Sharif university of Technology Title: Integration of Fuzzy Logic and Chaos Theory Supervisor: Dr.Bagheri Shuraki Presenters: Maryam Fereydoon, Sepideh Hajipour

Course: Fuzzy System and Sets Spring,2009 1 of Fuzzy Logic and Chaos Theory 2

Lotf Ali Asgar Zadeh 3 Brief Biography Born: February 4, 1921 ,age 88

Nationality: Iranian, American Fields: Mathematics Institutions: U.C. Berkeley Known for Founder of Fuzzy Mathematics,

fuzzy set theory, and fuzzy logic. He was born in Baku, of an Iranian father and mother. At the age of 10 the Zadeh family moved to Iran. Zadeh grew up in Iran, studied at Alborz High School. In 1942, he was graduated from the University of Tehran in electrical engineering (Fanni), and moved to the United States in 1944. He received an MS degree in electrical engineering from MIT in 1946, and a PhD in electrical engineering from Columbia University in 1949.

He was promoted to full professor in 1957. He published his seminal work on fuzzy sets in 1965. In 1973 he proposed his theory of fuzzy logic. Zadeh is married to Fay Zadeh and has two children, Stella Zadeh and Norman Zadeh. Zadeh has a long list of achievements 4

Edward Lorenz 5 Brief Biography Edward Norton Lorenz (May 23, 1917 - April 16, 2008) was an American mathematician and meteorologist, and a pioneer of chaos theory He discovered the strange attractor notion and coined the term butterfly effect.

Lorenz was born in West Hartford. He studied mathematics at both Dartmouth College and Harvard University.Lorenz was Professor of the Massachusetts Institute of Technology. Lorenz became skeptical of atmospheric phenomena. His work on the topic culminated in the publication his paper, the foundation of Chaos theory. His description of the Butterfly effect followed in 1969 Lorenz died at his home in Cambridge at the age of 90, having suffered from cancer.

Lorenz built a mathematical model of the way air moves around in the atmosphere. As Lorenz studied weather patterns he began to realize that they did not always change as predicted. Minute variations in the initial values of variables in his twelve variable computer weather model (c. 1960) would result in grossly divergent weather patterns. This sensitive dependence on initial conditions came to be known as the butterfly effect. 6 chaos can be important because its

presence means that long-term predictions are worthless and futile In fact, change and time are the two fundamental subjects that together make up the foundation of chaos. Chaos is a mathematical subject and therefore isn't for everybody. However, to understand the fundamental concepts, you don't need a background of anything more

than introductory courses in algebra, trigonometry, geometry, and statistics. That's as much as you'll need for this book. (on the other hand, does require integral calculus, partial differential equations, computer programming, and similar topics.) 7 A Simple Example Xt+1 = 1.9-(Xt)^2 Input value (xt) New value

(xt+1) 1.0 0.9 0.9 1.09 1.09 0.712

0.712 1.393 etc. 8 Complexity On the following pages we see three figures, two produced by the computer and one a photograph, Which one strikes you as being the most

complex? If you and a friend compare answers and do not agree, this is no cause for concern. The term "complexity" has almost as many definitions as "chaos." A small portion of the Mandelbrot set, with enough surrounding points to render it visible.(zn+1 = zn2 + c for n = 0, 1, 2, 3,with the starting value zwith the starting value z0 = 0 ) 9 Some wind streaks in a

field of packed snow A variant of the Japanese attractor 10 Fractality A fractal formed, except that in each retained square the corner to be discarded

has been chosen randomly. A fractal triangle, formed by dividing a square into four smaller squares and discarding the upper right square so produced, then dividing each remaining square into four still smaller squares and discarding each upper right square so produced, and repeating the process indefinitely. 11 A fractal tree, produced by first drawing a vertical segment, and then, after this

segment or any other one has been drawn, treating it as a "parent" segment and drawing two "offspring" segments, each six-tenths as long as the parent, and each extending at right angles from the end of the parent. 12 Butterfly Effect & Turbulance It says that a butterfly flapping its wings in, say Brazil, might create different "initial conditions" and trigger a later

tornado elsewhere (e.g. Texas), at least theoretically. not seem to be a clear distinction between turbulenc e and 13

fuzzy systems technology has achieved its maturity with widespread applications in many industrial, commercial, and technical fields, ranging from control, automation, and artificial intelligence to image/signal processing, pattern recognition, and electronic commerce. Chaos, on the other hand, was considered one of the three monumental discoveries of the twentieth century together with the theory of relativity and quantum mechanics. A deep seated reason to study the interactions

between fuzzy logic and chaos theory is that they are related at least within the context of human reasoning and information processing. 14 fuzzy definition of chaos, fuzzy modeling and control of chaotic systems using both Mamdani and TakagiSugeno models, fuzzy model identification using genetic algorithms and neural

