Articulatory Phonology

Transcription

1 Articulatory Phonology Catherine P. Browman Haskins Laboratories Louis Goldstein Yale University and Haskins Laboratories September 7, :52 Rough draft. 1

2 Chapter 6 Introduction to Dynamical Systems 6.1 Basic concepts The world is in a state of constant flux. Change over time is the rule. Nonetheless, it was the genius of people like Newton to see that if we narrow our focus to a little piece of the universe, and look at its behavior over a modest chunk of time, that we can sometimes discover that change is deterministic, in the sense that future states of the this little piece of the universe are predictable from the current state. Moreover, the rule or equation that expresses this determinism does not itself change over our modest chunk of time, and thus we have reduced the flux over time to something fixed. A dynamical system is one of these focussed phenomena for which we can find deterministic rules that hold over the time period. Here some everyday examples of dynamical systems. Consider an hourglass. The state of the hourglass can be described by the volume of sand in its upper chamber and the rate of flow of sand (e.g., in cm 3 /sec) into the bottom chamber. Given the state of the system at a given time (t i ), it is possible to predict what its state will be at an arbitrary future time (t i +n), or to predict the time at which the volume of sand will reduce to 0. Of course, the predictability of the time required to empty a full upper chamber is what makes the hourglass useful as a timer. The dynamical system remains valid even when the hourglass is empty; its volume is 0, and its flow rate is 0, and we can predict that its volume will still be 0 at some future time. But the validity of this dynamical system ends when someone turns the hourglass over again. A new dynamical system is now in place, very similar to the old one, but with a different initial time (t 0 ), at which the volume is at its maximum. The hourglass dynamical system itself, of course, cannot predict when someone is going to pick it up and turn it over, so the chunk of time over which it is valid is limited to the interval between one turn of the hourglass and the next. 2

3 CHAPTER 6. INTRODUCTION TO DYNAMICAL SYSTEMS 3 Now let us consider another type of dynamical system: the population of seagulls that live in a particular area of seashore. Let s assume that, in a given chunk of time, corresponding to a generation of seagulls, a certain proportion of the seagulls give birth on average, and a certain proportion of them die. If these proportions are reasonably steady, the population of seagulls can be viewed as a dynamical system, in which we predict the state of the system (the size of the population) at a future time from its current state. Let s examine the form of an equation that could accomplish this prediction. If b is the proportion that give birth, and d is the proportion that die, then we can predict the population of the next generation from the population of the current generation using an equation like that in (6.1): (6.1) P n+1 = P n + bp n dp n The equation states that population size at the next generation (P n+1 ) is equal to the populartion of the current generation P n, plus the increase in population due to birth rate, minus the decrease in population due to death rate. We simplify this equation by defining a growth rate parameter, r = 1 + b d. The new equation is: (6.2) P n+1 = rp n For example, if birth rate is.05 (5% of the seagulls have offspring) and the death rate is.01 ( 1% of the seagulls die), then r = = 1.04 We can predict how the seagull population will change over several generations by iterating equation 6.2. To do so, we must know the initial value of the system, the population at n = 0. Let us assume that value is 100. We can then calculate the number of seagulls at successive generations. For r = 1.04, the results would be as follows: n P n We can plot the population of seagulls as function of generation as in Fig The population of seagulls exhibits exponential growth, growing without bound. Several properties of dynamical systems can be learned from this very simple example: Specification of a dynamical system. System specification involves two parts: the definition of the state of the system, and the rule for change. In this example, the state of the system is defined by a single variable, the population of seagulls (P n ), and the rule for

