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Chapter 7 Dynamical Systems - UZH · 2016-06-23 · 7-2 CHAPTER 7. DYNAMICAL SYSTEMS This equation...
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Chapter 7 Dynamical Systems 7.1 Introduction Cellular automata, which we encountered in the previous chapter, are examples of dynamical systems. In the present chapter, we will explore two additional dynami- cal systems that are not only very famous, but also nicely illustrate one fascinating property of such systems: even very simple nonlinear dynamical systems can ex- hibit, despite being completely deterministic, a completely unpredictable behavior that seem to be random – a behavior called chaos. 7.2 Definition A dynamical system is described by the state x(t) of the system at time t, as well as by an evolution function Φ that describes the time evolution of the system, i.e. how the state of the system changes with time. The state x(t) usually consists of multiple components describing different variables of the system. Map When the time is discrete, the evolution function is usually called a map. It is also not uncommon to place the time index t as a subscript of the state x to indicate that the time is discrete. x t+1 = Φ(x t ) (7.1) 7-1
Transcript of Chapter 7 Dynamical Systems - UZH · 2016-06-23 · 7-2 CHAPTER 7. DYNAMICAL SYSTEMS This equation...
Chapter 7
Dynamical Systems
7.1 IntroductionCellular automata, which we encountered in the previous chapter, are examples ofdynamical systems. In the present chapter, we will explore two additional dynami-cal systems that are not only very famous, but also nicely illustrate one fascinatingproperty of such systems: even very simple nonlinear dynamical systems can ex-hibit, despite being completely deterministic, a completely unpredictable behaviorthat seem to be random – a behavior called chaos.
7.2 Definition
A dynamical system is described by the state x(t) of the system at time t, as wellas by an evolution function ! that describes the time evolution of the system, i.e.how the state of the system changes with time.
The state x(t) usually consists of multiple components describing differentvariables of the system.
Map
When the time is discrete, the evolution function is usually called a map. It is alsonot uncommon to place the time index t as a subscript of the state x to indicatethat the time is discrete.
xt+1 = !(xt) (7.1)
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This equation is sometimes also written as a difference equation:
"xt = !(xt)
where xt+1 = xt + "xt.
Flow
When the time is continuous, the evolution function is usually called a flow. Thedynamical system is thus described by
x(t) = !(x(t)) (7.2)
where x(t) = ddtx(t) is the time derivative of the system’s state x(t) at time t.
Note that Equation 7.2 has usually multiple components, and thus constitute awhole set of differential equations.
Non-Linearity
A dynamical system is said to be linear when the evolution function is linear, i.e.when ! satisfies the two following properties:
1. Additivity: !(x + y) = !(x) + !(y)
2. Homogeneity: ! · !(x) = !(! · x) with ! ! R.
Linear systems are obviously much more convenient for mathematical analy-sis. Often, non-linear systems are approximated by linear systems (they are lin-earized), assuming very small variations of the state of the system.
We are however interested in the behavior of a less restricted set of systemsthat might somehow better model phenomena observed in the “real” world. Wewill thus focus on non-linear systems.
7.3 The Logistic MapLet us consider a very simple model of population growth. The size of the pop-ulation at year t is denote with xt, a number between 0 (no individual) and 1(maximum number of individuals). We assume the following properties:
7.3. THE LOGISTIC MAP 7-3
1. (Reproduction) For small population, we assume that the population willincrease at a rate proportional to the current population. In other words, thepopulation at year t + 1 is proportional to the population at year t.
2. (Starvation) As the population grows towards 1 (i.e. starts reaching the max-imum number of individuals), the effects of limiting factors such as diseasesor finite food supply are felt. The population will decrease at a rate propor-tional to the maximum population less the current population, i.e. 1" xt.
The evolution of the population can be written as the following map:
xt+1 = r xt (1" xt) (7.3)
where r > 0 is a constant proportionality factor representing the combined ratefor reproduction and starvation. Figure 7.1 illustrates, for a particular value of r,how the population varies from one year to the next.
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Figure 7.1: Phase plot representing the evolution of the population, i.e. next year’spopulation xt+1 as a function of the current population xt, according to Equa-tion 7.3 (with here r = 2). The dashed lines represent the reproduction and star-vation limits.
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We will now explore the behavior of the system for different values of theproportionality factor 0 # r # 4. (It can easily be shown that for r > 4, thepopulation can exceed the maximum value of xt = 1).
