) lasers –, can synchronize to each other. Synchronization means that the irregular time evolution of the master laser, either in the optical power or in the wavelength, can be exactly reproduced by the slave laser.In this work, the solid-state laser Nd:YAG equations are chosen as chaos generator implemented from this new perspective and their possibilities for synchronization are confirmed. Synchronization of chaotic Nd:YAG lasers is applied to transmit encrypted digital confidential information. In addition, effects on synchronization and recovered information due to parameter mismatch and additive channel noise are discussed.The rest of this work is organized as follows: In Section 2, we give a brief review on complex systems and their synchronization, including level coupling concept, globally coupled configuration for identical cells, and necessary conditions for exponential synchronization. In Section 3, we give a mathematical model which describes the chaotic dynamics of a single solid-state Nd:YAG laser, which will be used as fundamental cell.In Section 4, we show synchronization of two chaotic solid-state Nd:YAG lasers, using the globally coupled configuration. In Section 5, we illustrate a potential application to chaotic communications, where we transmit encrypted digital confidential information, and we show the effects of parameter mismatch and channel noise on the recovered information. Finally, some conclusions are given in Section 6.2. Synchronization of complex systems2.1. Dynamical networksWe consider a complex network composes of N identical cells (or nodes), linearly and diffusively coupledcells through the first state variable of each cell. In thisdynamical network, each cell constitutes a n-dimensional dynamical system, described as follows(),1,2,...,,i i i x f x u i N =+= (1) where 12(,,...,)T n i i i im x x x x R π=∈are the state variables of the cell or node i , 1i i u u R =∈is the input signal of the cell i, and is defined by1,1,2,...,ij Ni j j u c a X i N ==Γ=∑ (2)the constant c > 0 represents the coupling strength of the dynamical network, and Γ*n n R ∈is a constant 0–1matrix linking coupled state variables. For simplicity, assume that Γ=diag （12,,...,n r r r ） is a diagonal matrix with 1i r = for a particular i and 1i r =for j i ≠This means that two coupled cells are linked through their i th state variables. Whereas, *()N N ij A a R =∈is the coupling matrix, which represents the coupling configurationFig. 1. Cell i with inputs 1i u , states i x , and outputs 1i y .of the dynamical network. If there is a connection between cell i and cell j, then 1ij a =; otherwise, 0ij a = for i j ≠. The diagonal elements of coupling matrix A are defined as1,1,,1,2,...,N N ii ij ji j j i j j i a a a i N =≠=≠=-=-=∑∑ (3)If the degree of cell i is i d , then,1,2,...,.ii i a d i N =-= (4) Now, suppose that the dynamical network is connected in the sense that there are no isolated clusters.Then, the coupling matrix A is a symmetric irreducible matrix. In this case, it can be shown that zero is an eigenvalue of A with multiplicity 1 and all the other eigenvalues of A are strictly negative [22,23].Fig. 1 illustrates the structure of an isolated cell with a single input 1i u and single output 1i y ; this output can be defined as 1()i i i y g X =, where (.)i g is in general a nonlinear function of the state vector. Nevertheless, in many cases, the output signalof interest coincides with one of the state variables. In this work, we consider as output of interest to 11i i y x = the first state of each cell i.Synchronization state of cells in complex systems, can be characterized by the nonzero eigenvalues of the coupling matrix A. The dynamical network (1) is said to achieve (asymptotically) synchronization, if [23]:12()()...(),N x t x t x t ===as t →∞. (5)The diffusive coupling condition (3) guarantees that the synchronization state is a solution,()n s t R ∈, of an isolated cell, that is,.()(())s t f s t =， (6)where ()s t can be an equilibrium point, a periodic orbit, or a chaotic attractor. Thus, stability of the synchronization state,12()()...