Chaos and Dynamical Systems
|
|
- Darleen Spencer
- 5 years ago
- Views:
Transcription
1 Chaos and Dynamical Systems y Megan Richards Astract: In this paper, we will discuss the notion of chaos. We will start y introducing certain mathematical concepts needed in the understanding of chaos, such as iterates of functions and stale and unstale fixed points. We will discuss the graphical analysis of an orit and show a couple examples of an orit diagram. We will riefly show how the Lyapunov exponent can e found and used to determine whether a system is chaotic and if so where it is chaotic. Then we will egin discussing the aker map, which this paper focuses on the most. In particular, we will discuss the values at which the aker map exhiits chaotic ehavior and why it is chaotic at those values. This has een focused on ecause, while it can e easy for someone to pick up a ook and read aout chaos, it may not e so easy to research the aker map and find the values at which it is chaotic and why. Finally, we will look at an example of a known chaotic system the doule-pendulum. We will use a computer program to model the ehavior of the pendulum and oserve how it is sensitive to initial conditions. In this paper, we will assume that the reader has a mathematical ackground up to the calculus level and can understand certain proof techniques, such as a proof y induction or y contradiction. Introduction Chaos is a word we all know usually meaning a lack of order or predictaility. Most people might associate the utterfly effect with the notion of chaos. This utterfly effect descries how a utterfly flapping its wings in some part of the world might e largely responsile for a huge storm in another part of the world several weeks later, ecause of the weather s extreme sensitivity to initial conditions. The weather is indeed one example of a chaotic system. A weather reporter might make nearly identical forecasts for a city in Europe as for a city in the United States, meaning that the initial conditions of the system are quite similar, while y the next week the weather patterns are completely different. Because of this sensitive dependence on initial conditions, weather forecasters have a very difficult time predicting the weather far in advance. Another example of chaos is the doule pendulum, and later we will use a computer program to model the motion of a doule pendulum. We will place the doule pendulum at a certain initial point and again 1
2 at another initial point just slightly different than the first to oserve how the resulting motion will e very different. What makes certain systems chaotic? Chaos descries the ehavior of a system that is highly sensitive to initial conditions. Chaotic systems are not predictale over a long period of time and are typically associated with fractal structures. Understanding chaos will help us understand why some systems exhiit seemingly erratic and random ehavior yet are still deterministic systems (that is, systems determined y their initial conditions). It will show us why chaos is not complete disorder, ut rather is associated with a geometrical structure. Understanding chaos is somewhat complicated, so we first need to e ale to understand certain mathematical concepts and the instaility and staility of non-chaotic systems. Hence, this paper will egin y explaining iterates of functions, unstale and stale fixed points, and some examples of maps and orit diagrams. We will then move on to discuss the aker map and at which points the aker map exhiits chaotic ehavior. Finally we will show the motion of a doule-pendulum using a computer program. Maps First we will look at the iterates of functions. Suppose that f is a function and x 0 is an element of the domain of f. The iterates of x 0 will consist of x 0, f(x 0 ), f(f(x 0 )), f(f(f(x 0 ))),... These iterates together are called the orit of x 0. Note we can write x n+1 = f(x n ), which we often refer to as a map, so that the orit of x ecomes x 0, x 1, x, x,... As an example, consider the map x n+1 = cos(x n ), and let x 0 = 0.5. To find the iterates of x 0, we enter 0.5 into a calculator. Then we enter cos(ans) repeatedly. The first iterate of x 0 for f is x 1 = f(x 0 ) = The second iterate is x = f(f(x 0 )) = , and so on. The orit of x 0 will start looking like 0.5, , , , ,... If we press cos(ans) enough times, we end up repeatedly getting the calculator to spit out the numer This numer descries the value of x for which x = cos(x). This value of x is called a fixed point of f. In general, if we have a function f and a numer c in the domain of f, c is a fixed point of f if
3 f(c) = c. We can visualize this graphically y plotting the line y = x with the graph of f and finding where they intersect. Fixed points can e either stale or unstale. An unstale fixed point is a fixed point for which iterates that start neary the fixed point will move away from the point. A stale fixed point is a fixed point for which iterates neary the fixed point approach the point. In our previous example, since our iterates of 0.5 approached the fixed point , we have reason to assume that this fixed point is stale (of course we would need to make sure that our fixed point is approached y iterates on oth sides of it). We can check this assumption using the following procedure. Assuming that f is a differentiale function at a fixed point c, we can use the following to determine whether or not c is stale or unstale: If df dx x=c < 1, then c is stale, whereas if df dx x=c > 1, then c is unstale. If df dx x=c = 1, then c can e either stale, unstale, or neither. Note, in our example aove, df = sin(x) x= = < 1, which proves dx x=c our earlier assumption that x = is a stale fixed point of f(x) = cos(x). Figure 1: Graphical analysis of the orit of.5 for f(x) = cos(x), in the neighorhood of a stale fixed point Figure : Graphical analysis of the orit of.9 for f(x) = x, in the neighorhood of an unstale fixed point We can easily visualize the orit of a point x 0 for a function f graphically. First we plot the function f as well as the line given y y = x. We start at the point (x 0, x 0 ), which is along the line y = x, and draw a line vertically to the function f, stopping at the point (x 0, f(x 0 )). Then we draw a horizontal line to the line y = x, where we stop at the point (x 1, x 1 ), where x 1 = f(x 0 ) is the first iterate of x 0. This process is