An interesting property of the logistic map (and of many other dynamical sys-tems) is that the quantitative behavior does not depend on the particular initialcondition (in our case, the population at the initial year, x0) – as long as we don’tfall into marginal conditions (in our case, a zero population, which will alwaysstays to xt = 0).
We thus choose to investigate, in the following sections, the behavior of thesystem with an arbitrary initial population of x0 = 0.2.
7.3.1 Point AttractorLet us consider the evolution of the population with r = 2:
x0 x1 x2 x3 x4 . . .0.2 0.32 0.435 0.492 0.499 . . .
These values are represented graphically in Figure 7.2(a). Note that the phaseplot provides a convenient way of geometrically determining the evolution of thepopulation. Figure 7.2(b) illustrates the construction of the successive values ofxt using the dotted line xt+1 = xt.
For r = 2, we see that the population converges towards the value xt = 0.5.
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Figure 7.2: Evolution of the population for r = 2. (a) Population as function oftime. (b) Corresponding phase plot.
7.3. THE LOGISTIC MAP 7-5
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Figure 7.3: Evolution of the population for r = 2.7. (a) Population as function oftime. (b) Corresponding phase plot.
For r = 2.7, we see that the system converges to x $ 0.63 (see Figure 7.3):
x0 x1 x2 x3 x4 x5 x6 x7 . . .0.2 0.432 0.663 0.604 0.646 0.618 0.637 0.624 . . .
For small values of r (actually, for r < 1), the populations dies out and con-verge towards x = 0. Figure 7.4 and the following table demonstrates this forr = 0.5:
x0 x1 x2 x3 x4 x5 x6 x7 . . .0.2 0.08 0.037 0.018 0.009 0.004 0.002 0.001 . . .
In summary, as long as r < 3, the population is observed to converge to asingle point in the phase plot – a point attractor.
7.3.2 Periodic AttractorAs the value of r is increased beyond 3, we observe that the population does notconverge to a single limit anymore, but starts oscillating between two differentvalues. Such an attractor is called a periodic attractor.
Figure 7.5 illustrates how the population of the system converges towards sucha 2-point periodic attractor for r = 3.1.
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Figure 7.4: Evolution of the population for r = 0.5. (a) Population as function oftime. (b) Corresponding phase plot.
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Figure 7.5: Evolution of the population for r = 3.1. (a) Population as function oftime. (b) Corresponding phase plot.
7.3. THE LOGISTIC MAP 7-7
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Figure 7.6: Evolution of the population for r = 3.6. (a) Population as function oftime. (b) Corresponding phase plot.
7.3.3 Strange AttractorAs r increases further, the population oscillates between 4 values (for 3 < r <3.45), 8 values (for 3.45 < r < 3.54), 16 values and so on, successively doublingthe period as r comes closer to the approximate value of 3.57.
At r $ 3.57 is the onset of chaotic behavior. For most values of r > 3.57, thepopulation doesn’t converge anymore, but has each year a different value (withinsome interval). Figure 7.6 illustrates the chaotic evolution of the system’s popu-lation for r = 3.6.
Such behavior is referred to as a system with a strange or chaotic attractor.
7.3.4 Bifurcation DiagramThe various behaviors described so far can be summarized with a plot showing,for all values of the parameters r, the limit values of the system’s population. Sucha plot, called bifurcation diagram, is shown in Figure 7.7.
The similarity of behaviors observed with the logistic map – a simple systemconsisting of just one real number changing according to Equation 7.3 – is aston-ishingly similar to what we have encountered with cellular automata – systemsconsisting of many finite state machines, all following a particular update rule.
On the one hand, both systems are observed to display point attractors, peri-odic attractors as well as chaotic attractors. Furthermore, there is a definite sim-ilarity between the proportionality constant r of the logistic map and Langton’s
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Figure 7.7: Bifurcation diagram of the logistic map. For each value of r, thediagram indicates the limit population values.
! parameter of cellular automata: as the parameter increases, the attractor of thesystem changes from point to periodic to chaotic. Note however that there is no“complex” behavior of the logistic map.
7.3.5 FractalsThere is nevertheless something interesting happening at the “edge of chaos” inthe logistic map. Figure 7.8 shows a close-up of the bifurcation diagram for ap-proximately 2.8 < r < 4.0.