()()N x t x t x t s t ====， (7)of dynamical network (1) is determined by the dynamics of an isolated cell, i.e., – function f and solution ()s t –,the coupling strength c, the inner linking matrix C, and the coupling matrix A.Given the dynamics of an isolated cell and the inner linking structure, the synchronizability of dynamical network (1) with respect to a specific coupling configuration (or topology) is said to be strong, if the dynamical network can synchronize with a small coupling strength c.2.2. Synchronization conditionsTheorem ([22,23]). Consider the dynamical network (1). Let1230...N λλλλ=>≥≥≥， (8)be the eigenvalues of its coupling matrix A. Suppose that there exists an n x n diagonal matrix D > 0 and two constants d - < 0 and τ> 0, such that[(())][(())]T nD f s t d D D D f s T d I τ+Γ++Γ≤-， (9) for all d d -≤, where *n n n I R ∈ is a unit matrix. If, moreover,2c d λ-≤ (10) then, the synchronization state (7) of dynamical network (1) is exponentially stable. Since 20λ< and 0d -<, inequality (10) is equivalent to20|d d c λ--<≥| (11)A small value of 2λcorresponds to a large value of 2||λ, which implies that dynamical network (1) can synchronizewith a small coupling strength c. Therefore, synchronizability of dynamical network (1) with respect to a specific coupling configuration can be characterized by the second-largest eigenvalue of the corresponding coupling matrix A.2.3. Globally coupled networksThe coupling configurations commonly studied in synchronization of complex networks are the so-called:globally coupled networks, nearest-neighbor couplednetworks, and star coupled networks. In this work, we consider only complex networks of identical cells globally coupled. In the sequel, we will show the particular arrangement of the coupling matrix for this class of dynamical network.Let (,)G V E = ; EÞ be a graph, consisting of ||N V = nodes or cells, with 12(){,,...,}N V V G υυυ==the node set and ||M E = edges between nodes, where 12(){,,...}M E E G e e e ==denotes the link set. In this work, we consider complex networks where all nodes are interconnected to each other, these regular coupling networks are reported in the literature as global coupled networks [22,23]. In addition, for these particular complex networks,in the corresponding graphs all their nodes are connected and are finite, without self-loops, and without multiple edges between two nodes. Under the mentioned assumptions, there are two principal associated matrices of interest with a graph G [24,25]: (i) The familiar (0, 1) Adjacency matrix ():A G NXN matrix whose entries ij a are given by1(,)(){0,ij if i j E G a otherwise ∈=where (,)()i j E G ∈ means that node i connects with node j, i.e., are adjacent. For a single graph without selfloops, the adjacency matrix must have 0’s on the diagonal. And (ii) the diagonal matrix, Degree matrix ():D G NXN matrix whose entries ij d are given by(,)(),{0,i ij d if i j E G d otherwise ∈=where di is the degree of the node i, and given that in this topology, each node i is connected with 1N - nodes,then we have 12...1N d d d N ====-The Laplacian matrix of a graph ()L G with N nodes is a NXN matrix ()()L G D G =, with entries ij l expressed as1(,)(),{,0,ij i if i j E G l d if i j otherwise -∈== The Laplacian matrix is some times also called the Kirchhoff matrix of G because of its role in the Matrix-Tree Theorem, which is usually attributed to Kirchhoff[24,25]. Another name for this matrix is the admittanceof G, which originates in theory of electrical networks. Nevertheless, the name of Laplacian matrix is more appropriate since it is just the matrix of a discrete Laplacian operator. Let us now compute the Laplacian matrix ()()()L G D G A G =- for the mentioned network topology (globally coupled network), which corresponds to the coupling matrix gc A , that is,Fig. 2. Network of identical cells globally coupled.1000011101001011()000001111000011110gc N N A L G N -⎡⎤⎡⎤⎢⎥⎢⎥-⎢⎥⎢⎥⎢⎥⎢⎥==-⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥-⎣⎦⎣⎦， thus, the coupling matrix for globally coupled networks is given by1111111111111111gc N N A N ----⎡⎤⎢⎥----⎢⎥⎢⎥=⎢⎥----⎢⎥⎢⎥----⎣⎦. (12) Note that the Laplacian matrix have rows with sum equal to zero. This matrix has a single eigenvalue at 0 and all the others equal to -N. Hence, the second largest eigenvalue 2gc N λ=- decreases to -∞ as →∞,i.e.,2lim gc N λ→∞=-∞. (13)The globally coupled configuration means that any two different cells are connected directly, this is shown in Fig. 2 for example, for N = 6 globally coupled cells.3. Dynamics of a solid-state Nd:YAG laserIn this section, we present the chaos generator (Nd:YAG laser) used as fundamental cell in the dynamical network to be synchronized. We take a modification of the equations suggested in [21] for a single solid-state Nd:YAG laser with a sinusoidally modulated loss, described by the state equations.01((cos())),X F t X ααω=-+(14).20(),F A F FX γ=--where ()X t and ()F t constitute the states of the Nd:Y AG laser, physically represent the amplitude of the electronic field