4 CHAPTER 6. INTRODUCTION TO DYNAMICAL SYSTEMS Population Size Generation Figure 6.1: Segull population with P 0 = 100 and r = 1.04 change is expressed in (6.2). The rule for change has a single free parameter, r. It is important to see that this specification is fixed over the life of the system. Even though the population of seagulls is constantly changing, the system underlying it and the value of the parameter r do not change. Put another way, given a choice of value for the system parameters (in this case r), and initial conditions (in this case P 0 ), the behavior of the system is predicted for all future time during which the system holds. Of course, as with the hourglass example, the system is only valid during an interval of time in which the seagull population is free from external events that the system cannot predict, for example someone poisoning the seagulls with really bad junk food. And of course this dynamical system is an unrealistic description of an actual seagull population, in which availability of food will limit growth in some fashion. Such factors could however, be built into a more realistic dynamical system. Qualitative behavior as a function of parameter values. The behavior of the seagull dynamical system with r = 1.04, as shown in Fig. 6.1 grows without limits. It is also true that the value of the population after n generations will depend on the initial condition, P 0. So if P 0 =10, instead of 100, the value after 5 generations will be 12. The population will still grow without limit, but the there will forever be a difference between the population values for the different initial condition; the history is recoverable. The system behaves quite differently if the the death rate (d) > birth rate (b). Let s look at what happens if d =.1 and b =.15. The value of r will be.95. Since r is less than 1, the population will decrease every generation, until it eventually disappears entirely, as shown in Fig. 6.2, for P 0 =100. Once the value of P n is 0, the system is stable, in the sense repeated application of (6.2) will not change the value of P n. It is also the case that regardless of the initial condition, P 0, the size of the population eventually arrives at the stable value of zero. Eventually,

5 CHAPTER 6. INTRODUCTION TO DYNAMICAL SYSTEMS 5 history is lost. The behavior of the system with values of r < 1 is described as point attractor dynamics, and the value of P n =0 is the point attractor. Presence of an attractor means that the system is drawn to a stable value or sequence of values, regardless of initial conditions. If in fact it is a single value to which the attractor draws the system, then it is a point attractor. If the system is drawn to a stable (repeating) sequence of values, then it is a periodic attractor, examples of which will be discussed later Population Size Generation Figure 6.2: Segull population with P 0 = 100 and r =.95 Attractor systems are also stable in the face of certain kinds of externally induced perturbations to the system. For example, if a bunch of intruder seagulls are introduced artificially to the population, it will still eventually arrive at the attractor value. Whereas when r > 1, the intruders will forever change the history of the population. Of course, stability will be maintained in the face of only certain kinds of perturbation. Changing the nature of the shoreline habitat in such a way that changes the value of r would lead to a new system entirely, What happens if r = 1? In this case, the value of P n will never change from P 0. This is called neutral stability. There is no attractor, as the different initial conditions always lead to different long-term values, and perturbation to the system, such as adding some extra seagulls from outside the population, will forever change the value of P n. Iteration and analytical solutions. The rule for change in equation 6.2 allows the population in generation n + 1 to be computed from the population in generation n (not to be confused with generation X). The form of this equation is called a discrete difference equation (if we re-write it as P n+1 rp n = 0, it is clearer why it is a difference equation). In this example, generation number is intrinsically a discrete variable,

6 CHAPTER 6. INTRODUCTION TO DYNAMICAL SYSTEMS 6 so such an equation appropriately expresses the dependencies between successive generations. However even when modeling a system that changes in continuous time (for example those discussed in the next section), it may be possible to approximate their behavior using such a discrete difference equation. The exact system description, however, is a differential equation, as will be discussed in the next section. As long as a system can be expressed as a difference equation (or many types of differential equations) its behavior over time can be tracked by iteratively applying the equation. However, some systems also allow analytical solutions, which means that a function X(t) can be found, which gives the value the state(s) of the system (X ) at any arbitrary time. Before the advent of computers (either analog or digital) understanding of dynamical systems was pretty much limited to those with analytical solutions, and the interesting behaviors of nonlinear dynamical systems without analytical solutions (such as chaos and self-organization) were largely unknown. In fact our system in equation 6.2 does have a simple analytical solution, and it is pretty easy to see what that solution is. Consider the values of P 1, P 2, and P 3, expressed in terms of r and P 0, as shown below: (6.3) P 1 = r P 0 P 2 = r P 1 = r r P 0 P 3 = r P 2 = r r r P 0 As is obvious, to find the population P at an arbitrary generation n, one simply multiplies P 0 by a number of r s equal to n, or r n. Thus, the function that is an analytical solution for this system is: (6.4) P (n) = r n P Discrete kinematics and dynamics We turn our attention now to dynamical systems that model the motion of objects in space. The states of such systems at a given moment in time are typically the position of the object and its velocity. Position and velocity can be described in one, two, or three dimensions or even more (if we are dealing with abstract motions in multidimensional spaces). In the examples in this chapter, we will focus on motion along a single dimension. The description of states of the system as they change over time is called the kinematics of the system, or a kinematic description. Many physical objects moving in the world move continuously in time, they do not jump from position to position. Nonetheless, it is possible give a useful kinematic description that (effectively) observes the states of the system at discrete points in time, or samples, as long as they are close to one another in time.