4 then repeated. The graphical interpretation for the iterates of x 0 =.5 for f(x) = cos(x) is given in Figure 1. (Note the Matla codes used to produce each figure in this paper are given in the Appendix.) Rememer that these iterates converged to the stale fixed point x = We can also see this graphically in the figure. Note that for unstale fixed points, iterates will diverge from the fixed point, which can also e seen graphically. Consider the graphical analysis of the orit of x 0 =.9 for the function f(x) = x shown in Figure. First, note that 0 and 1 are fixed points for f since f(1) = 1 = 1 and f(0) = 0 = 0. Note 1 is unstale since df dx x=1 = d dx (x ) x=1 = x x=1 = > 1, while 0 is stale since df dx x=0 = x x=0 = 0 < 1. Thus, iterates of x 0 =.9 will move away from the unstale fixed point at 1 and converge to the stale fixed point at 0. Now consider iterates of x 0 = 1 for the function f(x) = 4x 4x. The orit of 1 will egin looking like {0., , , ,...}. We might e expecting these iterates to converge to a fixed point, as they did with f(x) = cos(x). However, this turns out not to e the case. It turns out that the iterates are numers in the interval (0, 1) that have no predictale pattern. This is an example of a function that exhiits chaos. Suppose f is a function and x 0 is in the domain of f. Then if x n = x 0 and if x 0, x 1, x,..., x n 1 are distinct, we say that x 0 has period n. Note if x 0 has period n, then the orit of x 0 is given y {x 0, x 1, x,..., x n 1 }. This orit is called a periodic orit or an n-cycle. Note that if a point has period 1, it is a fixed point. A periodic point is a value x for which some iterate is again x. For instance, if we consider the function f(x) = x 1, the orit of x 0 = 1 is given y 1, 0, 1, 0, 1,.... Note f(f(x 0 )) = 1, and so 1 is a periodic point. In fact, { 1, 0} forms a -cycle since f( 1) = 0 and f(f( 1)) = f(0) = 1. Now consider the map x n+1 = r cos(x n ), where r is a real numer. To see how this map ehaves for all values of r at once, we plot what is known as an orit diagram, shown in Figure. The darkened patches represent the areas where the iterates do not converge to a fixed point, ut spread out from (r, r). It is important to note that this spreading out does not indicate chaos rather it is the sensitivity to initial conditions that determines whether or not a system is chaotic. However, in this case sensitivity to initial conditions does occur in these darkened areas, and so they show the values of r for which the map 4
5 Figure : Orit diagram for x n+1 = r cos x n x n+1 = r cos(x n ) is exhiiting chaotic ehavior. A function that is parametrized, such as f(x) = r cos(x), is said to have a ifurcation at a point r 0 if the type (stale or unstale) or numer of periodic points change at that point. This point r 0 is then known as a ifurcation point for the function f. Thus, as can e seen from the plot in Figure, a ifurcation exists around r 0 = 1. since at that point our 1-cycle changes to a -cycle. This is known as a period-douling ifurcation. In general, a period douling ifurcation is a ifurcation where an n-cycle gives rise to a n-cycle. Period-douling is a common route to chaos. The Lyapunov Exponent The Lyapunov exponent determines whether or not a system is chaotic. The Lyapunov exponent for the orit of a function f starting at x 0 is given y { n 1 1 } λ = lim ln f (x i ). n n i=0 If λ > 0 we have chaotic dynamics. Otherwise, we have a non-chaotic situation. Calculating the Lyapunov exponent numerically is fairly simple. In general, the Lyapunov exponent is hard to find analytically, ut we will calculate it for a simple example. Consider the tent map given y f(x) = { rx 0 x 0.5 r rx 0.5 x 1 5
6 for 0 r and 0 x 1. The derivative of f is then f (x) = { r 0 x 0.5 r 0.5 x 1 The Lyapunov exponent is given y { n 1 1 λ = lim n n i=0 } { n 1 1 ln f (x i ) = lim n n i=0 } { } 1 ln ±r = lim ln r n = ln r. n n Thus, since λ > 0 when r > 1, and λ 0 when 0 < r 1, we get chaos when r > 1 and order when 0 < r 1. Figure 4 shows a plot of the Lyapunov exponent as a function of r for the tent map, while Figure 5 shows the orit diagram for the tent map. Figure 4: Liapunov exponent for the tent map Figure 5: Orit diagram for the tent map Note that from the plot of the Lyapunov exponent, we can see that λ > 0 when r > 1, as we found aove. As can e seen in Figure 5, chaotic ehavior starts occurring at r = 1, which agrees with the plot in Figure 4. We also note from the diagram that as r increases the chaotic ehavior for the iterates of the corresponding tent functions is also increasing. 6
7 The Baker Map One Dimensional Baker Function: Let us first introduce a version of the one-dimensional aker function B 1, given y { x 0 x 0.5 B 1 = x < x 1 First we will show that B 1 is extremely sensitive to initial conditions. Let us find some iterates of = and to see if they are much different. iterates Note that the tenth iterates of and are, respectively, and 0.008, which differ y over 0.6. Thus, even though we started with two numers fairly close together, the tenth iterates of those numers are already quite far apart. Hence, we cannot relate the higher iterates of to the corresponding iterates of. If we were to experiment further, we would find that choosing a different pair of numers close together will exhiit similar ehavior. Two Dimensional Baker Map: Now, consider the two-dimensional aker map, given y { (cx n, y n ) 0 y n 0.5 B(x n, y n ) = (1 + c(x n 1), y n 1) 0.5 < y n 1 where 0 < c < 1 First, note that, setting B(x 0, y 0 ) = (x 0, y 0 ), we otain two fixed points, namely (0, 0) and (1, 1). Thus, the points (0, 0) and (1, 1) remain unchanged under the mapping of B. Other points move differently depending on if they are aove or elow the line y n = 1. 7
8 If we map a fine grid of the unit square through B, we otain the plot shown in Figure 6. In other words, Figure 6 shows the first iteration of all points taken from the unit square (shown in lue). Consequently, putting these first-iteration points through B again gives the second iteration of these points, which are plotted in Figure 7. Similarly, Figure 8 shows the third iteration and Figure 9 shows the fourth iteration. Figure 6: First iteration of points y the aker map Figure 7: Second iteration of points y the aker map Figure 8: Third iteration of points y the aker map Figure 9: Fourth iteration of points y the aker map Now we know how the aker map acts on the unit square. So next we are interested in finding out whether or not the aker map exhiits chaotic ehavior and, if so, the points at which chaos occurs. In the next couple of pages, we will look solely at the y-component of the aker map and ignore the x-component, reducing the aker map to the one dimensional aker map. We will show and go into detail 8
9 aout what happens if y 0 is rational and conclude that in this case the iterations of y 0 repeat. Then we will show that if y 0 is irrational, then the aker map is chaotic. Finally, we will show that chaos in the aker map actually does not depend on the initial x-value ut rather only on the initial y-value. So, now we will prove that if y 0 is a rational numer, the iterations of y 0 will eventually repeat, which will result in a non-chaotic situation. More specifically, if y 0 = a, where a, Z (so y 0 is rational), then y 0 will have at most iterations efore an iteration is repeated, or, in other words, y 0 or an iteration of y 0 will have at most period. Choose y 0 = a, where a, Z and 0 < y 0 1. Then a. If y 0 0.5, then y 1 = y 0 = a. If y 0 > 0.5, then y 1 = y 0 1 = a 1 = a. Note in oth cases the numerator is an integer and the denominator is. I can repeat this process and the numerator will still e an integer, while the denominator will still e the integer. Since the numerator is an integer and is less than or equal to, we only have choices for the numerator. Thus, we will have at most iterations of y 0 efore the iterations egin to repeat. The first iteration that will e repeated will e either y 0 (if the denominator of y 0 is not divisile y ) or an iteration of y 0 (if the denominator is divisile y ). The previous statement was a it of a claim to make. Let us prove this. Our method of proof will e to introduce a function γ which mimics the aker function when y 0 is a rational numer etween 0 and 1 whose denominator is a fixed numer that is not divisile y. We will show that γ is injective and hence show that if is not divisile y, then y 0 will e repeated in a later iterate. Let γ (a) = B( a ), where 0 < a < and is an integer not divisile y. Note we can use our γ function to represent the numerator values in our iterations of y 0 = a when a and are integers and is not divisile y. Since in each iteration the denominator does not change, we can let y 0 = a0, y 1 = a1, and so on. Then we can represent our iterations {y 0, y 1, y,...} as { a0, a1, a,..., }. Thus, y 0 = a 0, y 1 = a 1, and so on. So, for instance, γ (a 0 ) = B( a0 ) = B(y 0) = y 1 = a 1. Similarly, we have that γ (a 1 ) = B( a1 ) = y = a, and so on. 9
