ChaoticsystemsandChua'sCircuit课件.ppt
chaos
II. CHAOTIC PHENOMENON
In the section,we investigate chaotic phenomenon in
fractional two-neuron network system with time–varying
delays. The system is given as Eq.(1). By the Grünwald-Letnikov[GL][15-18] definition:
(1) at time t is y(i)m . If f '(t) ≤ M (constant),
and a(i)hαi M / (1+ hαi ) < 1,(i=1,2) then iteration (5) is
convergent. So just the calculus step h being taken smaller enough, the iteration (5) will be convergent to a constant.
+ a2 *
(3)
⎪⎩ [ f ( y (1 ) m ) − b 2 f ( y (1 ) mτ ) ] .
where y(1)mτ , y(2)mτ at t + τ (t) .
From (3) we have
⎧ ⎪
y1(
m
)
= {a1hα1 [ f
( y2 m ) − b1 f
( y 2 mτ
)] −
=
(−1) j
⎛α
⎜ ⎝
j
⎞ ⎟
,
⎠
⎛α
⎜ ⎝
j
⎞ ⎟ ⎠
=∂(∂ຫໍສະໝຸດ −1)"(∂ j!
chua 电路
Figure 1 :The Lorenz attractor
The first observations of chaos in electronical circuits are in 1927. By Van der Pol and Van der Mark. They see it in nonlinear oscillators. In 1980 Ueda and Akamatsu find chaos in negative resistance oscillators. But the circuit of Chua is a bit more special. Because of its simplicity and university. Chua's circuit is dated from 1993. This circuit will be used in the experiments of controlling chaos.
何谓混沌
y(t) @ [x0,y0,z0]=[0, 1, 1.05]
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y(t) @ [x0,y0,z0]=[0, 1, 1.0501]
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10
0
-10
-20
-30
0
5
10
15
20
25
30
35
40
45
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蝴蝶应（Butterfly effect）
1972年，Lorenz在美国科学发展学会第139次会议上做了一个题 为「可预测性：巴西一只蝴蝶扇动一下翅膀，能否在德克萨斯州 掀起一场龙卷风」的演讲。他指出，一个微小的初始条件变化有 可能引起一连串逐渐被放大的改变，最终导致完全意外的不同结 果。
欧盟第五届科技框架计划OCCULT
• • • • • • • •
University of the Balearic Islands Consejo Superior de Investigaciones Cientí ficas University of Wales Georgia Tech Lorraine University of Athens University of Pavia University of Technology, Darmstadt Opto Speed
“我们可以认为，宇宙的现在是由它的过去决定的； 现在也是决定未来的原因。如果有一位伟大的智者在某 一时刻，获知了自然界一切物体的位置和相互作用力， 并且他具有超常的计算和分析能力，他就可以把宇宙这 个最庞大的物体直到原子这个最细微的颗粒，全都囊括 到一个公式中。对于这位智者来说，没有什么东西是不 确定的—宇宙的未来会像它的过去一样完全呈现在他的 眼前。”
IntroductionToNonlinearDynamicsAndChaos-Outline
Format: 2-2-3, EE4060 Introduction to Nonlinear Dynamics and Chaos1.DescriptionThis course introduces the student to the basic concepts of nonlinear dynamics and chaos via numerical simulations and electric circuits. The primary goal is to understand the bifurcations and steady-state behavior of nonlinear dynamical systems. The secondary goal is to study the phenomenon of chaos using Chua’s circuit and memristor-based chaotic circuits.2.PrerequisitesMA-235 [Differential Equations for Engineers]EE-2050 [Circuits I: Steady State] or EE-201 [Linear Networks: Steady State Analysis] 3.Materials (REQUIRED)1.Nonlinear Dynamics and Chaos With Applications to Physics, Biology, Chemistryand Engineering. Strogatz, Steven H. Perseus Books, Reading, Massachusetts.1994.2. A Route to Chaos Using Nonlinear Circuits. Muthuswamy, Bharathwaj. PDF will beprovided.4.Course Learning ObjectivesUpon successful completion of the course, the student will be able to:1.Describe the fundamental differences between linear and nonlinear dynamicalsystems and the importance of studying nonlinear dynamics.2.Define the different bifurcation phenomenon of nonlinear systems in one, two andthree dimensions3.Apply bifurcation analysis to study practical systems such as laser models, Josephsonjunctions, op-amp oscillator circuits and the nonlinear pendulum4.Understand limit cycles and Poincare Maps5.Understand basic concepts of chaos using Chua’s circuit6.Perform literature review7.Prepare