7 CHAPTER 6. INTRODUCTION TO DYNAMICAL SYSTEMS 7 For example, imagine a really, really hard ball rolling along the floor at a constant speed, until it hits an equally hard wall, causing it to reverse direction and roll at the same speed. The kinematics of a system like this, measured at discrete instants in time, are shown in the time functions of Fig The top panel of the figure shows the position of the ball (X ) on the floor, measured in e.g., cm, at a sequence of time samples (1-19). Note that the line connecting the points is not part of the description, it is just there to make the time funtion easier to see. I will use the upper case X to refer to position when it is discrete, as it is here, and lower case x to refer to position when it is a continuous variable. For now, we don t need to consider how much clock time expires between observations (the sampling period of the system). It is enough to know that observation occurs at some regular intervals of time X D(X) D(D(X)) Time Samples Figure 6.3: Position function (X), its velocity (D(X)) and its acceleration (D(D(X))) The fact that the ball is moving at a constant speed is evident in fact that the successive points in the time function fall on a straight line (except for when the wall event occurs). The change in position between any pair of points (ie, the distance travelled during one sampling interval) is always the same (and is equal to 5 cm when the ball is rolling towards the wall and -5 cm when it is rolling back). The change in position during a sample interval is the velocity of a discretely observed system like this one. So the velocity at a given sample can simply be defined as the change in position between the current sample and the preceding sample: (6.5) V (n) = D(X(n)) = X n X n 1 I will use the symbol D(X) to refer to the velocity state of a discrete system such as this, and ẋ to refer to velocity in a continuous system. The time function

8 CHAPTER 6. INTRODUCTION TO DYNAMICAL SYSTEMS 8 of D(X ), computed by equation 6.5 is shown in the middle panel of the figure. It shows what we could see from the position time function: velocity is constant from sample time to sample time, first 5 cm/sample then -5 cm/sample. Note that D(X ) is undefined for time sample 1, as there is no previous position sample (X n 1 ) to use in the equation. We may also be interested in how the velocity of a system changes over time. Change in velocity is acceleration, and in a discrete system, it can defined as the change in velocity between the current time sample and the previous time sample. However, since velocity is defined with respect to position, acceleration can also be defined directly in terms of position, as shown in 6.6. We will use D(D(X)) to refer to the acceleration of a discrete system, and ẍ for the acceleration in a continuous system. (6.6) A(n) = D(V (n)) = D(D(X(n))) = V n V n 1 = (X n X n 1 ) (X n 1 X n 2 ) = X n X n 1 X n 1 + X n 2 = X n 2X n 1 + X n 2 The bottom panel of Fig. 6.3 shows the acceleration computed in this way. Note that the first two samples are undefined for acceleration, because as Eq. 6.6 shows, three samples are required to calculate acceleration, the current one two past ones. For a system moving with constant velocity, the acceleration is 0, which is what the figure shows. The only non-zero value is when the ball bounces off the wall, when the velocity changes by -10 cm/sample, which is the value of acceleration shown. For the epoch of time during which the ball rolls toward the wall, a very simple dynamical system models how the system states change over time. A system that moves with constant velocity can be described as in 6.7, where V is a constant. The next position, X n+1 is equal to the previous position plus the velocity, which is, of course, the amount by which position changes during from one sample to the next. The particular kinematics in Fig.6.3 result from setting V=5 with the initial condition X 0 = 10. (6.7) X n+1 = X n + V As noted above, the complete description of a system like this includes both position and velocity, but if we know the position at each moment in time, 6.5 tells us the velocity. An analytical solution to this system can also be easily determined. It is shown in Eq. 6.8 (6.8) X(n) = X 0 + n V The position at some arbitrary sample n is equal to the initial position, plus the n times the velocity.