10 Claim 1: The function γ is injective. Proof. Let a 1, a Z. Suppose γ (a 1 ) = γ (a ). We consider the three possile cases and show in each case a 1 = a. Case 1: a 1, a a1. Note, a Simplifying, we get a 1 = a. Thus, γ is injective. Case : a 1, a > a1. Note, a 1. Since γ (a 1 ) = γ (a ), we get B( a1 a. Simplifying, we get a 1 = a. Thus, γ is injective. > 1. Since γ (a 1 ) = γ (a ), we get B( a1 a a1 ) = B( ). So = a. a a1 ) = B( ). So = Case : a 1 and a > a1. Note 1 and a > 1. Since γ (a 1 ) = γ (a ), we get B( a1 a ) = B( ). So a1 = a. So (a a 1 ) =, a contradiction since is not divisile y. Hence, we have proven that γ is injective. Claim : If y 0 = a when a and are integers and is not divisile y, then y 0 will e repeated in a later iterate. Proof. Suppose y 0 = a0 where is not divisile y, and suppose, for the sake of contradiction, that y 0 does not ecome repeated later, ut that another iterate y i is the first iterate repeated later, where i 1. Let y k = y i, where k > i. Then γ (a i 1 ) = a i = a k = γ(a k 1 ). Since γ is injective, a i 1 = a k 1, a contradiction since a i was the first iterate repeated. Claim : Let y 0 e in lowest terms. If y 0 = a, where a, Z and is divisile y, then y 0 will not e repeated ut rather an iterate of y 0 will e repeated. Proof. Suppose y 0 = a, where a, Z and is divisile y. Then we can write = m for some m Z. Thus, y 0 = a m. If y 0 0.5, then y 1 = y 0 = a m. If y 0 > 0.5, then y 1 = y 0 1 = a m 1 = a m m. In oth cases, the denominator reduces to m. If m is again divisile y, we repeat the process until we have an iterate where the denominator is no longer divisile y. Let us call this iterate y i. Then, since the denominator of y i is no longer divisile y and we still have an integer in the numerator, y our previous proof, y i is the first iterate that will e repeated later. Hence, our inital y 0 will not e repeated. Now let us look at some examples. 10
11 Suppose y 0 = 1 9. We expect, since 9 is not divisile y, that y 0 should e repeated later in at most nine iterations. As can e seen in the iterates of y 0 shown elow, we are correct. iterates y 1 y y y 4 y 5 y Note that the 6 th iteration returns y 0 and so y 0 = 1 9 has period 6. So here we have a non-chaotic situation. See Figure 10 for a plot of the iterations of (x 0, y 0 ) = ( 1, 1 9 ). Figure 10: Iterations of (x 0, y 0) = ( 1, 1 ): a non-chaotic orit 9 Now suppose y 0 = 10. In this case the denominator is divisile y, and this causes the fraction to e reduced in the second iterate. The initial value y 0 is therefore never repeated, ut the first iterate of y 0, namely y 1, is repeated. The iterates of y 0 are shown elow. iterates y 1 y y y 4 y Note y 5 = y 1, and so y 1 has period 4. The aker map thus repeatedly spits out the values of y 1 through 11
12 y 4 for the y-values and so in this case exhiits non-chaotic ehavior. We have shown that we will get a non-chaotic situation if y 0 is rational and have shown a couple examples of this. So what happens if y 0 is irrational? Let us look at when y 0 =. The tale elow shows the iterations of this value of y 0 up to 4 iterations. iterates y 1 y y y 4 y 5 y 6 y 7 y iterates y 9 y 10 y 11 y 1 y 1 y 14 y 15 y iterates y 17 y 18 y 19 y 0 y 1 y y y To see how the aker function is sensitive to initial conditions, we will also plot a tale of the iterations of y 0 = and oserve how they differ from the iterations of y 0 =. iterates y 1 y y y 4 y 5 y 6 y 7 y iterates y 9 y 10 y 11 y 1 y 1 y 14 y 15 y iterates y 17 y 18 y 19 y 0 y 1 y y y Comparing the iterations of the two initial points, we see that the iterations stay somewhat close together at the eginning, ut then diverge. By the 17 th iteration they differ y aout Note we also see no repetitions so far with the iterations and the ehavior of the iterations seem somewhat random, giving us reason to elieve that we have found a point at which the aker function exhiits chaotic ehavior. However, it is very hard to make this claim from just 4 iterates. Thus, we use a computer program to make a plot showing many more iterates. See Figure 11 for a plot of iterations of (x 0, y 0 ) = ( 1, ) up to n = 00. 1
13 Figure 11: Iterations of (x 0, y 0) = ( 1, ) The plot shows very irregular ehavior. Along with the data earlier showing how y 0 was sensitive to initial conditions, the plot certainly makes it seem like we have found a point at which the aker map is chaotic. However, we have not proven this we would need to calculate the Lyapunov exponent to do so. Although we will not do this here, numerically calcuating the Lyapunov exponent would prove that the aker map is chaotic at this point. Will all irrational numers y 0 cause the aker map to e chaotic? To answer this question, first we show that we can write any iterate of an irrational y 0 in a particular form and from then show that iterates of y 0 can not repeat. Claim 4: Suppose y 0 is an irrational numer etween 0 and 1, so that y 0 = r, where r is an irrational numer and is an integer and 0 < r < 1. Then the kth iterate of our irrational numer can e written as y k = k r m, where m Z. Proof. We will proceed y induction. For the ase case, let k = 1. Note 1
14 y 1 = B(y 0 ) = B( r ) = { r 0 < r 1 r r, where oth 1 < r < 1 m = 0 in the first case and m = 1 in the second case). Now suppose y k = k r m. Note that r and are in the form y 1 = 1 r m (note y k+1 = B(y k ) = B( k r m ) = { k+1 r m y k 1 k+1 r (m+1) y k > 1 In either case, y k+1 can e written in the form k+1 r m for some m Z. Thus, we have shown that the k th iterate of our irrational numer r can e written in the form y k = k r m, where m Z.. Claim 5: The iterates of an irrational numer y 0 do not repeat. Proof. Suppose, for the sake of contradiction, that two iterates of our irrational numer y 0 = r are equal. That is, suppose y n and y k are iterates of y 0, where n < k, such that y n = y k. By our previous proof, y k = k r m 1 and y n = n r m, where m 1, m Z. Since y n = y k, we have k r m 1 = n r m. Simplifying, we get r( k n ) = (m 1 m ). Thus, r = (m1 m) k n. Note that k n 0 (since n < k). Also, note that since, m 1, m Z and the integers are closed with respect to multiplication and addition, (m 1 m ) Z. And note that since k, n Z, we have that k n Z. Thus, r is rational, a contradiction since r is irrational. Hence, we have shown that if y 0 is irrational, our iterations will not repeat. Rather, as shown y an earlier example, they will jump ack and forth in a seemingly irregular fashion, indicating that we have chaotic ehavior on our hands. Next we want to show that the values for x 0 do not affect whether or not the aker map exhiits chaotic ehavior, nor do the values of c. It is solely the values we choose for y 0 that determine whether or not we get a chaotic situation. 14
