In this paper I review some of the basis principles of the theory of dynamical systems. I introduce the reader to the definition of chaos and strange attractors and discuss their implications

Tsonis, Anastasios A

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University of Wisconsin-Milwaukee

University of Wisconsin MilwaukeeUWM Digital CommonsField Station Bulletins UWM Field StationSpring 1989A discussion on chaosAnastasios A. TsonisUniversity of Wisconsin - MilwaukeeFollow this and additional works at: https://dc.uwm.edu/fieldstation_bulletinsPart of the Forest Biology Commons, and the Zoology CommonsThis Article is brought to you for free and open access by UWM Digital Commons. It has been accepted for inclusion in Field Station Bulletins by anauthorized administrator of UWM Digital Commons. For more information, please contact open-access@uwm.edu.Recommended CitationTsonis, A.A. 1989. A discussion on chaos. Field Station Bulletin 22(1): 17-26A DISCUSSION ON CHAOSAnastasios A. TsonisDepartment of GeosciencesUniversity ofWisconsin-MilwaukeeMilwaukee, Wisconsin 53201ABSTRACTIn this paper I review some of the basis principles of the theory ofdynamical systems. I introduce the reader to the definition of chaos and strangeattractors and discuss their implications.DEFINITIONSOver the last decade physicists, astronomers, biologists and scientists frommany other disciplines have developed a new way of looking at complexity innature. This new way has been termed 'chaos'. Chaos, which is defined asrandomness generated by simple dynamical systems, allows us to see order inprocesses that were thought to be completely random. It is the purpose of thispaper to introduce the reader to the concept and implication of chaos.In the preceding paragraph the term 'dynamical systems' was used. What is adynamical system? In simple terms it is a system whose evolution from someinitial state (which we know) can be described by some rule(s). These rules areconveniently expressed as mathematical equations. The evolution of such a systemis best described by the so-called state space.A pendulum is allowed to swing back and forth from some initial state asdescribed in Figure la. This initial state can be completely described by thespeed and position of the pendulum. The position of the pendulum at any timecan be given by the angle x. Under such an arrangement, Newtonian Physicsprovide the equations (rules) which describe the evolution of that initial statein tjjne.Let us assume that to begin with the pendulum is held at position 1. Thenits initial state will be x = xl, and velocity v = 0. The pendulum is then letfree. As the pendulum moves toward point 0, its speed increases due to gravityacceleration. Therefore, after a while (position 2) the pendulum will be closerto point 0 and will have a higher speed. Once the pendulum crosses point 0 itsspeed decreases because now gravity acts in a direction opposite to its motion.Therefore at some point (position 3) the pendulum's speed will become zero again.Immediately after the pendulum will begin to swing back, and after it crossespoint 0 it will once again attain, at some point, a zero speed (position 4).Because there is always some friction, however, the points at which the speedbecomes zero (to the right and left of point 0) are not fixed but are foundcloser and closer to point 0. Finally the pendulum will come to rest at point 0.17A.B.Fig. 1. (a) A dynamic system is a system whose evolution from initial state canbe determined by some rules. In the above figure the motion of the pendulum canbe completely described by the laws of physics if its initial position andvelocity are known. (b) A useful concept in studying the evolution of dynamicalsystems is the state space. The coordinates of the state space are necessaryvariables that are needed to completely describe the evolution of the dynamicalsystem in question. In our example these coordinates are the velocity and theangle x of the pendulum. As the pendulum swings back and forth it follows atrajectory in the state phase which converges to a fixed point. This point iscalled an attractor of the dynamical system.18Apparently, the evolution of the above dynamical system can be completelydescribed by two variables, namely velocity and angle x. These two variablesdefine the coordinates of the so-called state space. If one plots the velocity(v) as a function of the angle (x) of the pendulum one will get Figure Ib. Thesolid line is called the trajectory in the state space and apparently describesthe evolution of our dynamical system. As it can be seen, the trajectoryconverges to point 0. As a matter of fact, any other trajectory which willcorrespond to an evolution of the above dynamical system from a different initialstate (velocity and angle) will converge to point 0 (i.e. no matter what theinitial state, the pendulum will always come to rest at point 0). The point 0 inthe state space is called an attractor. It attracts all the trajectories in thestate space. Apparently, the behavior of predictability is guaranteed. Theevolution of that system can be accurately predicted. The pendulum will alwaysrest at point 0. Point attractors, therefore, correspond to systems that reachsteady state of no motion.So far we have discussed only one form of attractor (i.e. a point) . Thenext simplest form is the limit cycle (Fig. 2) . A limit cycle in the state spaceindicates a periodic motion. An example of a system whose attractor is a limitcycle is the grandfather clock where loss of kinetic energy due to friction iscompensated mechanically via a mainspring. No matter how the pendulum clock isset swinging a perpetual periodic motion will be achieved. This periodic motionmanifests itself in the state space as a limit cycle. Again, in the cases ofsystems which have a limit cycle as an attractor, long-term predictability isguaranteed.Fig. 2. Another form of an attractor