9 CHAPTER 6. INTRODUCTION TO DYNAMICAL SYSTEMS 9 A more challenging example is a system with a constant acceleration. An example would be an objects in free fall. They are subject to the force exerted by gravity that causes a constant acceleration, g = 9.8m/sec 2. What is the dynamical system that can account for the the changing states of such a system? If we look at the equation that defines acceleration, repeated here as 6.9, we see that acceleration is defined as a function of three successive sample values. Since acceleration is constant in this dynamical system, it is possible to rearrange terms as in 6.10 and add 1 to all the indices, so as to derive an equations that gives the next value of position (X n+1 ) as a function of the value of the current position, the preceding position and the acceleration, as in 6.11 (6.9) (6.10) a = X n 2X n 1 + X n 2 X n = 2X n 1 X n 2 + a (6.11)X n+1 = 2X n X n 1 + a To observe the behavior of this system, we can iterate To do so, we need to specify two initial values, to serve as X n and X n 1 for the first iteration. This can be done either by specifying two initial positions (X 0 and X 1 ), or by specifying an initial position X 1, and an initial velocity, V 1, and calculating X 0 as X 1 V 1. Here are the results of iterating the equation with a = 10cm/sample, X 1 = 0 and V 1 = 0. The resulting parabolic time function in Fig. 6.4 is characteristic of free falling bodies. In Fig. 6.5, the initial position is again 0, but the object is given an upward initial velocity of 300 cm/sample. The acceleration slows down the upward velocity, until the object begins to fall. 0 x x n Figure 6.4: Constant acceleration system with a = 10cm/sample, X 1 = 0 and V 1 = 0 We can also find an iterative solution to a system with constant acceleration. One way of thinking about how to do this is given in introductory physics

10 CHAPTER 6. INTRODUCTION TO DYNAMICAL SYSTEMS x n Figure 6.5: Constant acceleration system with a = 10cm/sample, X 1 V 1 = 300 = 0 and chapters. For continuous systems under constant acceleration (like that due to gravity), the position at some arbitrary time, t, can be computed as follows: (6.12)x t = x 0 + V t The position is equal to the initial position, plus the distance traveled, which is equal to the average velocity times the elapsed time. A discrete equivalent looks like this: (6.13)X(n) = X 1 + V (n 1) where (n 1) is the number of elapsed samples between sample 1 and sample n. Since velocity is increasing linearly (due to constant acceleration), the average velocity is the mean of the initial velocity and the velocity at sample n. (6.14)X(n) = X (V 1 + V (n)) (n 1) Velocity increases by A every sample. so the velocity at sample n is known: (6.15)V (n) = V 1 + A (n 1) Substituting this into 6.14 gives: (6.16)X(n) = X (V 1 + V 1 + A (n 1)) (n 1) (6.17)X(n) = X 1 + V 1 (n 1) + 1 A (n 1)2 2 This equation is the same as that for continuous dynamical systems, with n 1 replacing the elapsed time ( t), and gives the same results when the amount

11 CHAPTER 6. INTRODUCTION TO DYNAMICAL SYSTEMS 11 of time per sample is specified. Note that the form of the function (second order polynomial) is that of a parabola, which is what we saw when we iterated the discrete difference equation in Note also that the system has no attractors: X increases without limit regardless of the value of the parameter (A) or the initial conditions. 6.3 Elastic Systems Some physical objects or systems have the capacity to store energy when they are displaced from some preferred configuration and to (later) expel this energy to return to the preferred configuration. The classic example of such an elastic system is the spring. When one end of a spring is attached to some rigid body and the other end is stretched so the spring is now longer than it was, the spring stores the energy that was expended in stretching it. When the stretched end is released, the spring will return to its shorter, resting, length that it that exhibited before being stretched. We will talk about springs in this chapter, but there are many, many systems that behave in a functionally (and mathematically) similar way: the foam in cushion, a column of air in a tube, the plates of a circuit capacitor, the tissues making up the lungs, the muscles of animal s bodies. Even more abstract social systems such as aspects of the economy (e.g., response to a sudden oversupply of some commodity) exhibit the same functional behavior. In all cases, the elastic behavior is caused by the interaction among the microscopic units out of which the system is constructed. In some cases, like the economy, most of our everyday experience is with those units (individual buyers and sellers) rather than with the spring-like 1 Readers who actually plot the iterated difference equation and the analytical function in 6.17 will find that they are not exactly the same. The iterated difference equation is timeshifted half a sample earlier compared to the analytical function. In other words, if you plot the two functions, then slide the analytical one by one-half a sample to the left, the curves will line up. Here s one way of seeing why this is so. When defining velocity for the discrete difference equation, we defined it as follows: V (n) = X n X n 1 which means that the position at sample n can be calculated as follows: X n = X n 1 + V (n) This means that the distance travelled during from sample n-1 to sample n is equal to the velocity of sample n. However, when we derived the analytical function, we stated that the distance traveled was equal to the average velocity of the interval times the number of samples. Let s apply that assumption to the distance traveled in one sample when the velocity is changing linearly. In that case, the value of X n should be: X n (V (n) + V (n 1)), which is different than that given in the 2 previous equation. Basically, the velocity midway between n 1 and n should be used to get the position value for X n under this assumption. By using the velocity at sample n in the iterated version, we are actually getting the position that should occur half a sample later in other words, the iterated function is shifted half a sample earlier compared to the analytical version. Note that when the analytical function is derived algebraically from iterative function (derivation not shown here) rather than the way we derived it above by analogy to the continuous dynamical system, this time shifted function is obtained. The function is this: X(n) = X 1 + V 1 (n 1) A (n 1)2 + 1 A (n 1). The extra term in this equation 2 compared to 6.17 is 1 2 A (n 1), which is in fact, equal to 1 2 V n 1 and is therefore the distance traveled in the extra half sample. Note also that when the sample duration is short with respect to the time interval being examined, the half-sample shift is a very small difference