We see not only that there are some “islands of stability” within the chaoticregion, but also that the diagram shows a self-similar, fractal pattern!
Even though fractals are encountered in totally different representations (in thepatterns produced in time by cellular automata vs. patterns found in the bifurcationdiagram of the map of a dynamical system), there seems to be a profound relationbetween chaos and fractals.
7.3. THE LOGISTIC MAP 7-9
Figure 7.8: Close-up of the bifurcation diagram of the logistic map.
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7.3.6 Sensitivity to Initial Conditions
A prime characteristic of chaotic behavior is the sensitivity to initial conditions.Figure 7.9 illustrates how in the “ordered” regimes, a slight variation in the initialpopulation value is quickly “absorbed”. On the other hand, Figure 7.10 shows thatin the chaotic regime, a slight variation in the initial condition yields dramaticallydifferent results over time.
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Figure 7.10: Sensitivity to initial conditions in the chaotic regime. The evolutionof the population for two slightly different initial conditions, x0 = 0.2 (circles)and x0 = 0.22 (crosses), yield completely different results over time.
7.4. THE LORENZ ATTRACTOR 7-11
7.4 The Lorenz AttractorThe Lorenz attractor was first studied by the meteorologist Edward Lorenz in1963. It was derived from a simplified model of convection in the earth’s atmo-sphere. In this model, the atmosphere is modeled as a system with simply threeparameters coupled together through 3 non-linear differential equations.
Assigning the letters x, y and z to these three parameters, their time evolu-tion, according to Lorenz’s model, can be expressed by the following dynamicalsystem:
x = ay " ax
y = bx" y " xz
z = xy " cz
where a, b and c are constants. Clearly, the flow of this dynamical system (seeEquation 7.2) is non-linear because of the terms xz and xy.
Lorenz was using a computer to run his simulation of the weather. For somereason he interrupted one of his simulations. He then wanted to see a sequence ofdata again and to save time he started the simulation in the middle of its course. Hewas able to do this by entering data from a printout corresponding to conditionsin the middle of his simulation which he had previously calculated.
Figure 7.11: Two different trajectories of the Lorenz simulation. There is only atiny difference at the starting point. This difference is sufficient for the system tobehave completely differently after some time.
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To his surprise, the weather that the machine began to predict was completelydifferent from the weather calculated before (see Figure 7.11). Lorenz tracked thisdown to only bothering to enter 3-digit numbers into the simulation, whereas thecomputer had last time worked with 5-digit numbers. This difference seemed tinyand the consensus at the time would have been that it should have had practicallyno effect. However, Lorenz had discovered that small changes in initial conditionsproduced large changes in the long-term outcome.
Lorenz’s simulation is an example of a chaotic system. This illustrates onceagain that chaotic systems are characterized by their extreme sensitivity to initialconditions. If the initial state of the system is slightly different from a previoussituation, the simulation might end up producing a completely different behavior.That is what Lorenz stumbled upon experimenting with his weather equations.
Figure 7.12: A trajectory in the Lorenz attractor.
7.4.1 The Butterfly EffectThe notion of sensitivity to initial conditions – here in the context of weather sim-ulation, together with the typical shape of the trajectories obtained with this model(see Figure 7.12) – is also known as the “butterfly effect”. This phrase refers to
7.5. CHAPTER SUMMARY 7-13
the idea that a butterfly’s wings might create tiny changes in the atmosphere thatultimately cause a tornado to appear (or prevent a tornado from appearing). Theflapping wing represents a small change in the initial condition of the system,which causes a chain of events leading to large-scale phenomena. Had the but-terfly not flapped its wings, the trajectory of the system might have been vastlydifferent.
7.5 Chapter Summary• A dynamical system describes the time evolution of the state of a system.
• The evolution function of a dynamical system is called map when the timeis discrete, and flow when the time is continuous.
• The logistic map illustrates how a very simple, yet non-linear map can dis-play chaotic behavior. It is a simple model of population growth.
• The behavior of population in the logistic map can reveal point, periodic orstrange attractors. Furthermore, it illustrates how chaos and fractals can beclosely related.
• The Lorenz attractor is an example of time-continuous dynamical systemcapable of displaying chaotic behavior.
• The butterfly effect encapsulates the notion of sensitive dependence on ini-tial conditions in chaos theory.
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