short presentations5. Course Topics1.Introduction, differences between linear and nonlinear dynamics (1 class)2.Fixed Points and Stability (2 classes)3.Bifurcations in one dimensions (2 classes)4.Examples of bifurcations in one dimensions – laser models (1 class)5.Flows on a circle – oscillators (2 classes)6.Midterm review (1 class)7.Examples of oscillations–op-amp oscillator circuits,Josephson junctions andnonlinear pendulum (2 classes)8.The phase plane – linear systems, conservative systems, reversible systems, limitcycles and two-dimensional bifurcations (3 classes)9.Introduction to chaotic systems – Chua’s circuit (1 class)10.Memristor-based chaotic circuits (1 class)11.Tools for analyzing chaotic systems – period-doubling bifurcations, PoincareMaps revisit (1 class)12.Some properties of the strange attractor (1 class)13.Applications of Chaos (1 class)14.Final presentation guidelines and course wrap up (1 class)15.Group presentations (1 lab section)16.In-lab midterm (1 lab section)6.Prerequisites by Topic1.Understanding of linear constant coefficient ODEs2.Basic circuit analysisboratory Topicsboratory experiment details and requirements are described in [2].2.Students are expected to prepare for the lab by doing all required pre-lab activitiesand finishing all remaining requirements during the lab itself.3.Limited laboratory reports will be required.4.During weeks 9 and 10 of the course, students will have an opportunity to implementdifferent versions of Chua’s circuit. They will be required to perform a basic bifurcation analysis,visualize their system in Mathematica,simulate their circuit using MultiSim and implement their version of Chua’s circuit on a breadboard. They will also be required to give a final presentation on their work.Weekly Course Topics8.。
学年高中创新设计物理粤教选修第二章走进门电路了解集成电路精选PPT
图2
解析答案
五、复合门电路 1.与非门 一个 与 门电路和一个 非 门电路组合在一起，组成一个复合门电路， 称为与非门，如图3所示.
可简化为 图3
答案
真值表：
输入
输出
解析 ①为“非”门，输出为“0”，②为“或”门，输出为“1”，③为“与”门，输出为“0”，则只有选项C正确.
该逻辑电路是________门电路，要增大该控制电路的灵敏度，可使变阻器R的阻值________(填“变大”或“变小”).
(3)真值表：
与门电路
输入
S1
S2
0
0
0
1
1
0
1
1
输出 Q 0 0 0 1
答案
2.或门电路：实现 或 逻辑的电子电路.
(1)符号：
.
“≥1”表示当输入呈现“1”状态的个数大于等于1时，输出为“1”状态.
(2)或门电路的特点：有一为一，全零为零；
答案
(3)真值表：
或门电路
输入
S1
S2
0
0
0
1
1
0
1
0 0 1 (2)集成电路按集成度可分为六类：
将信号编成“ ”或“ ”的二进制代码，然后转换成
，中规模集成电路MSI，大规模集成电路LSI， 进行处理的技术.
，特大规模集成电路ULSI，巨大规模集成电路GSI.
初步了解集成电路的作用及发展情况.
0 1 __1_ 定义：集成电路(IC)是指把
、 、电容等元件，按电路结构的要求，制作在一块
成电路GSI.
答案
例1 如图1所示的A、B端均为输入端，如果A输入“1”、B输入“0”. 则下列选项正确的是( C )
Circuit Analysis Lecture 3
• And that
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Power and Energy
• And we know that Current is the rate of flow of charge (C/s or Amps)
• Gives,
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Topic 2.10
POWER
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Power
By direct substitution of Ohm’s law the equation for power can be obtained in two other forms:
IS1 = 6 A
IS2 = 4 A
IS = 10 A
RT
Same Polarity!
IS1 = 5 A
IS2 = 4 A
IS = 1 A
RT
Opposite Polarity!
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Topic 2.9
POWER AND ENERGY
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Power and Energy
• Power is an indication of how much work can be done in a specified time • Power is the rate of doing work • The rate at which energy is delivered
Vs 5 6 30V
RL experiences the same current flow (IL) and therefore the same voltage drop (VL) when connected between ‘a’ and ‘b’ in both circuits
RS is constant Note: the direction of the arrow and positive terminal indicate the direction of current flow.
