There are several important things you can learn from phase space visualizations. First, you can tell from the phase space what will eventually happen to a system’s state in the long run. For a deterministic dynamical system, its future state is uniquely determined by its current state (hence, the name “deterministic”). Trajectories of a deterministic dynam- ical system will never branch off in its phase space (though they could merge), because if they did, that would mean that multiple future states were possible, which would vio- late the deterministic nature of the system. No branching means that, once you specify an initial state of the system, the trajectory that follows is uniquely determined too. You can visually inspect where the trajectories are going in the phase space visualization. They may diverge to infinity, converge to a certain point, or remain dynamically chang- ing yet stay in a confined region in the phase space from which no outgoing trajectories are running out. Such a converging point or a region is called an attractor. The concept of attractors is particularly important for understanding the self-organization of complex systems. Even if it may look magical and mysterious, self-organization of complex sys- tems can be understood as a process whereby the system is simply falling into one of the attractors in a high-dimensional phase space.
Second, you can learn how a system’s fate depends on its initial state. For each at- tractor, you can find the set of all the initial states from which you will eventually end up falling into that attractor. This set is called the basin of attraction of that attractor. If you have more than one attractor in the phase space (and/or if the phase space also shows divergence to infinity), you can divide the phase space into several different regions. Such a “map” drawn on the phase space reveals how sensitive the system is to its initial con- ditions. If one region is dominating in the phase space, the system’s fate doesn’t depend much on its initial condition. But if there are several regions that are equally represented in the phase space, the system’s fate sensitively depends on its initial condition.
3.3. WHAT CAN WE LEARN? 33 Another thing you can learn from phase space visualizations is the stability of the system’s states. If you see that trajectories are converging to a certain point or area in the phase space, that means the system’s state is stable in that area. But if you see trajectories are diverging from a certain point or area, that means the system’s state is unstable in that area. Knowing system stability is often extremely important to understand, design, and/or control systems in real-world applications. The following chapters will put a particular emphasis on this stability issue.
Exercise 3.4 Where are the attractor(s) in the phase space of the bouncing ball example created in Exercise 3.3? Assume that every time the ball bounces it loses a bit of its kinetic energy.
Exercise 3.5 For each attractor obtained in Exercise 3.4 above, identify its basin of attraction.
Exercise 3.6 For each of the phase spaces shown below, identify the following:
• attractor(s)
• basin of attraction for each attractor
• stability of the system’s state at several locations in the phase space
Exercise 3.7 Consider a market where two equally good products, A and B, are competing with each other for market share. There is a customer review website
34 CHAPTER 3. BASICS OF DYNAMICAL SYSTEMS where users of each product submit their ratings. Since there is no actual differ- ence in the product quality, the average rating scores are about the same between the two products, but the customers can also see the total number of submitted ratings for each product, which shows how popular the product is in the market. Customers tend to adopt a more popular choice. Answer the following questions:
• What would the phase space of this system look like?
• Are there any attractors? Are there any basins of attraction?
• How does the system’s fate depend on its initial state?
Chapter 4
Discrete-Time Models I: Modeling
4.1
Discrete-Time Models with Difference Equations
Discrete-time modelsare easy to understand, develop and simulate. They are easily im- plementable for stepwise computer simulations, and they are often suitable for modeling experimental data that are almost always already discrete. Moreover, they can repre- sent abrupt changes in the system’s states, and possibly chaotic dynamics, using fewer variables than their continuous-time counterparts (this will be discussed more in Chapter 9).
The discrete-time models of dynamical systems are often calleddifference equations, because you can rewrite any first-order discrete-time dynamical system with a state vari- ablex(Eq. (3.1)), i.e.,
xt=F(xt−1, t) (4.1)
into a “difference” form
∆x=xt−xt−1 =F(xt−1, t)−xt−1, (4.2)
which is mathematically more similar to differential equations. But in this book, we mostly stick to the original form that directly specifies the next value ofx, which is more straight- forward and easier to understand.
Note that Eq. (4.1) can also be written as
xt+1 =F(xt, t), (4.3)
which is mathematically equivalent to Eq. (4.1) and perhaps more commonly used in the literature. But we will use the notation with xt, xt−1, xt−2, etc., in this textbook, because
36 CHAPTER 4. DISCRETE-TIME MODELS I: MODELING this notation makes it easier to see how many previous steps are needed to calculate the next step (e.g., if the right hand side contains xt−1 and xt−2, that means you will need to
know the system’s state in previous two steps to determine its next state).
From a difference equation, you can produce a series of values of the state variablex
over time, starting with initial conditionx0:
{x0, x1, x2, x3, . . .} (4.4)
This is calledtime series. In this case, it is a prediction made using the difference equa- tion model, but in other contexts, time series also means sequential values obtained by empirical observation of real-world systems as well.
Here is a very simple example of a discrete-time, discrete-state dynamical system. The system is made of two interacting components: A and B. Each component takes one of two possible states: Blue or red. Their behaviors are determined by the following rules:
• A tries to stay the same color as B.
• B tries to be the opposite color of A.
These rules are applied to their states simultaneously in discrete time steps.
Exercise 4.1 Write the state transition functionsFA(sA, sB)andFB(sA, sB)for this
system, wheresAandsB are the states of A and B, respectively.
Exercise 4.2 Produce a time series of (sA, sB) starting with an initial condition
with both components in blue, using the model you created. What kind of behavior will arise?