15 Suppose y 0 is a rational numer, and either y 0 or an iteration of y 0 has a certain period. Let y k e a repeated iteration. Claim 6: If y 0 1, we can write the kth iteration of x 0 as x k = c k x 0 + k 1 i=0 α ic i. Proof. We will proceed y induction. If k = 1, then x k = x 1 = cx 0, which is in the desired form. Now, suppose x k = c k x 0 + k 1 i=0 α ic i. Note { cx k y k 1 x k+1 = B(x k ) = 1 + c(x k 1) y k > 1 So x k+1 can e written in the desired form. = { c k+1 x 0 + k 1 i=0 α ic i+1 y k 1 1 c + c k+1 x 0 + k 1 i=0 α ic i+1 y k > 1 Claim 7: When k is large, x k x k. Proof. Case 1: y 0 1 Since y k is a repeated value for y 0 or an iteration of y 0, the aker map will do the same operation on x k+k as it will on x k. Note k 1 k 1 x k = c k x k + α i c i = c k (c k x 0 + α i c i ) + i=0 k 1 = c k x 0 + i=0 k 1 = c k x 0 + i=k k 1 α i c i+k + i=0 k 1 α i c i + i=0 i=0 k α i c i i=0 k 1 α i c i = c k x 0 + j=k k 1 α i c i = c k x 0 + i=k k 1 α j k c j + i=0 α i c i + x k c k x 0 Thus, x k = c k x 0 + k 1 i=k α ic i + x k c k x 0. Note when k is very large, all three terms except x k will e very small, and so x k x k. α i c i Case : y 0 > 1 Now we simply proceed until some i th iteration at which y i 1. We then treat y i as y 0 and x i and x 0 and proceed as we have in Case 1 to get the same result. Let s look at a couple of examples. First suppose y 0 = 1. Let us choose any c such that 0 < c < 1 and any x 0 such that 0 < x 0 < 1. Note y 1 =, y = 1, and so on. Note if i is even, then y i < 1, while if i is 15
16 odd, then y i > 1. Thus, we get x 1 = cx 0 x = 1 c + cx 1 = 1 c + c x 0 x = cx = c c + c x 0 x 4 = 1 c + cx = 1 c + c c + c 4 x 0 x 5 = c c + c c 4 + c 5 x 0 x 6 = 1 c + c c + c 4 c 4 + c 6 x 0 x 7 = c c + c c 4 + c 5 c 6 + c 7 x 0 x 8 = 1 c + c c + c 4 c 5 + c 6 c 7 + c 8 x 0 Note that, since 0 < c < 1, large powers of c will e very small. Thus, for large k, x k x k and x k+1 x k 1. To see this clearer, suppose c = 1 and x 0 =. Then x 1 = x = x = x 4 = x 5 = x 6 = x 7 = x 8 = x 9 = x 10 = x 0 = 0.75 x 1 = 0.5 x = 0.75 x = 0.5 Note we will still get the same convergence if we change x 0, ut leave y 0 and c the same. For instance, if instead we let x 0 = 1, we get the following iterations: 16
17 x 1 = x = x = x 4 = x 5 = x 6 = x 7 = x 8 = x 9 = x 10 = x = 0.75 x = 0.5 If we change c, ut leave y 0 = 1, then the even and odd iterations will converge to two different numers (not 0.5 and 0.75 ut something else). Meanwhile, if we change y 0 this will change how many distinct numers in our (approximately) repeating iterations we have. For instance, if y 0 = 1 5, say, then we will have five distinct numers that keep getting repeated in our iterations. So for large k, y k y k+5. So we will have, for instance, y 0 y 5 y 40 and y 1 y 6 y 41, and so on. Our final conclusion is that x 0 and c do not have any influence on whether or not we get chaotic ehavior. It is solely our choice for y 0 that determines this. Now suppose we create a plot showing the locations of many iterations of a single point (x 0, y 0 ), where y 0 is irrational, in the unit square. What will this look like? Consider the iterations of the point (x 0, y 0 ) = ( 1, ). Figures 1-17 show plots of 10, 50, 100, 500, 1000, and iterations, respectively. The initial four points are also shown and laeled in each plot. 17
18 Figure 1: 10 iterations of (x 0, y 0) = ( 1, ) Figure 1: 50 iterations of (x0, y0) = ( 1, ) Figure 14: 100 iterations of (x 0, y 0) = ( 1, ) Figure 15: 500 iterations of (x0, y0) = ( 1, ) Figure 16: 1000 iterations of (x 0, y 0) = ( 1, ) Figure 17: iterations of (x 0, y 0) = ( 1, ) 18
19 These plots emphasize the fact that the iterations of our point (x 0, y 0 ) = ( 1, ) really do spread out over many areas of the unit square and do not settle down. But, we do see that there are wide gaps where the later iterations of the points do not touch. In fact, note that our plot of iterations in Figure 17 is strikingly similar to the plot in Figure 9 of the fourth iteration of a fine grid of points taken from the unit square. Both are ecause of the way the aker map acts on a point. The plot in Figure 17 resemles what is known as a fractal pattern. This is an important aspect of chaotic systems. We can actually find geometrical structures associated with chaos, which tells us that chaos is not total disorder. If it were, the iterations of the point (x 0, y 0 ) = ( 1, ) would fill up the whole unit square instead of creating a pattern. We will not go into detail aout fractals, ut they are important in understanding chaos, and whole ooks can e devoted just to the study of fractals or the relationship etween fractals and chaos. The Doule Pendulum Now we will show how a doule pendulum exhiits chaos. A doule pendulum consists of two pendulums, one hanging from a fixed point and another pendulum hanging from the first. We will consider a doule pendulum in which the masses and lengths of the two pendulums are equal. We also do not consider friction in the system. To talk aout the position of the doule pendulum, we will say that the first pendulum is at an angle θ 1 from the vertical and the second pendulum is at an angle θ from the vertical, as shown in the drawing in Figure 17. Note that the red circle in the drawing indicates the fixed point at which the first pendulum is attached. 19
20 Figure 18: Doule Pendulum We will position two doule pendulums at nearly identical locations and oserve how their motion differs over just a short period of time. We position our first doule pendulum with its first pendulum at θ 1 = π 4 radians, and its second pendulum at θ = π radians. Our second doule pendulum is placed with its first pendulum at θ 1 =.14 4 radians and its second pendulum at θ =.14 radians; that is, it has een placed in nearly the same initial position as that of the first doule pendulum except that we have approximated π as.14. Figure 19 shows how the motion of each doule pendulum varies over a period of five seconds. Our first doule pendulum is shown in the right side of each figure, while the second doule pendulum (with π.14) is shown on the left side. As can e seen in the figure, the pendulums are still in nearly the same positions after three seconds. By four seconds, they have started to change slightly, and y five seconds they are in completely different locations, and their locations thereafter will not e related to one another. 0
21 Figure 19: Motion of two doule pendulums over a period of 5 seconds. Conclusion Chaos is everywhere in the world. This paper is meant to e a rief introduction to chaos and dynamical systems in hopes that the reader can egin to understand what chaos is and how it can e studied. We started y looking at the iterates of functions, stale and unstale fixed points, the graphical analysis of an orit, and some examples of maps and orit diagrams. Then we discussed the Lyapunov exponent and its importance in determining whether or not a system is chaotic. Next we went into a discussion of the aker map, including how it acts on the unit square and at what points it exhiits chaotic ehavior. Finally, we riefly discussed an example of a more well-known ut chaotic system that we can all visualize the doule pendulum and showed an example of how it is sensitive to initial conditions. 1
22 Resources Gulick, Denny. Encounters with Chaos. New York: McGraw-Hill, 199. Tél, Tamás and Márton Gruiz. Chaotic Dynamics. New York: Camridge University Press, 006.