of the laser and its gain, respectively. The parameters 0α and 0A denote the rates of intra cavity loss and pump strength, respectively. While 1α represents the strength of modulation of the intra cavity loss at a frequency ω, and γ is a ratio of the time scale of light in the laser cavity, and the upper level spontaneous emission lifetime of the lasing media.Fig. 3. Projection of chaotic attractor on the (X,F)-plane.We performed our simulations using 210γ-=to avoid stiffness problems that arise with smaller values of γ.It is known that for suitable values of parameters 0α and 1α, the Nd:YAG laser (14) exhibits chaotic oscillations,in Fig. 3 is shown the projection of chaotic attractor on the (X,F)-plane; where we have taken the following set of parameter values:00.9α=,100.2, 1.2,0.01.A and αγ===For the particular case, where all losses are modulated equally at the rate [21], 0.90.2cos(0.045)t +, the pump parameters were equal to 1.2.The single solid-state Nd:YAG laser is modulated with a depth a1 relative to its mean losses 0α. In absence of modulation, the Nd:YAG laser is stable and exhibits damped oscillations to their fixed-point values [23]. All numerical results presented here and in next sections were obtained by means of the familiar MATLAB Software Package, using a fourth-order Runge –Kutta integration algorithm with step time of size 0.0001.4. Chaotic secure communicationIn this final part, we apply the synchronization of two chaotic Nd:YAG lasers, to transmit encrypted confidential information through a public channel. The securetransmission of digital information is achieved by chaotic switching technique [9,10]. In this technique, the confidential information t m is a binary signal, and is used to modulate one parameter of the master/transmitter laser (11) and (12), i.e., t m controls a switch whose action changes the parameter values of the signal modulate in the master/transmitter laser. Thus, according to the value of t m at any given time t, the chaotic master/transmitter (13) and (14) has either the parameter value 0α, or the parameter value 0α-. At the slave/receiver end (9) and (12), t m is decoded by using the synchronization error to decide whether the received signal corresponds to one parameter value, or the other, which can be interpreted as a bit ‘‘1’’ or ‘‘0’’. In our case, to transmit encrypted information t m via chaotic switching, the parameter a0 will be modulated in01cos(),t ααω+of master laser (14). To this purpose, we use a ‘‘modulation rule’’ to modulate t m as follows00().(),t r m t αα=+The following figures illustrate the binary encoding transmission and the recovery of t m at the receiver, when the parameter 0αis switched between 0(1)1α= and 0(0)0.9α=. Fig. 7 depicts: (a) The confidential information t m to be transmitted, (b) the transmitted chaotic signal 11()X t to receiver laser through of insecure channel, and (c) the synchronization error detection 1121()()()e t X t X t =-, and the recovered information ,()m t after a filtering stage. Decoding: 1 when ()0e t ≠ and 0 when ()0e t =. Fig. 4 clearly shows that the original information t m is completely recovered at the receiver end.Fig. 4 depicts: (a) The confidential information to be transmitted, (b) the transmittedchaotic signal to receiver laser through of insecure channel, and (c) thesynchronization error detection , and the recovered information after a filtering stage.Thanks to this robustness property of the synchronization between Nd:YAG lasers, we can still transmit encrypted information and recover it at the receiver end (faithfully). We have shown that, even with the presence of parameter mismatch and additive channel noise, relatively large, information encoding into the chaotic laser for transmission and decoding of original information m(t) is relatively easy, after a filtering stage.5. ConclusionsIn this work, we have presented synchronization of an array of two chaotic solid-state Nd:YAG lasers. By means of computer simulations, we have illustrated the synchronization of two Nd:YAG lasers, in master–slave coupling configuration. In particular, this synchronization was achieved by using a communication network globally coupled of lasers. Given that the applied synchronization method is more general than those developed earlier [1,9] where a stable subsystem with negative Lyapunov exponents is necessary. Thus, the approach used for synchronization can be easily extended to synchronize complex dynamical networks composed by many large Nd:YAG lasers as coupled nodes.In addition, the approach can be implemented