12 CHAPTER 6. INTRODUCTION TO DYNAMICAL SYSTEMS 12 behavior, while in other cases, like a physical spring, the particulate structure is transparent to our everyday experience (the spring is a thing ). But the cause underlying the behavior is nonetheless similar in all cases (though exact nature of units and laws governing their interaction may differ). For our purposes (modeling speech gestures), we are not so interested in the internal particulate structure of the system, but rather the elastic behavior, and it is that we focus on here. To that end we will use a physical spring as an example to investigate the behavior of such systems, because our everyday experience with them is at the appropriate level. To begin our exploration of springs, let us consider a mass attached to one end of an ideal spring that conserves all the energy applied to it and does not lose any to friction. The behavior of such a system at five time points is illustrated in Figure 6.6. At time (1), the end of the spring with the mass is stretched its Figure 6.6: Behavior of a spring at five time points (one cycle) rest position (X 0 ) to a new position, X 0 +A. The stretching of the spring exerts a force on the mass and when it is released, it heads towards the rest position, which reaches at time (2). However, because it is moving when it reaches X 0, its inertia causes it to continue traveling in the same direction. One it passes X 0, it begins to compress the spring, which exerts a force slowing it down. This force is however, stored in potential energy in the spring. When the spring reaches the position X 0 A, at time (3), it stops moving in that direction, and the force exerted by the spring now causes it to move in the reverse direction, reaching its rest position at time (4). Again it continues to move through its rest position until it winds up back at X 0 + A at time (5), at which point all the steps will repeat. Thus, the spring will oscillate back and forth between positions X 0 + A and X 0 A. The time points in Fig. 6.6 represent one cycle of the spring s motion. The time function of the spring s motion is plotted in Fig It shows the

13 CHAPTER 6. INTRODUCTION TO DYNAMICAL SYSTEMS 13 position of the spring (x) as a function of time. The five time points displayed Figure 6.7: Time function of spring, with five time positions marked in Fig. 6.6 are also marked in this figure: (1) X 0 +A, (2) X 0, (3) X 0 A, (4) X 0, (5) X 0 + A. This figure also shows the positions intermediate between these five positions (as well as additional cycles). Note the the shape of the time function during each cycle is a sinusoid: the function between positions 1 and 5 has the shape of one cycle of the cosine function. Why it is that the time function of the spring is sinusoidal will be explained in following sections. Another way of visualizing the behavior of the spring is to plot its history as points in a two-dimensional plot in which one axis represents the position along x at some moment in time, and on the other represents the velocity of motion at that same point in time. As the spring oscillates, the points trace out an ellipse (in this case a circle) in this kind of plot, as shown in Fig The mass is at rest (zero velocity) at the times of maximum displacement from its rest position (1,3,5) and has maximal speed as it passes through the rest position at times (2) and (4). Velocity takes into account both speed and direction, and since the mass is moving in the negative x direction at time (2), we say that its velocity is at a minimum there, and at time (4) it is moving in the positive x direction, and is therefore at a maximum velocity. Thus, the velocity of motion also oscillates between minimal (2) and maximal (4) values, just as does the position, and the times at which the velocity maxima and minimal occur are intermediate between those times at which the position maxima and minima occur. Time is not explicitly represented in this plot, rather it shows the cumulative history of the states that system the system visits, where the each system state is a pair of {position, velocity} values. Because the system is displaying cyclic behavior, each successive cycle after the first traces out the same points in the display. Plotting the position against velocity is usually called a phase plane display (for reasons that will be become clearer in the following paragraphs), and the cumulative history of the system as plotted in that display is its phase portrait. Because the points in the system s phase portrait are always on the closed