23 Appendix This appendix contains the Matla codes that were written to produce the figures in this paper.
24 4
25 5
26 The following function and code were used to produce Figures 6-9: 6
27 Note Figure 11 was made simply y changing the first line x0 = sym(1/); y0 = 1/9; in the previous code to x0 = sym(1/); y0 = sqrt()/; 7
28 8
29 9
Polynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson, you Learn the terminology associated with polynomials Use the finite differences method to determine the degree of a polynomial
More informationSection 8.5. z(t) = be ix(t). (8.5.1) Figure A pendulum. ż = ibẋe ix (8.5.2) (8.5.3) = ( bẋ 2 cos(x) bẍ sin(x)) + i( bẋ 2 sin(x) + bẍ cos(x)).
Difference Equations to Differential Equations Section 8.5 Applications: Pendulums Mass-Spring Systems In this section we will investigate two applications of our work in Section 8.4. First, we will consider
More information#A50 INTEGERS 14 (2014) ON RATS SEQUENCES IN GENERAL BASES
#A50 INTEGERS 14 (014) ON RATS SEQUENCES IN GENERAL BASES Johann Thiel Dept. of Mathematics, New York City College of Technology, Brooklyn, New York jthiel@citytech.cuny.edu Received: 6/11/13, Revised:
More informationResearch Article Coexistence of the Bandcount-Adding and Bandcount-Increment Scenarios
Discrete Dynamics in Nature and Society Volume 211, Article ID 681565, 3 pages doi:1.1155/211/681565 Research Article Coexistence of the Bandcount-Adding and Bandcount-Increment Scenarios Viktor Avrutin,
More informationChaos in the Dynamics of the Family of Mappings f c (x) = x 2 x + c
IOSR Journal of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 319-765X. Volume 10, Issue 4 Ver. IV (Jul-Aug. 014), PP 108-116 Chaos in the Dynamics of the Family of Mappings f c (x) = x x + c Mr. Kulkarni
More informationModule 9: Further Numbers and Equations. Numbers and Indices. The aim of this lesson is to enable you to: work with rational and irrational numbers
Module 9: Further Numers and Equations Lesson Aims The aim of this lesson is to enale you to: wor with rational and irrational numers wor with surds to rationalise the denominator when calculating interest,
More information1Number ONLINE PAGE PROOFS. systems: real and complex. 1.1 Kick off with CAS
1Numer systems: real and complex 1.1 Kick off with CAS 1. Review of set notation 1.3 Properties of surds 1. The set of complex numers 1.5 Multiplication and division of complex numers 1.6 Representing
More informationLecture 6 January 15, 2014
Advanced Graph Algorithms Jan-Apr 2014 Lecture 6 January 15, 2014 Lecturer: Saket Sourah Scrie: Prafullkumar P Tale 1 Overview In the last lecture we defined simple tree decomposition and stated that for
More information2 Problem Set 2 Graphical Analysis
2 PROBLEM SET 2 GRAPHICAL ANALYSIS 2 Problem Set 2 Graphical Analysis 1. Use graphical analysis to describe all orbits of the functions below. Also draw their phase portraits. (a) F(x) = 2x There is only
More informationTHE INFLATION-RESTRICTION SEQUENCE : AN INTRODUCTION TO SPECTRAL SEQUENCES
THE INFLATION-RESTRICTION SEQUENCE : AN INTRODUCTION TO SPECTRAL SEQUENCES TOM WESTON. Example We egin with aelian groups for every p, q and maps : + (here, as in all of homological algera, all maps are
More informationThe function graphed below is continuous everywhere. The function graphed below is NOT continuous everywhere, it is discontinuous at x 2 and
Section 1.4 Continuity A function is a continuous at a point if its graph has no gaps, holes, breaks or jumps at that point. If a function is not continuous at a point, then we say it is discontinuous
More informationChaos and Liapunov exponents
PHYS347 INTRODUCTION TO NONLINEAR PHYSICS - 2/22 Chaos and Liapunov exponents Definition of chaos In the lectures we followed Strogatz and defined chaos as aperiodic long-term behaviour in a deterministic
More information... it may happen that small differences in the initial conditions produce very great ones in the final phenomena. Henri Poincaré
Chapter 2 Dynamical Systems... it may happen that small differences in the initial conditions produce very great ones in the final phenomena. Henri Poincaré One of the exciting new fields to arise out
More informationDynamical Systems and Chaos Part I: Theoretical Techniques. Lecture 4: Discrete systems + Chaos. Ilya Potapov Mathematics Department, TUT Room TD325
Dynamical Systems and Chaos Part I: Theoretical Techniques Lecture 4: Discrete systems + Chaos Ilya Potapov Mathematics Department, TUT Room TD325 Discrete maps x n+1 = f(x n ) Discrete time steps. x 0
More informationNon-Linear Regression Samuel L. Baker
NON-LINEAR REGRESSION 1 Non-Linear Regression 2006-2008 Samuel L. Baker The linear least squares method that you have een using fits a straight line or a flat plane to a unch of data points. Sometimes
More informationChapter 7. 1 a The length is a function of time, so we are looking for the value of the function when t = 2:
Practice questions Solution Paper type a The length is a function of time, so we are looking for the value of the function when t = : L( ) = 0 + cos ( ) = 0 + cos ( ) = 0 + = cm We are looking for the
More informationPHY411 Lecture notes Part 5
PHY411 Lecture notes Part 5 Alice Quillen January 27, 2016 Contents 0.1 Introduction.................................... 1 1 Symbolic Dynamics 2 1.1 The Shift map.................................. 3 1.2
More informationSolving Systems of Linear Equations Symbolically
" Solving Systems of Linear Equations Symolically Every day of the year, thousands of airline flights crisscross the United States to connect large and small cities. Each flight follows a plan filed with
More informationExploring Lucas s Theorem. Abstract: Lucas s Theorem is used to express the remainder of the binomial coefficient of any two
Delia Ierugan Exploring Lucas s Theorem Astract: Lucas s Theorem is used to express the remainder of the inomial coefficient of any two integers m and n when divided y any prime integer p. The remainder
More informationMATH 415, WEEKS 14 & 15: 1 Recurrence Relations / Difference Equations
MATH 415, WEEKS 14 & 15: Recurrence Relations / Difference Equations 1 Recurrence Relations / Difference Equations In many applications, the systems are updated in discrete jumps rather than continuous
More informationMCS 115 Exam 2 Solutions Apr 26, 2018
MCS 11 Exam Solutions Apr 6, 018 1 (10 pts) Suppose you have an infinitely large arrel and a pile of infinitely many ping-pong alls, laeled with the positive integers 1,,3,, much like in the ping-pong
More informationCS 4120 Lecture 3 Automating lexical analysis 29 August 2011 Lecturer: Andrew Myers. 1 DFAs
CS 42 Lecture 3 Automating lexical analysis 29 August 2 Lecturer: Andrew Myers A lexer generator converts a lexical specification consisting of a list of regular expressions and corresponding actions into
More informationAt first numbers were used only for counting, and 1, 2, 3,... were all that was needed. These are called positive integers.