第四章 混沌时间序列分析及相空间重构
Lyapunov Exponents
f
• Quantifies separation in time between trajectories, assuming rate of growth (or decay) is exponential in time, as: n
1 i lim ln( eig J(p)) n n p 0
估计吸引子维数的算法，需要大量的数据点作为输入，当这些点的 输入被选择为最大化的包含吸引子信息情况下，输入数据点的数量可以减 少。（由Holzfuss和Mayer—kress 1986年提出） 重构相空间所需要解决的关键问题，就是确定重构维数m。 在重构相空间维数未知的情况下，可用以下方法获得： 令 nr 为重构空间的维数。首先把nr （或m）设置为1，计算重构吸引子 的维数Dcap，然后增加 nr （或m）的大小，并重复计算重构吸引子的维数 Dcap,直到Dcap不再改变为止（如曹书p103），最后的Dcap是正确的相 关维数，产生正确的Dcap的最小 nr (m) 即重构空间的最小维数m.
Time delay embedding
Differs from traditional experimental measurements
Provides detailed information about degrees of freedom beyond the scalar measured Rests on probabilistic assumptions - though not guaranteed to be valid for any particular system Reconstructed dynamics are seen through an unknown “smooth transformation” Therefore allows precise questions only about invariants under “smooth transformations” It can still be used for forecasting a time series and “characterizing essential features of the dynamics that produced it”
基于蔡氏电路的通信保密系统的设计(1)
基于蔡氏电路的通信保密系统的设计(1)混沌理论自上世纪70年代兴起以来于和各学科相互渗透，成为了各个领域内研究的热点。
在信息科学高度发达的今天，信息安全也与人们的生活息息相关，信息安全、无损的传输不仅对于军事有重要的意义，对于人们生活的影响也是巨大的。
混沌系统所具有的系统对于初始参数、系统参数极为敏感、混沌信号类似噪声等特点均适用于通信保密。
利用混沌系统产生的混沌信号对信号进行掩盖保密传输具有可行性与实用性。
随着现代科学技术的发展，计算机仿真技术得到广泛的运用使得系统的设计分析更加容易，本文通过采用Multisim以及Matlab仿真技术对混沌电路以及通信保密系统的特性进行验证、分析。
蔡氏电路是混沌理论转化为实际电路模型的典型电路，蔡氏电路具有完整的混沌系统的特性，因此蔡氏电路得到广泛的研究与运用。
本文首先通过对蔡氏电路的微分方程组利用Matlab进行数值求解，绘出对应状态变量的相轨迹图。
利用Multisim搭建仿真电路原理图，同样绘出相应状态变量的相轨迹图，并与Matlab的结果进行对比，确保仿真原理图所选的元件参数能够满足蔡氏电路微分方程的特性。
在Multisim提供的仿真环境下，无法直接观察电感电流的波形图，本文通过串联一个微小电阻，通过观测电阻两侧的电压作为电感电流信号。
调整蔡氏电路的参数，研究不同参数下的电路特性，分析系统参数对混沌信号的影响情况。
混沌电路对电路的参数变化极为敏感，为增强通信保密系统的工作稳定性，采用有源元件对无源电感进行等效。
对等效后的蔡氏电路的电压信号进行调制、耦合同步等关键技术处理，并对耦合的情况进行分析，以此来说明基于蔡氏电路的通信掩盖保密系统的工作原理，以及信号的耦合同步对于本系统的必要性。
在对完成了电路的改进、信号调制、耦合同步后的主从结构的蔡氏电路，通过增加减法器、反相器等基本模块构成的信号通道的实现传输信号与混沌掩盖保密信号的叠加、消去。
为了检验设计的模拟信号的通信保密系统的运行效能及可靠性，选取了正弦信号、chirp电压信号等模拟信号作为测试信号，测试系统对模拟量的保密传输性能；选取了锯齿信号、方波信号作为数字信号的测试信号，测试系统对数字信号的保密传输性能。
英语科技论文写作PPT课件