on experimental setup, and shows great potential for actual optical communication systems in which the encoding is required to be secure.A possible experimental implementation of the proposed synchronization scheme (and its application to secure communication) could be realized with any laser or electronic system that generates a chaotic sequence of pulses. In particular, a loss modulated external-cavity semiconductor laser could be used to transmit encrypted information at much higher bit rates. All lasers of the complex dynamical network are taken to be similar, which are mutually coupled by injection of the transmitted electronic field into each laser in the (global) arrangement.Thanks to robustness of synchronization, the (significantly) presence of parameter mismatch and additive noise in the channel; the recovery of the encrypted digital information at the receiver is faithfully.In a forthcoming work we will be concerned with a complete robustness analysis of parameter mismatch,variation in coupling strength, and SNR. AcknowledgementsThis work was supported by the CONACYT,Me´xico under Research Grant Nos. J49593-Y and P50051-Y.And by UABC, Me´xico, under Research Grant No. 0460.References[1] Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett1990;64(8):821–4.[2] Special Issue on Chaos synchronization and control: Theory andapplications.IEEETrans Circ Syst I 1997; 44 (10).Special Issue on Control and synchronization of chaos. Int J Bifurct Chaos 2000;10 (3–4).[3] Pikovsky A, RosenblumM, Kurths J. Synchronization: A universal concept innonlinear sciences. Cambridge University Press; 2001.[4] Cruz-Herna´ndez C, Nijmeijer H. Synchronization through filtering. Int J BifurctChaos 2000;10(4):763–75. Synchronization through extended Kalman filtering.And In: Nijmeijer H, Fossen TI. New Trends in Nonlinear Observer Design, Lecture Notes in Control and Information Sciences 244 Springer-Verlag; 1999, p.469–90.[5] Sira-Ramı´rez H, Cruz-Herna´ndez C. Synchronization of chaotic systems: Ageneralized Hamiltonian systems approach. Int J Bifurct Chaos 2001;11(5):1381–95. And in Procs American Control Conference (ACC2000) Chicago, USA. p. 769–73.[6] Aguilar AY, Cruz-Herna´ndez C. Synchronization of two hyperchaotic Ro¨ sslersystems: Model-matching approach. WSEAS Trans Syst 2002;1(2):198–203;Lo´pez-Mancilla D, Cruz-Herna´ndez C. Output synchronization of chaotic systems: Model-matching approach with application to secure communication. Nonlinear Dyn Syst Theory 2005;5(2):141–56.[7] Feldmann U, Hasler M, Schwarz W. Communication by chaotic signals: Theinverse system approach. Int J Circ Theory Appl 1996;24(5):551–79.[8] Nijmeijer H, Mareels IMY. An observer looks at synchronization. IEEE Trans CircSyst I 1997;44(10):882–90.[9] Cuomo KM, Oppenheim A V. Circuit implementation of synchronized chaos withapplications to communications. Phys Rev Lett 1993;71(1):65–8.[10] Cuomo KM, Oppenheim A V, Strogatz SH. Synchronization of Lorenz-basedchaotic circuits with application to communications.IEEE Trans Circ Syst I 1993;40(10):626–33.[11] Posadas-Castillo C, Cruz-Herna´ndez C, Nu´n˜ez R. Experi mental realization ofbinary signals transmission based on synchronized Lorenz circuits. J Appl Res Technol 2004;2(2):127–37.[12] Cruz-Herna´ndez C, Lo´pez-Mancilla D, Garcı´a V, Serrano H, Nu´n˜ez R.Experimental realization of binary signal transmission using chaos. J Circ Syst Computers 2005;14(3):453–68.[13] Cruz-Herna´ndez C. Synchronization of time-delay Chua’s oscillator withapplication to secure communication. Nonlinear Dyn Syst Theory 2004;4(1):1–13.[14] Cruz-Herna´ndez C, Romero-Haros N. Communicating via synchronizedtime-delay Chua’s circuits. Commun Nonlinear Sci Numer Simul, in press.doi:10.1016/sns.2006.06.010.[15] Colet P, Roy R. Digital communications with synchronized chaotic lasers. OptLett 1994;19(24):2056–8.[16] Van Wiggeren GD, Roy R. Communication with chaotic lasers. Science1998;279:1198–200; Van Wiggeren GD, Roy R. Int J Bifurct Chaos 1999;9(11):2129–56.混沌的同步固态Nd:YAG激光器:应用程序安全通信电子和电信部门、科研和先进的研究中心恩塞纳达港(CICESE) 2006年9月15日收稿,在修订稿于2006年2月21日收到网上于2006年11月23日收到摘要本文立足于同步耦合混沌激光在主从配置问题数值研究。