1 Numers One thread in the history of mathematics has een the extension of what is meant y a numer. This has led to the invention of new symols and techniques of calculation. When you have completed this
More informationRATIONAL EXPECTATIONS AND THE COURNOT-THEOCHARIS PROBLEM
RATIONAL EXPECTATIONS AND THE COURNOT-THEOCHARIS PROBLEM TÖNU PUU Received 18 April 006; Accepted 1 May 006 In dynamic models in economics, often rational expectations are assumed. These are meant to show
More informationThe Mean Version One way to write the One True Regression Line is: Equation 1 - The One True Line
Chapter 27: Inferences for Regression And so, there is one more thing which might vary one more thing aout which we might want to make some inference: the slope of the least squares regression line. The
More informationMAT335H1F Lec0101 Burbulla
Fall 2012 4.1 Graphical Analysis 4.2 Orbit Analysis Functional Iteration If F : R R, then we shall write F 2 (x) = (F F )(x) = F (F (x)) F 3 (x) = (F F 2 )(x) = F (F 2 (x)) = F (F (F (x))) F n (x) = (F
More informationSolutions to Exam 2, Math 10560
Solutions to Exam, Math 6. Which of the following expressions gives the partial fraction decomposition of the function x + x + f(x = (x (x (x +? Solution: Notice that (x is not an irreducile factor. If
More informationfunction independent dependent domain range graph of the function The Vertical Line Test
Functions A quantity y is a function of another quantity x if there is some rule (an algebraic equation, a graph, a table, or as an English description) by which a unique value is assigned to y by a corresponding
More informationUpper Bounds for Stern s Diatomic Sequence and Related Sequences
Upper Bounds for Stern s Diatomic Sequence and Related Sequences Colin Defant Department of Mathematics University of Florida, U.S.A. cdefant@ufl.edu Sumitted: Jun 18, 01; Accepted: Oct, 016; Pulished:
More informationCalculus 221 worksheet
Calculus 221 worksheet Graphing A function has a global maximum at some a in its domain if f(x) f(a) for all other x in the domain of f. Global maxima are sometimes also called absolute maxima. A function
More informationIdentifying the Graphs of Polynomial Functions
Identifying the Graphs of Polynomial Functions Many of the functions on the Math IIC are polynomial functions. Although they can be difficult to sketch and identify, there are a few tricks to make it easier.
More informationLecture 1: Period Three Implies Chaos
Math 7h Professor: Padraic Bartlett Lecture 1: Period Three Implies Chaos Week 1 UCSB 2014 (Source materials: Period three implies chaos, by Li and Yorke, and From Intermediate Value Theorem To Chaos,
More informationContents. 1 Introduction to Dynamics. 1.1 Examples of Dynamical Systems
Dynamics, Chaos, and Fractals (part 1): Introduction to Dynamics (by Evan Dummit, 2015, v. 1.07) Contents 1 Introduction to Dynamics 1 1.1 Examples of Dynamical Systems......................................
More informationFinding Complex Solutions of Quadratic Equations
y - y - - - x x Locker LESSON.3 Finding Complex Solutions of Quadratic Equations Texas Math Standards The student is expected to: A..F Solve quadratic and square root equations. Mathematical Processes
More informationUnit 2: Solving Scalar Equations. Notes prepared by: Amos Ron, Yunpeng Li, Mark Cowlishaw, Steve Wright Instructor: Steve Wright
cs416: introduction to scientific computing 01/9/07 Unit : Solving Scalar Equations Notes prepared by: Amos Ron, Yunpeng Li, Mark Cowlishaw, Steve Wright Instructor: Steve Wright 1 Introduction We now
More informationMATH 1130 Exam 1 Review Sheet
MATH 1130 Exam 1 Review Sheet The Cartesian Coordinate Plane The Cartesian Coordinate Plane is a visual representation of the collection of all ordered pairs (x, y) where x and y are real numbers. This
More informationSTEP Support Programme. Hints and Partial Solutions for Assignment 2
STEP Support Programme Hints and Partial Solutions for Assignment 2 Warm-up 1 (i) You could either expand the rackets and then factorise, or use the difference of two squares to get [(2x 3) + (x 1)][(2x
More informationFinQuiz Notes
Reading 9 A time series is any series of data that varies over time e.g. the quarterly sales for a company during the past five years or daily returns of a security. When assumptions of the regression
More informationConvergence of sequences and series
Convergence of sequences and series A sequence f is a map from N the positive integers to a set. We often write the map outputs as f n rather than f(n). Often we just list the outputs in order and leave
More informationTHREE DIMENSIONAL SYSTEMS. Lecture 6: The Lorenz Equations
THREE DIMENSIONAL SYSTEMS Lecture 6: The Lorenz Equations 6. The Lorenz (1963) Equations The Lorenz equations were originally derived by Saltzman (1962) as a minimalist model of thermal convection in a
More informationA function is actually a simple concept; if it were not, history would have replaced it with a simpler one by now! Here is the definition:
1.2 Functions and Their Properties A function is actually a simple concept; if it were not, history would have replaced it with a simpler one by now! Here is the definition: Definition: Function, Domain,
More informationMATH 116, LECTURE 13, 14 & 15: Derivatives
MATH 116, LECTURE 13, 14 & 15: Derivatives 1 Formal Definition of the Derivative We have seen plenty of limits so far, but very few applications. In particular, we have seen very few functions for which
More informationGeneralized Geometric Series, The Ratio Comparison Test and Raabe s Test
Generalized Geometric Series The Ratio Comparison Test and Raae s Test William M. Faucette Decemer 2003 The goal of this paper is to examine the convergence of a type of infinite series in which the summands
More informationEach element of this set is assigned a probability. There are three basic rules for probabilities:
XIV. BASICS OF ROBABILITY Somewhere out there is a set of all possile event (or all possile sequences of events which I call Ω. This is called a sample space. Out of this we consider susets of events which
More informationDEFINITE INTEGRALS & NUMERIC INTEGRATION
DEFINITE INTEGRALS & NUMERIC INTEGRATION Calculus answers two very important questions. The first, how to find the instantaneous rate of change, we answered with our study of the derivative. We are now
More informationarxiv: v1 [cs.fl] 24 Nov 2017
(Biased) Majority Rule Cellular Automata Bernd Gärtner and Ahad N. Zehmakan Department of Computer Science, ETH Zurich arxiv:1711.1090v1 [cs.fl] 4 Nov 017 Astract Consider a graph G = (V, E) and a random
More informationStructuring Unreliable Radio Networks
Structuring Unreliale Radio Networks Keren Censor-Hillel Seth Gilert Faian Kuhn Nancy Lynch Calvin Newport March 29, 2011 Astract In this paper we study the prolem of uilding a connected dominating set
More information3.5 Solving Quadratic Equations by the
www.ck1.org Chapter 3. Quadratic Equations and Quadratic Functions 3.5 Solving Quadratic Equations y the Quadratic Formula Learning ojectives Solve quadratic equations using the quadratic formula. Identify
More information8.5 Taylor Polynomials and Taylor Series
8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES 50 8.5 Taylor Polynomials and Taylor Series Motivating Questions In this section, we strive to understand the ideas generated by the following important questions:
More informationMath 216 Second Midterm 28 March, 2013
Math 26 Second Midterm 28 March, 23 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that
More information1.17 Triangle Numbers
.7 riangle Numers he n triangle numer is nn ( ). he first few are, 3,, 0, 5,, he difference etween each numer and e next goes up y each time. he formula nn ( ) gives e ( n ) triangle numer for n. he n
More informationEssential Maths 1. Macquarie University MAFC_Essential_Maths Page 1 of These notes were prepared by Anne Cooper and Catriona March.