所做工作提出今后应如何做的意见或方案。一般来说，结 论部分一两小段的长度。 有时作者列出一些结果，这时标题用results; 有时则就 某一方面的问题加以讨论，标题可用discussion，这种情况 下文字可能比较长。
6
科技论文本身内容的构成
Special gratitude is owed to… I’m grateful to … I greatly appreciate … I would like to express my appreciation of … I’m greatly indebted to … I’m in indebtedness to … Thanks go to … I wish to thank … Particular recognition is due to …
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引言_基本内容
引言（或导论）应该清楚地说明以下四个项目 主要的研究工作 研究工作的目的 进行研究的动机或背景 本研究工作在相关领域中的地位
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引言_组织形式
研究论文的引言常常至少包含下列的四个基本步骤： 步骤一 背景资料。
先介绍作者的研究领域，叙述有关该研究领域的一般 信息，并针对研究论文将要探讨的问题或现象提供背景知 识。 步骤二 文献回顾。
2
科技论文本身内容的构成
二、正文（Body） 论文的关键部分。 包括一些设想、实验情况、实验装置、获得的数据、证实 的理论等等。作者可通过定义、描写、说明、举例、实验、 论证、比较、分析综合、判断推理等不同的研究过程来证 实作者在引言中所提及的主题。
3
科技论文本身内容的构成
三、结论或结束语（Conclusion(s) or Concluding Remarks） 是作者通过正文的论述而形成的总的观念，是一种很
分数阶时滞忆阻混沌电路的动力学分析及电路仿真
2］区间变化，见图 4。从分岔图可以很明显地看出
和 C1 = 1.232μFC 2 = 1.835μFC 3 = 1.10μF 。 运 算
时，出现 Hopf 分叉，最后随着 a 的增加变为混沌状
供 ±15 的 电 压 和 R = 11.24kΩ ，整 体 电 路 图 如 图 6
系统（2）的轨道从周期状态开始，然后当经过阈值
数阶忆阻器 ，将其替换图 1 中的电容得到分数阶磁
控忆阻器，数学表达式为
ì dx q 2
ï
= 1 q x1
dt
R 0C 0
ï
（1）
í
æ1
g1 g 2
2ö
ï
1
ï f ( x1 x 2 ) = ç R - R + R [ x 2] ÷ x1
2
è 1
ø
î
其中 x1、x2 和 （
f x1，x2）分别是忆阻器的输入、内部状
态。各状态的相位图和时域图如图 3 所示。当
（a）a = 1.62
放 大 器 和 乘 法 器 采 用 AD711KN 和 AD633JN ，提
（c）所示。
（b）a = 1.68
（d）a = 1.62
（e）a = 1.73
图5
相位图与时域图
（c） a = 1.73
统进入混沌状态，如图 5（b）和（e）所示。当 a = 1.73
时，系统展现出双涡旋的混沌吸引子，如图 5（c）和
（f）所示。
出一个引理来讨论式（3）的根的分布。
引理 1 对于式（3），以下结果成立：
1）如 果 ψ k > 0(k = 1234) 且 A 3 + A 4 ¹ 0 ，则
方程（3）在时滞 τ ³ 0 时没有实部为零的根。
【重庆大学本科四门专业课PPT】组成原理3.Arithmetic+for+Computers
3.1 Introduction
Computer words are composed of bits; thus words can be represented as binary numbers.
Binary, Decimal, and Hexadecimal Equivalents
Converting Binary to Hexadecimal
❖ Each hexadecimal digit corresponds to 4 binary bits ❖ Example:
Convert the 32-bit binary number to hexadecimal 1110 1011 0001 0110 1010 0111 1001 0100 ❖ Solution:
Unsigned and signed integersMost
Characters
Significant Bit
Least Significant Bit
Floating-point numbers Images, sound, etc.