Instruments and Experimental Techniques, Vol. 45, No. 2, 2002, pp. 231–236. Translated from Pribory i Tekhnika Eksperimenta, No. 2, 2002, pp. 94–99. Original Russian Text Copyright © 2002 by Lebedev, Ivanov.The phenomenon of dynamic chaos, which substan-tially determines the behavior of a complex self-sus-tained oscillation system, obviously manifests itself in coupled oscillators. While in-sync modes in coupled oscillators are investigated quite well, chaotic dynam-ics of such systems continues to attract attention of researchers. This is explained by the prospects of their implementation for producing coupled oscillators (noise generators) as well as in noise radio communica-tion devices and in information technologies for read-ing, writing, and the protection of information [1–4]. The numerical analysis of such systems, especially in the microwave range, is difficult, because constructing mathematical models described by systems of low-order equations is impossible; the latter can be regarded as qualitative models of their low-frequency analogs with respect to actual systems.For the purpose of producing chaotic dynamic devices with predetermined statistical and performance characteristics, experimental investigations of two self-sustained oscillation system were conducted in this work: the first one is based on oscillators with identical parameters, and the second is based on oscillators with substantially different parameters.The factors, which determine chaotic dynamics of self-sustained oscillation systems in general and cou-pled oscillators in particular, were taken into account when producing the experimental specimen. These fac-tors are the active-element operating mode, the nonlin-earity of its dynamic characteristics, and parameters of the self-sustained oscillation system: the passband, feedback coefficient, sluggishness, and the signal delay in the feedback circuit.Microwave bipolar transistors 2T610, 2T640A-2, and 2T647A-2 are used as active elements. The possi-bility of changing the transistor operating mode within wide ranges is provided. An oscillating system of the nonresonance type and a broadband feedback circuit are features of the investigated oscillators, which pro-vide conditions for exciting a series of fundamental oscillations and their harmonic components of higher orders and undertones.The system of coupled oscillators with identical parameters (Fig. 1a) is based on a multitransistor design using a microstrip technique and contains simi-lar active elements 1, oscillatory circuits 2, and a com-mon delayed-feedback circuit. The oscillators are cou-pled with the help of capacitive elements 3. The partial oscillators differ only by the operating frequencies. This difference is determined by their location on the board relatively to the delayed feedback circuit.The system of oscillators with different parameters contains two coupled oscillators (Fig. 1b). The first one with delayed feedback and lagged self-bias is a master oscillator. It sets the system’s natural frequencies and contains a nonlinear amplifier 1, oscillating system with distributed parameters 2, delayed feedback circuit 3, and sluggish self-bias circuit 5. The second oscillator is a slave oscillator and works in the mode of external start-up from the first oscillator. It contains a nonlinear amplifier 1, oscillating system 2, and adjustable delayed-feedback circuit 3, with which the positions of the natural frequencies of this oscillator can be varied relative to those of the first oscillator. The capacitive coupler 4 connects the first oscillator output to the sec-ond-oscillator input.Before considering systems of coupled oscillators as a whole, we note the features of performance of a single oscillator, because, in this case, chaotic modes are also possible, being caused by several reasons, in particular, by sluggishness and delay. An altered char-acter of the oscillatory processes in the investigated systems upon changes of a parameter is reflected in a bifurcational diagram (Fig. 2) and spectrograms (Fig. 3).Chaotic OscillatorsM. N. Lebedev and V. P. IvanovInstitute of Radio Engineering and Electronics, Fryazino Branch, Russian Academy of Science, pl. Akademika Vvedenskogo 1, Fryazino, Moscow oblast, 141120 RussiaReceived March 26, 2001; in final form, July 10, 2001Abstract— The results of the experimental research of various kinds of coupled oscillators in the meter and centimeter wavelength ranges are presented. Chaotic dynamics of a system with identical partial oscillators and a system of oscillators with essentially different parameters are investigated. General scenarios of the transition of oscillations to chaos, as well as particularities of their behavior in the autonomous operating mode and under external action, are determined. A RF masking device circuit based on a system of coupled oscillators is pre-sented.0020-4412/02/4502-$27.00 © 2002 åAIK “Nauka/Interperiodica”0231232INSTRUMENTS AND EXPERIMENTAL TECHNIQUES V ol. 45 No. 2 