Essential Maths 1 The information in this document is the minimum assumed knowledge for students undertaking the Macquarie University Masters of Applied Finance, Graduate Diploma of Applied Finance, and
More informationDynamical Systems Solutions to Exercises
Dynamical Systems Part 5-6 Dr G Bowtell Dynamical Systems Solutions to Exercises. Figure : Phase diagrams for i, ii and iii respectively. Only fixed point is at the origin since the equations are linear
More information17 Exponential and Logarithmic Functions
17 Exponential and Logarithmic Functions Concepts: Exponential Functions Power Functions vs. Exponential Functions The Definition of an Exponential Function Graphing Exponential Functions Exponential Growth
More informationERASMUS UNIVERSITY ROTTERDAM Information concerning the Entrance examination Mathematics level 2 for International Business Administration (IBA)
ERASMUS UNIVERSITY ROTTERDAM Information concerning the Entrance examination Mathematics level 2 for International Business Administration (IBA) General information Availale time: 2.5 hours (150 minutes).
More informationf(x) = lim x 0 + x = lim f(x) =
Infinite Limits Having discussed in detail its as x ±, we would like to discuss in more detail its where f(x) ±. Once again we would like to emphasize that ± are not numbers, so if we write f(x) = we are
More informationLECTURE 8: DYNAMICAL SYSTEMS 7
15-382 COLLECTIVE INTELLIGENCE S18 LECTURE 8: DYNAMICAL SYSTEMS 7 INSTRUCTOR: GIANNI A. DI CARO GEOMETRIES IN THE PHASE SPACE Damped pendulum One cp in the region between two separatrix Separatrix Basin
More informationMain topics for the First Midterm Exam
Main topics for the First Midterm Exam The final will cover Sections.-.0, 2.-2.5, and 4.. This is roughly the material from first three homeworks and three quizzes, in addition to the lecture on Monday,
More informationSlope Fields: Graphing Solutions Without the Solutions
8 Slope Fields: Graphing Solutions Without the Solutions Up to now, our efforts have been directed mainly towards finding formulas or equations describing solutions to given differential equations. Then,
More information1 Functions, Graphs and Limits
1 Functions, Graphs and Limits 1.1 The Cartesian Plane In this course we will be dealing a lot with the Cartesian plane (also called the xy-plane), so this section should serve as a review of it and its
More informationMath Analysis CP WS 4.X- Section Review A
Math Analysis CP WS 4.X- Section 4.-4.4 Review Complete each question without the use of a graphing calculator.. Compare the meaning of the words: roots, zeros and factors.. Determine whether - is a root
More informationChapter 3: The Derivative in Graphing and Applications
Chapter 3: The Derivative in Graphing and Applications Summary: The main purpose of this chapter is to use the derivative as a tool to assist in the graphing of functions and for solving optimization problems.
More informationMATH 103 Pre-Calculus Mathematics Test #3 Fall 2008 Dr. McCloskey Sample Solutions
MATH 103 Pre-Calculus Mathematics Test #3 Fall 008 Dr. McCloskey Sample Solutions 1. Let P (x) = 3x 4 + x 3 x + and D(x) = x + x 1. Find polynomials Q(x) and R(x) such that P (x) = Q(x) D(x) + R(x). (That
More informationa b a b ab b b b Math 154B Elementary Algebra Spring 2012
Math 154B Elementar Algera Spring 01 Stud Guide for Eam 4 Eam 4 is scheduled for Thursda, Ma rd. You ma use a " 5" note card (oth sides) and a scientific calculator. You are epected to know (or have written
More informationSOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 2 NOVEMBER 2016
SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY NOVEMBER 06 Mark Scheme: Each part of Question is worth 4 marks which are awarded solely for the correct answer.
More informationUniversity of Toronto MAT137Y1 Calculus! Test 2 1 December 2017 Time: 110 minutes
University of Toronto MAT137Y1 Calculus! Test 2 1 December 2017 Time: 110 minutes Please complete this cover page with ALL CAPITAL LETTERS. Last name......................................................................................
More informationHomework 7 Solutions to Selected Problems
Homework 7 Solutions to Selected Prolems May 9, 01 1 Chapter 16, Prolem 17 Let D e an integral domain and f(x) = a n x n +... + a 0 and g(x) = m x m +... + 0 e polynomials with coecients in D, where a
More informationA Toy Model of Consciousness and the Connectome, First Approximation, Second Part
A Toy Model of Consciousness and the Connectome, First Approximation, Second Part Aility Potential of A Neural Network As A Prognostic Connective Indicator Stan Bumle, Ph. D. Astract: The paper vixra:1305.0128
More information7.5 Partial Fractions and Integration
650 CHPTER 7. DVNCED INTEGRTION TECHNIQUES 7.5 Partial Fractions and Integration In this section we are interested in techniques for computing integrals of the form P(x) dx, (7.49) Q(x) where P(x) and
More information1 + lim. n n+1. f(x) = x + 1, x 1. and we check that f is increasing, instead. Using the quotient rule, we easily find that. 1 (x + 1) 1 x (x + 1) 2 =
Chapter 5 Sequences and series 5. Sequences Definition 5. (Sequence). A sequence is a function which is defined on the set N of natural numbers. Since such a function is uniquely determined by its values
More informationSequences and Series
Sequences and Series What do you think of when you read the title of our next unit? In case your answers are leading us off track, let's review the following IB problems. 1 November 2013 HL 2 3 November
More informationDR.RUPNATHJI( DR.RUPAK NATH )
Contents 1 Sets 1 2 The Real Numbers 9 3 Sequences 29 4 Series 59 5 Functions 81 6 Power Series 105 7 The elementary functions 111 Chapter 1 Sets It is very convenient to introduce some notation and terminology
More informationEnergy-Rate Method and Stability Chart of Parametric Vibrating Systems
Reza N. Jazar Reza.Jazar@manhattan.edu Department of Mechanical Engineering Manhattan College New York, NY, A M. Mahinfalah Mahinfalah@msoe.edu Department of Mechanical Engineering Milwaukee chool of Engineering
More informationMath 3450 Homework Solutions
Math 3450 Homework Solutions I have decided to write up all the solutions to prolems NOT assigned from the textook first. There are three more sets to write up and I am doing those now. Once I get the
More informationSolved Examples. Given are two sets A {1, 2, -2, 3} and B = {1, 2, 3, 5}. Is the function f(x) = 2x - 1 defined from A to B?
