Chaos & Recursive. Ehsan Tahami. (Properties, Dynamics, and Applications ) PHD student of biomedical engineering

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1 Chaos & Recursive Equations (Properties, Dynamics, and Applications ) Ehsan Tahami PHD student of biomedical engineering Tahami@mshdiau.a.ir

2 Index What is Chaos theory? History of Chaos Introduction of Recursive equations Dynamics of Recursive equations(1&2 dimensional maps) Famous Family of Recursive equations Applications of Recursive e equations Conclusion

3 What is Chaos theory? The name Chaos theory" comes from the fact that the systems that the theory describes are apparently disordered. Chaos theory describes the behavior of certain dynamical systems that is, systems whose state evolves with time and may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears to be random.

4 The chaos happens even though these systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos.

5 Example Chua Circuit

7 Chaotic behavior is observed in: Natural systems such as: -Weather and climate. - Action potentials in neurons. - Population growth in ecology. - Molecular Vibrations. - Magnetic field of celestial bodies Variety of systems in the laboratory such as: - Electrical circuits - Lasers - Oscillating chemical reactions - Fluid dynamics - Mechanical and magneto-mechanical devices Plate tectonics and in Economics

8 History of Chaos Henri Poincaré. In 1890 while studying the three-body yp problem, he found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point.

9 Later studies in , also on the topic of nonlinear differential equations, were carried out by G.D. Birkhoff, A. N. Kolmogorov. Cartwright and J.E. Littlewood,and Stephen Smale. In 1927 by van der Pol and din 1958 by R.L. Ives, Chaos was observed by a number of experimenters before it was recognized; e.g.,

Yoshisoke Ueda Seems to have been the first experimenter to have identified a chaotic phenomenon as such by using an analog computer on November 27, 1961.

10 Yoshisoke Ueda Seems to have been the first experimenter to have identified a chaotic phenomenon as such by using an analog computer on November 27, In 1967, Benoit Mandelbrot published "How long is the coast of Britain? Statistical selfsimilarity and fractional dimension," showing that a coastline's length varies with the scale of the measuring instrument, resembles itself at all scales, and is infinite in length for an infinitesimally small measuring device. In 1975 he published The fractal geometry of nature, which h became a classic of chaos theory.

11 In 1977, Mitchell Feigenbaum published the noted article Quantitative Universality for a Class of Nonlinear Transformations", where he described logistic maps. Feigenbaum notably discovered the universality in chaos, permitting an application of chaos theory to many different phenomena.

Edward Lorenz whose interest in chaos came

initial conditions produced large changes in

Title of a paper given by Edward Lorenz in

Flap of a Butterfly s Wings in Brazil set off

12 Edward Lorenz whose interest in chaos came about accidentally through his work on weather prediction in Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome. Title of a paper given by Edward Lorenz in 1972 to entitled Predictability: Does the Flap of a Butterfly s Wings in Brazil set off a Tornado in Texas? And the Essence of chaos has published d the ssence of chaos s pub s ed in 1993

13 Henri Poincaré ( ) and others showed, the three-body problem is impossible to solve in the general case; that is, given three bodies in a random configuration, the resulting motion nearly always turns out to be chaotic: no one can predict precisely what paths those bodies would follow. However, the problem becomes tractable in certain special cases.

14 Recursive Equation The recursive equations, which form a class of computable equations, take their name from the process of "recurrence" or "recursion".

15 Poincaré é&three body problem The problem is to determine the possible motions of three point masses m1,m2,and m3, which attract each other according to Newton's law of inverse squares.

the state space of a continuous dynamical system with a certain lower

16 Poincare first return map A first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of p p,, f a periodic orbit in the state space of a continuous dynamical system with a certain lower dimensional subspace, called the Poincaré section, transversal to the flow of the system.

17 Poincaré map can be interpreted as discrete dynamical systems with a state space that is one dimension smaller than the original continuous dynamical system. Poincaré map preserves many properties of periodic and quasiperiodic orbits of the original system and has a lower dimensional state space it is often used for analyzing the original system.

18 F is called an Recursive function

20 Poincare & return map for a signals Plot x(t+1) versus x(t) to find the poincare map of the signals.

21 The Poincare plot depicts the nature of R R interval fluctuations. And the shape of the plot varies for different heart conditions and indicates the degree of the heart failure in a subject The standard deviation of the points perpendicular to the line-of identity is denoted by SD1. The standard deviation along the line-of-identity is denoted by SD2.

22 Euler Method r = 1+ h y n = ry n (1 y n ) Logistic Map

23 f = Fixed point of the map Difference between and

24 Recursive vs Differential i equations Dimension of problem Constraints of continuity Dynamic properties Creation of finformation and Evolutionary

25 Famous Family of Recursive Equations a) Logistic Recursive Equation b) Host-parasitoid model c) One-parameter family of Recursive Equations d) Two dimensional Recursive Equations

26 a) Logistic i Recursive equation x = rx (1 x n+ 1 n n ) xn is a number between zero and one, and represents the population at year n. x0 represents the initial population (at year 0) r is a positive number, and represents a combined rate for reproduction and starvation. Reproduction: where the population will increase at a rate proportional to the current population when the population size is small. Starvation: (density-dependent mortality) where the growth rate will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment less the current population.