2002LEBEDEV, IVANOVThe supply voltage is chosen here as a parameter ( U / U 0 in Fig. 2, where U 0 = 1V), which determines the gain in the system. Adjusting the supply voltage allows one to observe all the variety of oscillation processes. In case of a small gain, when the amplitude generationconditions are fulfilled only at a single frequency, a mono-chromatic oscillation is excited (zone A , Fig. 2) at a rela-tive frequency equal to 960 F / F 0 , where F 0 = 1000 MHz (conventional frequency).As the gain increases ( U / U 0 ≅ 2.5 ), conditions for exiting oscillations (modes) at several natural frequen-cies (zone Ç , Fig. 2) separated by ∆ f = 1/ T , where í is a signal delay in the feedback circuit, are met. In this case, the transistor operates in the overvoltage mode. In this mode, the gain of a small signal exceeds the gain of a large signal for their simultaneous application to the amplifier input, and minor disturbances in the system grow from one signal pass to another in the oscillator feedback circuit [5, 6].A synchronous mode, characterized by the locking of some oscillation modes by the appropriate frequency components of interaction of other modes, establishes in the system if the number of exited modes in the oscil-lator is not large. The amplitude conditions of excita-tion at a larger number of natural frequencies are ensured by increasing the oscillator supply voltage. In this case, self-mode locking turns out to be impossible (right edge of zone Ç , Fig. 2), since, even at a small dis-persion in the delayed-feedback circuit, the detuning of separately exited oscillations relative to the correspond-ing synchronizing components of interaction of other modes increases occurs.In this case, oscillations are unstable, and each mode is entrained by other modes in different ways. Such an asynchronous mode is characterized by chaotically altering phase differences between oscillations at dif-ferent natural frequencies. A chaotic pulsation of amplitudes then takes place, because the gain at any natural frequency is a complex function of amplitudes of all other asynchronously interacting oscillations [7].Besides that, an additional nonlinear signal transfor-mation using a self-bias circuit occurs [5, 8]. A control voltage produced by a self-bias circuit is determined by the amplitude of preceding oscillations; i.e., the posi-tion of the working point and the gain of the nonlinear oscillator element with a delayed feedback changes from travel to travel of the signal over the delayed feed-back circuit.Since, as the result of the avalanche multiplication of intermodulation components, chaotic oscillations establish in the oscillator (zone ë , Fig. 2), the self-bias circuit also produces a chaotic low-frequency control voltage, which arrives at the oscillator input and ran-domly changes the position of the working point of the nonlinear amplifier. This leads to an additional modula-tion of the resulting signal, and the oscillation spectrum extends to lower frequencies.Hence, three zones can be distinguished in the bifur-cation diagram: Ä , monochromatic oscillation zone; Ç ,multifrequency oscillation zone; and ë , chaotic (sto-chastic) oscillation zone.(b)0.11.01.924U / U 0F / F 0 ABCFig. 1. Block diagram of the investigated oscillators: (a)system of coupled oscillators with identical parameters; (b)system of coupled oscillators with different parameters; ( 1 )active element; ( 2 ) oscillatory circuit; ( 3 ) delayed-feedback circuit; ( 4 ) coupling element; and ( 5 ) sluggish self-bias cir-cuit.Fig. 2. Bifurcation diagram: A, B , and Cmonochromatic,multifrequency, and chaotic oscillation zones, respectively.INSTRUMENTS AND EXPERIMENTAL TECHNIQUES V ol. 45 No. 2 2002CHAOTIC OSCILLATORS233Qualitatively, scenarios of change to chaotic oscilla-tions in the oscillators considered are analogous, and the width of the generation zones and the number of bifurcations depend on the parameters of circuit com-ponents and the transistor operating modes. Spectro-grams of the oscillations for zone Ç (Fig. 2) of I oscil-lators, which are included in the investigated systems at equal values of parameters U / U 0 = 3.5 and í = 5.6 ns,are presented in Figs. 3a and 3b. It is obvious that the oscillator in Fig. 1a is characterized by a deterministic dynamics with a multifrequency oscillation spectrum (Fig. 3a), whereas the dynamics of the oscillator in Fig. 1b is close to chaotic (Fig. 3b), and its oscillation spectrum has many nonequidistant components. This is due to the fact that a sluggish self-bias circuit intro-duced into the oscillator