network schemes, bifurcation phenomena and self-referencing in fuzzy systems, complex fuzzy systems and their collective behaviors, as well as some applications of combining fuzzy logic and chaotic dynamics, such as fuzzychaos hybrid controllers for nonlinear dynamic systems, and fuzzy model based chaotic cryptosystems. 15 2 papers in these interdisciplines are discussing: 1- Chaotic Dynamics with Fuzzy Systems:

explorations have been carried on mainly in three directions: the fuzzy control of chaotic systems, the definition of an adaptive fuzzy system by data from a chaotic time series, and the study of the theoretical relations 16 The peculiarities of a chaotic system can be listed as follows: 1. Strong dependence of the behavior on initial conditions 2. The sensitivity to the changes of system

parameters 3. Presence of strong harmonics in the signals 4. Fractional dimension of space state trajectories 5. Presence of a stretch direction, represented by a positive Lyapunov exponent 17 Famous artificial chaotic systems are Chuas circuit:

x(k + 1) = ax(k)(1 x(k)) one or more nonlinear elements one or more locally active resistors three or more 18 Fuzzy Sets:

19 20 21 2 D 22 Rule

Base: 23 Conclusions 1. Simple fuzzy systems are able to generate complex dynamics. 2. The precision in the approximation of the time series depends

only on the number and the shape of the fuzzy sets for x. 3. The chaoticity of the system depends only on the shape of the fuzzy sets for d. 4. The analysis of a chaotic system via a linguistic description allows a better understanding of the system itself. 5.

Accurate generators of chaos with desired characteristics can be built using the fuzzy model. 6. Multidimensional chaotic maps in some cases do not need a large number of rules in order to be represented. 24 2- Chaos Control Using Fuzzy Controllers (Mamdani Model): There

are two good reasons for using the fuzzy control: first, mathematical model is not required for the process, and second, the nonlinear controller can be developed empirically, without complicated mathematics. The fuzzy logic controller can be used when there is no mathematical model available for the process, this gives the robustness behavior for the proposed fuzzy logic controller design.

25 Membership Functions: 26 27 28 System response with some possible fuzzy rule in the rule-based system:

29 Uncontroll ed simulation s of Lorenz equation: (Chuas Circuit) 30

Controlled system to get on a periodic solution using the fuzzy controller: 31 Controlled system to get on a constant solution using the fuzzy controller: 32 The mathematical model of the lorenz system:

mathematical model of the Chuas circuit: 33 3- Bifurcation Phenomena in Elementary TakagiSugeno Fuzzy Systems: Bifurcation: A small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden qualitative or

topological change in its behavior. Local bifurcations Changes in the dynamics of a small region of the phase space, typically, a neighborhood of an equilibrium point. Global bifurcations For instance, when the interaction of a limit

cycle with a saddle point is produced. 34 The bifurcation diagram of the logistic map x K 1 Ax K (1 x K ) A 2.9 A 3.3 A 4

35 Takagi-Sugeno System with Affine Consequents 36 6 4 2

37 0.06 0 0.028 38 Conclusion Even

very elementary T-S systems can display local and global bifurcations 39 4- Self-Reference, Chaos, and Fuzzy Logic: The Liar paradox: This sentence is false TF TF TF TF TF

x n 1 1 x n 40 Fuzzy Definitions Lukasiewi cz: (p)

1 ( p ) (q ) ( p q ) min(1,1 ( p ) (q )) O .W . 1 ( ( p ) (q )) ( p q ) (( p q ) (q p )) ( p q ) 1 abs ( ( p ) (q )) 41 The Liar paradox with fuzzy

membership functions This sentence is A seedfalse value of 0.3: x n 1 1 x n 42 Attractor and Repeller Fixed Points in the Phenomena of Self-Reference Modest Liar:

This sentence is fairly false x n 1 1 x n 43 Emphatic Liar: This sentence is very false

x n 1 (1 x n ) 2 44 True Teller This sentence is true Modest True-Teller True-Teller This

sentence is fairly true x n 1 x n x n 1 x n Emphatic This sentence is very true 2 x n 1 (x n )

45 Fuzzy Chaos The Chaotic Liar This sentence is true if and only if it isThis false sentence is as true as it x n 1

is false 1 abs ((1 x n ) x n ) 0 x n 0.5 2x n x n 1 2(1 x n ) 0.5 x n 1 46 It is very false that this

sentence is true iff it is false. x n 1 (1 (1 abs ((1 x n ) x n ))) 2 47 Fuzzy Self-Reference in Two Dimensions The Dualist:

Socrates: What Plato is about to say is true. Plato: Socrates speaks falsely. X: Y is true Y: X is Fuzzy variations on the Dualist: false X: X is true if and only if Y is true. Y: Yxis true if and

only if X is 1 abs ( x y ) n 1 n

n very false. 2 y n 1 1 abs ( y n (1 x n ) ) 48 Sequential calculation Escape-time diagram for a Fuzzy Dualist with

simultaneous calculation 49 50 Thanks for Your Attention 51