is the limit cycle. In this case alltrajectories are attracted by the limit cycle which represents a periodevolution. The grandfather clock is a system which possesses a limit cycle as anattractor. Another familiar system with a limit cycle as its attractor is theheart.19Another form of an attractor is a torus. The torus looks like the surfaceof a donut (Fig. 3). In this case all the trajectories in the state space areattracted to and remain on that surface. Systems that possess a torus as anattractor are quasi-periodic. In quasi-periodic evolution a periodic motion ismodulated by a second motion, itself periodic, but with another frequency. Thecombination of frequencies will produce a time series whose regularity is notclear. The power spectrum, however, should consist of sharp peaks at each of thebasic frequencies with all its other prominent features being combinations of thebasic frequencies. Geometrically a quasi-periodic trajectory fills the surfaceof a torus in the appropriate state space. An important characteristic of suchan attractor is that two trajectories which represent the evolution of the systemfrom different initial conditions and which are close to each other when theyapproach the attracting surface will remain close to each other forever (Fig.3). This characteristic can be translated as follows. The two points in thestate space where the trajectories enter the attractor can be two measurements(initial states) which differ by some amount. Since these trajectories remainclose to each other it means that the states of the system at a later time aregoing to differ by the same amount that they differed initially. Thus, if weknow the evolution of such a system from an initial condition we can predict theevolution of the system from some other initial condition accurately. Again inthis case long-term predictability is guaranteed.Fig. 3. Another form for an attractor is the torus. In this case the evolutionof the corresponding dynamical system from any initial condition will follow^atrajectory in the state space which will eventually be attracted and remainforever on the torus. The most important characteristic of a system whichexhibits such an attractor is that if the two involved frequencies have no commondivisor two initially nearby trajectories on the attractor remain nearby forever.20The above-mentioned forms of attractors are 'well behaved1 attractors andusually correspond to a system whose evolution is predictable. Often they arecalled non-chaotic attractors. In mathematical terms the above-mentionedattractors are topological submanifolds of the available state space.SUttNGE AT1RACDQRS - CHBOSIn 1963, Lorenz, in a paper which started everything, discovered a systemwhich under certain circumstances possessed an attractor that did not look likeanything described above. Lorenz was experimenting with a very simple model ofthree differential equations that describe the motion of an individual moleculein a fluid flow which travels over a heated surface. The warmer fluid formed atthe bottom is lighter and it tends to rise, creating convection. The attractorof this dynamical system is shown in Figure 4a. Since the state of the system isdescribed by three equations the state space has three coordinates. Not onlydoes this attractor not look like anything described above, but also it has twovery important properties: (i) the evolution described by a trajectory isdeterministic but strictly nonperiodic (never repeats itself); and (ii) as withall attractors all trajectories converge on the attractor but two nearbytrajectories do not stay close to each other but they very soon diverge andfollow totally different paths in the attractor. That means that the evolutionof the system from two slightly different initial conditions will be completelydifferent. The above is very effectively demonstrated in Figure 4a and b. Thedot in Figure 4a represents 10,000 initial conditions that are so close to eachother in the attractor that they are indistinguishable. They may be viewed as10,000 initial situations that differ only slightly from each other. If we allowthese initial conditions to evolve according to the rules (equations) thatdescribe the system, we see (Fig. 4b) that after some time the 10,000 dots canbe anywhere in the attractor. In other words, the state of our system after sometime can be anything despite the fact that the initial conditions were very closeto each other. Apparently, the evolution of the system is very sensitive toinitial conditions. In this case we say that our system has generatedrandomness. We then see that there exist systems which, even though describedby sinple deterministic rules, can generate randomness. Randomness generatedthis way has been termed chaos. These systems are called chaotic dynamicalsystems and their attractors are called strange or chaotic attractors. Theseattractors are not topological submanifolds of the total available space(Mandelbrot, 1983; Tsonis and Tsonis, 1987; Tsonis and Eisner, 1988).21A. B.Fig. 4 (a) An example of a strange attractor. This structure in the state space represents the attractor of a fluid flowwhich travels over a heated surface. All trajectories (which will represent the evolution of that system for differentinitial conditions) will eventually converge and remain on that structure. However, any two initially nearby trajectoriesin the attractor do not remain nearby but they diverge. (b) The effect of the divergence of initially nearbytrajectories in the attractor. The dot in (a) represents 10,000 measurements (initial conditions) which are so very closeto each other that they are practically indistinguishable. If we allow each one of these states to evolve according to therules, because their trajectories diverge irregularly, after a while their states can be practically anywhere. (Figurescourtesy of Dr. James Crutchfield.)A SIMPLE EXAMPLEI now proceed in