in Fig. 1b causes an additional instability of this oscillator and determines the differ-ences in energy spectra.A combined operation of two coupled oscillators is characterized by a more complex oscillation dynamics.In the case of nonmultiple partial oscillation frequen-cies, when a synchronous operation is impossible or unstable, a beats mode takes place. It is accompanied by self-modulation phenomena with a subsequent tran-sition to chaos via a sequence of period-doubling bifur-cations with an increase in the supply voltage.Scenarios of the transition to chaos may contain var-ious numbers of bifurcations depending on the whole totality of the system’s parameters: the signal delays í in the feedback circuit, the ratios of partial oscillation frequencies, the coupling between oscillators, etc.Along with this, disseminations of chaotic oscillations (stochastic spikes), determining scenarios of the transi-tion to chaos via intermittency [9], are available in zone Ç (Fig. 2). The intermittency is based on the system’s inability to attain the phase locking as a result of mode competition.Oscillation spectrograms for the joint operation of two coupled oscillators (Fig. 3c for system in Fig. 1a and Fig. 3d, for system Fig. 1b) are presented in Figs. 3c and 3d for the following main parameters: the delay is T = 5.6 ns, the ratio of partial frequencies is100.1S 20304050 1.01.9 F 0(a)(b)(d)Fig. 3. Oscillation spectrograms: (a, b) I partial oscillators (Fig. 1), included in the investigated systems at U / U 0 = 3.5, í = 5.6 ns;(c, d) systems of coupled oscillators with identical and different parameters, respectively, í = 5.6 ns, F 1 / F 2 = 1.13, U / U 0= 5.0.234INSTRUMENTS AND EXPERIMENTAL TECHNIQUES V ol. 45 No. 2 2002LEBEDEV, IVANOV F 1 / F 2 = 1.13, and U / U 0 = 5.0. Comparing the dynamics of the oscillation processes in the investigated systems,we note that the chaos zone for the system of oscillators with different parameters is wider at the expense of the narrowing of the first two zones and may occupy at most half the range of U / U 0 variation. The system of oscillators with identical parameters is more stable with respect to self-mode-locking regimes, and a transition to chaos occurs at higher voltage values ( U / U 0 > 4.0).Moreover, partial self-mode locking in a system of identical oscillators can be observed in the chaotic oscillation mode. This is testified by the stable charac-ter of chaotic oscillations, which holds under both changes of the supply voltage (in a range of 3–5% U 0 )and an external action. This means that, by analogy with ordinary mutual locking of sine-wave oscillators, certain lock-in range (synchronization range) exists for chaotic systems, within which a significant increment of the sys-tem output power is observed: ê out = 0.8 – 0.9( ê 1 + ê 2 ) ,where ê 1 and ê 2 are the output powers of the partial oscillators. In this case, partial mutual locking takes place. This phenomenon is of special interest and is beyond the framework of this study [10, 11].External noise nuisance is an additional factor for increasing the chaotic oscillation stability. As is known,an external noise signal normalizes the oscillation pro-cess in a chaotic dynamic system and reduces its sensitiv-ity to changes of its parameters [12]. For example, a decrease in the spectral-power-density nonuniformity of a high-frequency noise signal by 3–5 dB is observed under the action of external low-frequency noise ( U ext = 1.5 V).As a whole, the investigations performed allow us to conclude that systems of coupled oscillators both with identical and different parameters generate chaotic oscillations in certain operating modes via a sequence of bifurcations of period doubling. It is expedient toSpectral density, µV/(m · kHz 1/2)Hz1010101010Fig. 4. Simplified circuit of the RF masking device: (1) source of low-frequency noise (Q 1, Q 2 – KT3172A9, Q 3 – KT665A9, D –2É4016); (2) noise generator (Q 4 – KT610A, Q 5 – 2T939A); (3) antenna; and (4) serviceability test circuit.Fig. 5. Normalized levels of masking and informative signals: (1) RF masking device for a SONY monitor (Trinitron); (2) 800 × 6003 SVGA; and (3) 1024 × 768 × 85 SVGA.CHAOTIC OSCILLATORS235apply systems of coupled oscillators with different parameters while constructing broadband noise gener-ators, because chaotic oscillations formed by them are less critical to external and internal destabilizing factors (supply voltage, temperature, spread of parameters of active elements, and change in the load), and static characteristics are close to those of white