76 543 210 10011101
27 26 25 24 23 22 21 20
Each bit represents a power of 2 Every binary number is a sum of powers of 2 Decimal Value = (dn-1 2n-1) + ... + (d1 21) + (d0 20) Binary (10011101)2 = 27 + 24 + 23 + 22 + 1 = 157
《自动化专业英语》PPT课件
标题的书写特征 （续）
所有字母全大写
REVIEW AND PROSPECT OF COMPUTER CONTROL SYSTEM FOR ALUMIUM SMELTERS IN CHINA
标题的内容特征
关键词，反映论文的性质和主要内容 研究的手段或方法
Research on Controllability of Magnetic Suspension with High Temperature Superconducting Electromagnet
示例2
A model predictive control strategy for supply chain optimization
This paper describes a model predictive control strategy to find the optimal decision variables to maximize profit in supply chains with multiproduct, multiechelon distribution networks with multiproduct batch plants. The key features of this paper are: (1) a discrete time MILP dynamic model that considers the flow of material and information within the system; (2) a general dynamic optimization framework that simultaneously considers all the elements of the supply chain and their interactions; and (3), a rolling horizon approach to update the decision variables whenever changes affecting the supply chain arise. The paper compares the behavior of a supply chain under centralized and decentralized management approaches, and shows that the former yields better results, with profit increases of up to 15% as shown in an example problem. Keywords: Supply chain management; Scheduling; Dynamic scheduling; Optimal control; Decision support tools; Hybrid systems
unit 1 computerscience_2 计算机系统概论(双语课件)专业英语课件
Boolean Operation
• Bits are stored and manipulated inside computer .
• 1=true • 0=False • Manipulating bits as called Boolean
NOT Gate
AND Gate
Gates
• A device that produces the output of a Boolean operation when give the operation’s input values is called a “Gate”.
Data Storage
By Suresh Foreign Teacher East China Institute of Technology
Data Storage
• Data Representation and storage of data within a computer.
The Types of data we will consider
• Gates provide the building blocks from which computer are constructed one important step in this direction .
Flip-Flops
• Collection of circuit are know as flip-flops. • A flip-flop is a circuit that produces an
• Indeed, technology know as Very Large Scale Integration(VLSI).
新混沌系统与变形蔡氏电路系统的异结构同步
新混沌系统与变形蔡氏电路系统的异结构同步作者：郭玉祥吴然超来源：《现代电子技术》2009年第02期摘要：研究一个新的混沌系统与变形蔡氏电路系统的异结构同步问题。
基于Lyapunov稳定性理论，分步构造Lyapunov函数，并在响应系统中采用设计单个非线性控制器的方式，实现了这两个不同混沌系统之间的异结构同步，并证明误差变量随时间演变时是逐渐趋于零的。
数值模拟验证了这种方法的可行性和有效性，所设计的控制器具有可操作性强，同步效果好，易于推广等优点。
关键词：新混沌系统；变形蔡氏电路系统；混沌同步；Lyapunov函数中图分类号：TN918文献标识码：B文章编号：1004 373X(2009)02 079 03Synchronization of New Chaotic Systemand Modified Chua′s Circuit System with Different StructureGUO Yuxiang,WU Ranchao(School of Mathematical Sciences,Anhui University,Hefei,230039,China)Abstract：Synchronization of new chaotic system and modified Chua′s circuit system with different structure is studied.The Lyapunov function is deduced based on the Lyapunov stabilization theory,a nonlinear controller is designed to realize the synchronization between chaotic systems with different structure.Conclusion about the error variable approaching to zero smoothly and quickly is also testified with the evolution of the time.Numerical simulations prove that the approach is effective and feasible.The designed controller processes the merits of highly operating,getting better results on synchronization and generalizing easily.Keywords：new chaotic system;modified Chua′s ci rcuit system;chaoticsynchronization;Lyapunov function0 引言近年来，混沌及其应用是非线性科学研究领域中的一个热门课题。
Chaotic Oscillators