14 CHAPTER 6. INTRODUCTION TO DYNAMICAL SYSTEMS 14 Figure 6.8: Phase plane ellipse, the changing states of the system over time can be modeled by a rotating vector attached to the center of the phase plane. At any given time, the state of the system can then be represented by a a single variable, ϕ, which represents the angle of this rotating vector. Given the angle, the position and velocity of the system are known (they are the states where the ellipse intersects the rotating vector), as shown in Figure 6.9. States of the system at times (1)-(5) 180 o Figure 6.9: Phase plane with phase angles marked would be represented by angles 0, 90, 180, 270, and 360 (which = 0 ). The angle of the rotating vector is also called the phase angle, which is why the plot is called a phase plane. The phase angles can also be noted on the time function

15 CHAPTER 6. INTRODUCTION TO DYNAMICAL SYSTEMS 15 display, as in Fig Note that the time function can be obtained by plotted Figure 6.10: Time function with phase angles marked the x value associated with each ϕ as the vector rotates in the phase plane. The speed at which the vector rotates around will determine the actual amount of time (number of samples in a discrete system) required to complete one cycle around the circle. This time is usually referred to as τ, the period of the spring. The frequency of oscillation f, represents the number of cycles completed in a unit of time, usually one second for continuous systems (1 Hz is one cycle per second) or one sample for a discrete system (which we will be considering in the next two sections). Frequency is simply related to period f = 1 τ. The speed at which the vector rotates around the circle is usually represented by ω. In a discrete system, ω has the units of angle traversed per sample. Angle traversed can be represented in degrees or radians, where 2π radians is equal to 360. Thus, the frequency f of oscillation of the spring (in cycles per sample) can be derived from the angular speed, f = ω 360 (where ω is given in degrees per sample), or f = ω 2π (where ω is given in radians per sample). Given the rotating vector analysis, it is easy to find an analytical expression that gives the position of the spring any given time, t, or time sample n. If the vector is rotating at ω radians per sample, then the elapsed angle after n samples is: (6.18) ϕ = ωn If we know ϕ, then we can calculate the corresponding position along the x-axis. It is the projection of the point in the phase plot on the x-axis. If the radius of the phase plot, which is equal to the amplitude of oscillation, equals A, then the length of that projection is A cos ϕ. Putting this together with the equation 6.18 which tells us how phase angle depends on time sample, we get the following

16 CHAPTER 6. INTRODUCTION TO DYNAMICAL SYSTEMS 16 expression for the value of x as a function of sample n: (6.19) X(n) = A cos(ωn) As we will see below, this equation is the analytical solution of the dynamical system corresponding to a type of spring. Finally, note that the equation for the velocity of the spring at any instant is very similar in form to that of the position. Since velocity is plotted on the y-axis, the value of velocity will be the y-projection of the point in the phase plot, and therefore, it will be use the sine instead of the cosine: (6.20) V (n) = A sin(ωn) The negative sign can be understood with reference to Fig When the spring is near time 1, and beginning to go around the phase plot in a counterclockwise direction, the value of position is positive (on the positive side of the x-axis), while the the value of the y-projection is negative (on the negative side of the y-axis) Thus, velocity is in the opposite direction from the position at this point the velocity is leading the spring back towards the rest position Second order difference equation undamped Let s now develop the dynamical system that characterizes the behavior of a spring like that one in the previous section, one that oscillates indefinitely without losing energy. Technically (in a sense that we be clarified below), such a system is undamped. We will attempt to define a rule for change that we can iterate to give us history of the system over time. The key element of elastic systems is that when such a system is displaced from its preferred configuration (e.g. rest position of a spring, or x 0 ), an elastic force develops that attempts to restore the spring back to the rest position. However, once this force is imparted to the spring, inertial forces cause it to it overshoot its rest position, and to stretch the spring in the opposite direction causing new elastic forces to develop. The magnitude of the elastic force is, in many systems, a simple linear function of the amount of displacement from rest, as in The proportionality constant, k, is usually called the stiffness of the system. The greater the stiffness, the greater force is generated for a given displacement. (6.21) F = k(x x 0 ) We also know that the force on the spring will produce an acceleration in the direction of the force, and that the acceleration will be a linear function of the F, with a proportionality constant of 1/m: (6.22) F = ma a = 1 m F