noise.As an example of particular implementation, Fig. 4 presents a circuit of a device for RF masking subsidiary electromagnetic radiations and pickups (SERAP) of computer aids while processing confidential data [13]. This device was developed on the basis of the results of this study. Taking into consideration the SERAP levels, necessary spectral and energy characteristics of the device, which forms the noise electromagnetic masking field in a given frequency range, were determined. The RF masking device contains a noise generator, a broad-band antenna, a low-frequency noise source, and a ser-viceability test circuit.The noise generator represents a system of two cou-pled oscillators on transistors Q4 and Q5. The first oscil-lator on Q4 contains a delayed feedback circuit (T = 5.5 ns) and a sluggish self-bias circuit R3,C2. The interval between the natural frequencies of this oscillator amounts to ~180 MHz. The position of the Q4 transistor operating point is determined by a voltage divider on resistors R1 and R2 and by the voltage drop across the elements of the self-bias circuit, which depends on the emitter current flowing through the transistor, the ratio between the charge and discharge time constants of capacitor C2, and the signal delay time in the feedback circuit.The second oscillator is based on Q5 and contains an adjustable delayed feedback circuit (T = 3.0 ns) built as a microstrip line. As the capacitance of ë3 changes, the natural frequency of this oscillator is adjusted in a fre-quency range of 270–350 MHz. The oscillators are cou-pled with capacitor C1.A noise diode D operating in the mode of the ava-lanche breakdown of the p–n junction and a three-stage amplifier on transistors Q1–Q3 are used as a low-fre-quency noise source. This source forms noise signals in a frequency band from a few kilohertz to ~6 MHz. The noise signal arrives at the input of the first oscillator from the Q3 output.A loop broadband antenna WA (magnetic dipole) is included in the Q5 collector circuit in such a way that the total collector current of this transistor flows through the antenna. The current through the antenna and, consequently, the integral level of the formed noise electromagnetic field can be adjusted with resistor R4.The serviceability test circuit, based on the double signal detection principle, makes it possible to perma-nently analyze the generated signal incoming from the antenna to its input and appropriately indicates the absence of the noise signal at the antenna input. The RF masking device is supplied from a 12-V dc voltage source.Measurements of the spectra levels of the electro-magnetic field formed by the RF masking device in a frequency band of 0.01–1000 MHz were carried out using SMV-6.5 and SMV-8.5 (Germany) selective microvoltmeters and showed that, in the entire fre-quency range of informative radiations of computer aids, the masking signal intensity exceeds the SERAP intensity of computer aids (printer, SVGA and VGA monitors, and plotter) and, thus, the reliable masking and protection of the processed information are pro-vided (Fig. 5). The entropy coefficient of the masking signal quality [14, 15] measured for three masking-device specimens using an X6-5 (Russia) device for investigating the correlation characteristic, which was at least 0.95, satisfies the qualifying standards for such devices.One device ensures the masking the SERAP of com-puter aids located in a room with an area of ~40 m2. It is necessary to use several sets of masking devices, placing them over the periphery of the protected object, for protecting computer aids in large computing cen-ters, terminal halls, etc. The maximal distance between neighbor RF masking devices should not exceed 20 m. The RF masking device is certified according to the safety requirements by the State Technical Commission and Ministry of Public Health of the Russian Federa-tion and is presently delivered on order of organiza-tions.REFERENCES1.Vakin, S.A. and Shustov, L.I., Osnovy radioprotivo-deistviya i radiotekhnicheskoi razvedki (Fundamentals of Radio Counteraction and Radiotechnical Surveil-lance), Moscow: Sovetskoe Radio, 1968.2.Dmitriev, A.S. and Kislov, V.Ya., Stokhasticheskie kole-baniya v radiofizike i elektronike (Stochastic Oscilla-tions in Radiophysics and Electronics), Moscow: Nauka, 1989.3.Rabinovich, M.I. and Trubetskov, D.I., Vvedenie v teo-riyu kolebanii i voln (Introduction to the Theory of Oscillations and Waves), Moscow: Nauka, 1984.4.Parker, T.S. and Chua, L.O., Proc. 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