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. IEEE, 1987, vol. 75,no. 8.5.Sudakov, Yu.I., Amplitudnaya modulyatsiya i avtomodu-lyatsiya tranzistornykh generatorov (teoriya i raschet) (Amplitude Modulation and Self-Modulation of Transis-tor Generators: Theory and Calculation), Moscow: Energiya, 1969.6.Kaganov, V.I., SVCh poluprovodnikovye radiopere-datchiki (Microwave Semiconductor Transmitters), Moscow: Radio i Svyaz’, 1981.7.Kal’yanov, E.V., Ivanov, V.P., and Lebedev, M.N.,Radiotekh. Elektron. (Moscow), 1982, vol. 27, no. 5, p. 982.INSTRUMENTS AND EXPERIMENTAL TECHNIQUES V ol. 45No. 22002236INSTRUMENTS AND EXPERIMENTAL TECHNIQUES V ol. 45 No. 2 2002LEBEDEV, IVANOV8.Dmitriev, A.S., Ivanov, V .P., and Lebedev, M.N.,Radiotekh. Elektron. (Moscow), 1988, vol. 23, no. 5,p. 1085.9.Anishchenko, V .S., Stokhasticheskie kolebaniya v radiofizicheskikh sistemakh (Stochastic Oscillations in Radiophysical Systems), Saratov: Saratov. Gos. Univ.,1986, part 2.10.Anishchenko, V .S., Astakhov, V .V ., and Shabunin, A.V .,Radiotekh. Elektron. (Moscow), 2000, vol. 45, no. 2,p. 196. 11. Dmitriev, A.S., Kyarginskii, B.E., Maksimov, N.A., et al.,Radiotekhnika (Moscow), 2000, no. 3, p. 9.12.Kal’yanov, E.V ., Ivanov, V .P., and Lebedev, M.N.,Radiotekh. Elektron. (Moscow), 1990, vol. 35, no. 8,p. 1682.13.Bezrukov, V .A., Ivanov, V .P., Kalashnikov, V .S., andLebedev, M.N., RF Patent 2170493, Byull. Izobret.,2001, no. 19, p. 337.14.Nicolis, J.S., Dynamics of Hierarchical Systems: AnEvolutionary Approach , Berlin: Springer, 1986. Trans-lated under the title Dinamika ierarkhicheskikh sistem.Evolyutsionnoe predstavlenie , Moscow: Mir, 1989.15.Kharkevich, A.A., Ocherki obshchei teorii svyazi(Sketches on General Communication Theory), Mos-cow: GINTL, 1955.。
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by
Dao Tran
Missouri State University KME Alpha Chapter
Presented at KME Regional Meeting, Emporia State University, KANSAS
Buffer
Vc(t) +
r(t) Inverter
Message signal
Motivation/Application
Receiver
r(t) Chaos generator (Chua’s circuit)
Buffer
Vc(t)
s’(t)
-
What is Chaotic System?
►Phenomenon that occurs widely in dynamical systems
►Readily constructed at low cost using standard electronic components
Linear systems
► Linear System of D.E
.
X
AX
; X Rn
X (0) X 0
► General solution
Outline
►Linear systems ►Nonlinear systems
Local behavior Global behavior
►Chaos and Chua’s Circuit
Bifurcation Periodic orbits Strange attractors
Linear systems (cont.)
►Stability
►Stability of linear systems is determined by eigenvalues of matrix A.
►Invariant Sets
►(Generalized) eigenvectors corresponding to eigenvalues λ with negative, zero, or positive real part form the stable, center, and unstable subspaces, respectively.
known as linearization.
► In this work, we consider:
►Linearization around equilibrium points.
►Linearization around periodic orbits.
dI3
dt
dV2
dt
1 L
1 C2
V2 I3
G C2
, V2
where G
1/
R
Linear System (cont.)
With the fixed values of R, L, and C, using MATLAB, we obtained the solution
What is Chua’s Circuit?
►Autonomous circuit consisting two capacitors, inductor, resistor, and nonlinear resistor.
►Exhibits a variety of chaotic phenomena exhibited by more complex circuits, which makes it popular.
Nonlinear systems
.
X
F(X ); X
Rn
F Rn Rn
► Even for F smooth and bounded for all t є R, the solution X (t) may become unpredictable or unbounded after some finite time t.
► We divide the study of nonlinear systems into local and global behavior.
Local Behavior
► Idea: use linear systems theory to study nonlinear systems, at least locally, around some special sets, a technique
►Considered to be complex and no simple analysis
►Study of chaos can be used in real-world applications: secure communication, medical field, fractal theory, electrical circuits, etc.
X (t) e At X 0
► The solution is explicitly known for any t.
Linear systems (cont.)
► Stability
.
X AX
► Equilibrium points
► If Re(λ)<0 => Stable ► If Re(λ)>0 => Unstable
ቤተ መጻሕፍቲ ባይዱ
Linear Systems (cont.)
Consider the linear RLC circuit
Applying KCL law and choosing V2 and IL as state variables ,we obtain the differential equation:
Motivation/Application
Secure Communication
S(t)
Information signal
Transmitter (Chaotic)
y(t)
Transmitted signal
Receiver
S’(t)
Retrieved signal
Transmitter
Chaos generator (Chua’s circuit)