27 Some properties of Logistic Recursive equation The fixed points are the solutions of x n = rxn( 1 xn) xn = 0 is asymptotically stable and xn = (r 1)/r, which does not belong to [0,1], is unstable. xn = 0 is unstable and xn= (r 1)/r is asymptotically stable. The map has four fixed points that are the x n 2 solutions of the equation f ( xn, r) = x n two unstable fixed points, x n = 0 and x n = (r 1)/r, of f. The remaining twosolutions o s are x n

28 Period doubling bifurcation in Logistic model

29 Chaos in Logistic model

30 Bifurcation diagram & Creation of Information

31 Feigenbaum number & Universality of Chaos

32 b) Host-parasitoid idmodel A parasitoid is an insect having a lifestyle intermediate between a parasite and a usual predator. Parasitoid larvae live inside their hosts, feeding on the host tissues and generally consuming them almost completely. where Ht and Pt are, respectively, the host and parasitoid populations At time t, and r, a, and c are positive constants. The model has two equilibrium points, (0, 0) and (H, P ), where

34 c) One-parameter family of maps

35 1 Equilibrium points are solutions to the equation No equilibrium points Two equilibrium points is asymptotically stable is unstable This type of bifurcation is called a saddle-node bifurcation. A ddl d bif i ilbif i i l l bif i i hi h A saddle-node bifurcation or tangential bifurcation is a local bifurcation in which two fixed point of a dynamical system collide and annihilate each other.

36 1 Equilibrium points are solutions to the equation For all values of μ, there exist two fixed points: 0 and μ. 0 is stable and μ unstable for μ > 0, 0 becomes unstable and μ stable. This type of bifurcation is called a transcritical ii lbifurcation. i A transcritical bifurcation is bifurcation in which before and after the bifurcation, there is one unstable and one stable fixed point. However, their stability is exchanged when they collide. So the unstable fixed point becomes stable and vice versa.

37 1 Equilibrium points are solutions to the equation 0 is the only fixed point and it is asymptotically y stable Three fixed points, 0 is unstable, and ± μ are both asymptotically y stable. The curve exists only on one side of (0, 0) and is tangent at this point to the line μ = 0. This type of bifurcation is called a pitchfork bifurcation. A pitchfork bifurcation is a particular type of local bifurcation.

38 1 One fixed point is (0,0)and the other fixed points are the solutions to the equation 0 is the fixed point and it is stable 0 is the fixed point and it is unstable Two other fixed points, that are both unstable. There are three fixed points all of them being unstable This new type of bifurcation i is called a period-doubling idd bifurcation. A Period doubling bifurcation is a bifurcation in which the system switches to a new behavior with twice the period of the original system.

39 d) Two dimensional Recursive Equations Heno'n equation Coupled logistic equation Standard equation Dl Delayed dlogistic i model dl

Henon equation The Henon map is the most

difference equations For other values of a

40 Henon equation The Henon map is the most studied two-dimensional map with chaotic behaviour. The map can also be written as a system of difference equations For other values of a and b the map may be chaotic, intermittent, or converge to a periodic orbit.

41 Coupled logistic equation 1 The four fixed points are given by: where Depending on the initial conditions and the parameter values one can find the following behaviour: (i) orbits tend to a fixed point, (ii) periodic behaviour, (iii) quasiperiodic behaviour, (iv) chaotic behaviour, (v) hyperchaotic behaviour

42 Standard equation 1 This map displays all three types of orbits: periodic cycles, quasi periodic orbits and chaotic orbits. Quasi periodic motion is in rough terms the type of motion executed by a dynamical system containing a finite number (two or more) of incommensurable frequencies.

43 Dl Delayed dlogistic i model dl 1 let The delayed logistic equation then takes the form That has two fixed points Fixed point is asymptotically stable, Stable limit cycle Bifurcation point

45 Application of Recursive equations (Nonlinear( dynamics of cardiac excitation-contraction nonlinear recursive map model used to investigate the dynamics of cardiac excitation ti contraction ti coupling in a periodically stimulated cell.

46 Bifurcation diagram showing action potential Duration (ADP) versus pacing period T

47 Application of Recursive equations (Image encryption) In cryptography, encryption is the process of transforming information (referred to as plaintext) using an algorithm (called cipher) to make it unreadable to anyone except those possessing special knowledge, usually referred to as a key. The proposed image encryption procedure is highly hl key sensitive. Advantages: chaos based encryption techniques are considered good for practical use as these techniques provide a good combination of speed, high security, complexity, reasonable computational overheads and computational power etc

48 Original Image Encrypted Image Decrypted Image Encrypted A139FD52FC87CD1E4406 The secret key A039FD52FC87CD1E4406 (in hexadecimal) Encrypted A039FD52FC87CD1E4406

49 Application of Recursive equations (Chaotic Neural Network) n y i( t + 1) = kyi ( t) + α w ijxj( t) + Ii zx i i ( t ) j= 1 x i( t) = y ( t)/ε 1+ e 1 i

50 Conclusion Whereas a real dynamical system, such as the motion of the planets, is described by differential equations and continuous time, it is often convenient to consider simpler mathematical models, called recursive equations, where the system evolves through a set of discrete time steps. Recursive equations show a much greater range of dynamical behavior than do differential equation systems because the recursive equations are free from the constraints of continuity. Recursive equations can also used to exhibit the complex dynamical behaviors, such as the period orbits, cascade of period-doubling bifurcation, quasi-periodic orbits and the chaotic sets. These results reveal far richer dynamics of the Recursive equation models compared with the differential equation models.

51 Recursive equations are one of the essential compartment of Creation of information and evolutionary process.

52 Thank you

Math 345 Intro to Math Biology Lecture 7: Models of System of Nonlinear Difference Equations

Math 345 Intro to Math Biology Lecture 7: Models of System of Nonlinear Difference Equations Junping Shi College of William and Mary, USA Equilibrium Model: x n+1 = f (x n ), here f is a nonlinear function