17 CHAPTER 6. INTRODUCTION TO DYNAMICAL SYSTEMS 17 Now putting setting F equal to the spring force, we get: (6.23) a = k m (x x 0) But we know that acceleration is just the second derivative of displacement, so we can replace a with the time derivative of displacement, thus producing an equation in which all the terms are defined with respect to displacement and its time derivatives. Here we use the discrete version of equation for acceleration (6.6), as we are attempting to derive an equation for the system that we can iterate to derive its time history. To keep the equation looking simpler, we will define a new discrete displacement variable X, which is the distance of the spring position from its rest position, X = x x 0. This gives us: (6.24) X n 2X n 1 + X n 2 = k m X As we did earlier in 6.9, we can add 1 to all the indices on the left, as we want to derive a difference equation for the next sample (X n+1 ) from the current sample (X n ), and the previous one (X n 1 ): (6.25) X n+1 2X n + X n 1 = k m X X on the right hand side is the current value of displacement, which is, in fact, X n, so the equation becomes: (6.26) X n+1 2X n + X n 1 = k m X n We can now rearrange terms, giving us an equation we can iterate, with X n+1 on the left. (6.27) X n+1 = (2 k m )X n X n 1 To iterate Eq. 6.27, we have to supply the value of the system parameters: k, the stiffness of the spring, and m, the mass of the object at the end of the spring. The equation tell us that it is the ratio of stiffness to mass that controls the behavior of the system. If we were to observe a spring that has twice the stiffness of some other spring, the behavior of that system would identical to a third system with the same stiffness as the first, but with half the mass. As we experiment with the system, therefore, we will manipulate the ratio k/m directly. We will also have to specify initial conditions, the values of X n and X n 1. Figure 6.11 shows the results of iterating the undamped spring system equation, with three different values for k/m:.1,.3, and.5. The value of X for 40 successive samples is shown in each graph. For k/m =.1,.3, the initial conditions were the same, X n = X n 1 = 10. That is, the system begins 10 units from its rest position, with zero velocity. For k/m =.5, results from two initial conditions are shown: X n = X n 1 = 10 and X n = X n 1 = 5. The result in

18 CHAPTER 6. INTRODUCTION TO DYNAMICAL SYSTEMS 18 each case is oscillation between a minimum and maximum displacement from the rest position. As k/m increases, so does the frequency: the stiffer the spring, the higher the frequency (if mass is kept constant); the smaller the mass, the higher the frequency (if stiffness is held constant). In fact, as we will derive in a later section, the angular frequency of the system in radians per sample (ω) is equal to k/m k/m =.1 k/m = X 0 X n n k/m = =.5.5 k/m = X 0 X n n Figure 6.11: Time function of undamped spring, varying values of k/m. Bottom two figures differ in initial conditions. Forty time samples are shown. When we change the initial conditions, the scale of the oscillations changes, but nothing else does. That is, for X n = X n 1 = 10 we see oscillation between 10 and -10, while for X n = X n 1 = 5, we see oscillation between 5 and -5. This system has no attractors. The initial conditions will effect the resulting state forever. In fact, we could say that this system codes its own history. If we perturb the system (by adding a push or a pull), the system will then oscillate between a new set of displacement values, and will not recover from

19 CHAPTER 6. INTRODUCTION TO DYNAMICAL SYSTEMS 19 the perturbation to return to its previous oscillatory pattern. non-attractor pattern is sometimes called neutrally stable. This kind of Second order difference equation damped (6.28) F = bẋ (6.29) a = bẋ k m X (6.30) X n 2X n 1 + X n 2 = b(x n X n 1 ) k m X (6.31) X n+1 2X n + X n 1 = b(x n+1 X n ) k m X n (6.32) X n+1 = (b + 2 k)x n X n b Second order differential equation Ẍ = a Ẍ = k m (X X 0) x = X X 0 ẍ = k m (x) x = cos(ωt) d 2 dt cos(ωt) = k m cos(ωt) ω d dt sin(ωt) = k m cos(ωt) ω 2 cos(ωt) = k m cos(ωt) ω 2 = k m ω = k m