presenting an example of a biological system with a varietyof periodic and chaotic solutions. The system is described by the following'logistic1 difference equation which is used to model population dynamics (May,1976):Xt+1 = aXt (1 - Xt), 0 < X < 1, (1 < a < 4)This equation relates the population of a given generation X. to thepopulation of the next generation Xfc+1. The parameter a is called thenonlinearity parameter and it represents a growth rate which may be related tothe food supply to fertility, etc. The philosophy behind this equation is thatit represents a function which increases when the population is small, reducesgrowth at intermediate values and decreases as the population becomes large. Byiterating (repeating) the above equation one can obtain the population'sevolution from some initial value for a given choice of the parameter a. Thedynamics of the logistic equation have been studied extensively by May (1976) whodiscovered an amazing variety of possible evolutions as the parameter a wasvaried. For a value of a < 3.0 the population settles into a steady state (nochange) . As the parameter a is varied slightly > 3.0 something surprisinghappens. The population now settles into a period-two (repeating every othergeneration) oscillation. A further increase of a and the evolution becomes aperiod-four oscillation (repeating every four generations). The 'magic1continues as the nonlinearity parameter increases by giving rise to periodsincreasing in powers of 2(23, 24, 2 ...). And then this period doubling comes toan end when for a value of a > 3.5700 the evolution becomes chaotic (or strictlyspeaking periodic with a period 2°°).This means that the evolution even though deterministic is for all practicalpurposes nonperiodic. The system is said in this case to oscillate with norecognizable frequency (chaotic evolution). As was the case with the Lorenzsystem, in this case as well, small uncertainties in the initial conditions candrive the system to completely different evolutions (see Figure 5 for anillustration of the above).23(a)(c)TIMEFig. 5. Based on the equation Xt + ^ = aXt(l - Xt) this figure shows thepopulation evolution from some initial condition XQ and for different choices ofthe parameter a: (a) for X = 0.2 and a = 2.707 the population finally reaches asteady state of no change; (b) for X =0.2 and a = 3.35 the population becomesperiodic of period two (i.e. it repeats every two generations); (c) for XQ = 0.2and a = 3.5 the population becomes periodic of period four; (d) for a = 3.829 theevolution becomes chaotic with no distinct periodicity. The solid line shows theevolution from an initial value of X =0.1 and the broken line from an initialovalue X =0.101. After some time the two evolutions are completely different.24TIME(b)(d)TIMEThis demonstrates the very inportant characteristic of chaotic dynamics referredto as the divergence of initially .nearby trajectories. Dae to the underlyingchaotic dynamics the initially close trajectories diverge irregularly. Suchdivergence cannot be observed for the periodic cases which are not sensitive tofluctuations in the initial conditions. The above results can be obtained bypressing a button in the programmable calculator. Type in f(x) = ax(l - x)choose your (a) and enter your first value (initial condition). Press thebutton. The number X will then appear on the display. Press the button again.The number X will appear and so on.25TIMEODNCEDSICNThe implications of chaos are profound. If one knows exactly the initialconditions, one can follow the trajectory that corresponds to the evolution ofthe system from those initial conditions and basically predict the evolutionforever. The problem, however, is that, because of the always present minuterandom fluctuations, any initial condition is only approximately known. In sucha case, even if we completely know the physical laws that govern our system, dueto the action of the underlying attractor, the state of the system at a latertime can be totally different than it would have been if we knew exactly theinitial condition. Simply, due to the nature of the system, initial microscopicerrors are amplified to a macroscopic scale. In this case we say that theprediction power of the system is very limited. Therefore, we see that theexistence of a strange or chaotic attractor, coupled with the fact that we canonly know approximately an initial situation, naturally imposes predictionlimits to the system. However, the macroscopic randomness is confined in a verywell-defined region of the total available state (the attractor) and, thus, inchaos there is some underlying order. This, together with the fact that achaotic trajectory is quite deterministic, suggests that processes that lookcompletely random may be in fact chaotic and thus more predictable anddescribable than they were thought to be.Chaos has opened new horizons in science and it is already considered bymany the third most important discovery in the twentieth century, afterrelativity and quantum mechanics. Philosophically speaking, chaos has broughtsome pessimism since it imposes limits on prediction. At the same time, however,it has offered a new forum for the understanding and description of irregularity,complexity and unpredictability in Nature.REFERENCESLorenz, E. N. 1963. Deterministic nonperiodic flow. J. Atmos. Sci. 20: ISO-MI.Mandelbrot, B. B. 1983. Fractal Geometry of Nature. W.H. Freeman, New York.May, R. 1976. Simple dynamical models with very complicated dynamics. Nature,261: 459-467.Tsonis, A. A. and Eisner, J. B. 1988. The weather attractor over very shorttime scales. Nature 333: 545-547.Tsonis, A. A. and Tsonis, P. A. 1987. Fractals: A new look at biological shapeand patterning. Perspec. Biol. Medic. 30: 355-361.26

A discussion on chaos

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A discussion on chaos

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