Solved Examples Example 1: Given are two sets A {1, 2, -2, 3} and B = {1, 2, 3, 5}. Is the function f(x) = 2x - 1 defined from A to B? Solution : Out of all the ordered pairs, the ordered pairs which are
More informationThe Growth of Functions. A Practical Introduction with as Little Theory as possible
The Growth of Functions A Practical Introduction with as Little Theory as possible Complexity of Algorithms (1) Before we talk about the growth of functions and the concept of order, let s discuss why
More informationWeekly Activities Ma 110
Weekly Activities Ma 110 Fall 2008 As of October 27, 2008 We give detailed suggestions of what to learn during each week. This includes a reading assignment as well as a brief description of the main points
More informationCool Results on Primes
Cool Results on Primes LA Math Circle (Advanced) January 24, 2016 Recall that last week we learned an algorithm that seemed to magically spit out greatest common divisors, but we weren t quite sure why
More informationName: AK-Nummer: Ergänzungsprüfung January 29, 2016
INSTRUCTIONS: The test has a total of 32 pages including this title page and 9 questions which are marked out of 10 points; ensure that you do not omit a page by mistake. Please write your name and AK-Nummer
More information1.10 Continuity Brian E. Veitch
1.10 Continuity Definition 1.5. A function is continuous at x = a if 1. f(a) exists 2. lim x a f(x) exists 3. lim x a f(x) = f(a) If any of these conditions fail, f is discontinuous. Note: From algebra
More information0.4 Combinations of Functions
30 welcome to calculus 0.4 Combinations of Functions Sometimes a physical or economic situation behaves differently depending on various circumstances. In these situations, a more complicated formula may
More informationSequence. A list of numbers written in a definite order.
Sequence A list of numbers written in a definite order. Terms of a Sequence a n = 2 n 2 1, 2 2, 2 3, 2 4, 2 n, 2, 4, 8, 16, 2 n We are going to be mainly concerned with infinite sequences. This means we
More informationFast inverse for big numbers: Picarte s iteration
Fast inverse for ig numers: Picarte s iteration Claudio Gutierrez and Mauricio Monsalve Computer Science Department, Universidad de Chile cgutierr,mnmonsal@dcc.uchile.cl Astract. This paper presents an
More informationOne dimensional Maps
Chapter 4 One dimensional Maps The ordinary differential equation studied in chapters 1-3 provide a close link to actual physical systems it is easy to believe these equations provide at least an approximate
More information10/22/16. 1 Math HL - Santowski SKILLS REVIEW. Lesson 15 Graphs of Rational Functions. Lesson Objectives. (A) Rational Functions
Lesson 15 Graphs of Rational Functions SKILLS REVIEW! Use function composition to prove that the following two funtions are inverses of each other. 2x 3 f(x) = g(x) = 5 2 x 1 1 2 Lesson Objectives! The
More informationInfinite series, improper integrals, and Taylor series
Chapter 2 Infinite series, improper integrals, and Taylor series 2. Introduction to series In studying calculus, we have explored a variety of functions. Among the most basic are polynomials, i.e. functions
More informationIntroduction to Dynamical Systems Basic Concepts of Dynamics
Introduction to Dynamical Systems Basic Concepts of Dynamics A dynamical system: Has a notion of state, which contains all the information upon which the dynamical system acts. A simple set of deterministic
More information(Refer Slide Time: 0:21)
Theory of Computation Prof. Somenath Biswas Department of Computer Science and Engineering Indian Institute of Technology Kanpur Lecture 7 A generalisation of pumping lemma, Non-deterministic finite automata
More information3 Forces and pressure Answer all questions and show your working out for maximum credit Time allowed : 30 mins Total points available : 32
1 3 Forces and pressure Answer all questions and show your working out for maximum credit Time allowed : 30 mins Total points availale : 32 Core curriculum 1 A icycle pump has its outlet sealed with a
More informationMath 106 Calculus 1 Topics for first exam
Chapter 2: Limits and Continuity Rates of change and its: Math 06 Calculus Topics for first exam Limit of a function f at a point a = the value the function should take at the point = the value that the
More informationMTH101 Calculus And Analytical Geometry Lecture Wise Questions and Answers For Final Term Exam Preparation
MTH101 Calculus And Analytical Geometry Lecture Wise Questions and Answers For Final Term Exam Preparation Lecture No 23 to 45 Complete and Important Question and answer 1. What is the difference between
More informationExploring the relationship between a fluid container s geometry and when it will balance on edge
Exploring the relationship eteen a fluid container s geometry and hen it ill alance on edge Ryan J. Moriarty California Polytechnic State University Contents 1 Rectangular container 1 1.1 The first geometric
More informationSVETLANA KATOK AND ILIE UGARCOVICI (Communicated by Jens Marklof)
JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 4, 010, 637 691 doi: 10.3934/jmd.010.4.637 STRUCTURE OF ATTRACTORS FOR (a, )-CONTINUED FRACTION TRANSFORMATIONS SVETLANA KATOK AND ILIE UGARCOVICI (Communicated
More informationHomework 3 Solutions(Part 2) Due Friday Sept. 8
MATH 315 Differential Equations (Fall 017) Homework 3 Solutions(Part ) Due Frida Sept. 8 Part : These will e graded in detail. Be sure to start each of these prolems on a new sheet of paper, summarize
More informationSample Solutions from the Student Solution Manual
1 Sample Solutions from the Student Solution Manual 1213 If all the entries are, then the matrix is certainly not invertile; if you multiply the matrix y anything, you get the matrix, not the identity
More information6.1 Polynomial Functions
6.1 Polynomial Functions Definition. A polynomial function is any function p(x) of the form p(x) = p n x n + p n 1 x n 1 + + p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers and
More informationDemographic noise slows down cycles of dominance in ecological models
J. Math. Biol manuscript No. (will e inserted y the editor) Demographic noise slows down cycles of dominance in ecological models Qian Yang Tim Rogers Jonathan H.P. Dawes Received: date / Accepted: date
More information1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
More informationC-N M151 Lecture Notes (part 1) Based on Stewart s Calculus (2013) B. A. Starnes
Lecture Calculus is the study of infinite Mathematics. In essence, it is the extension of algebraic concepts to their limfinity(l). What does that even mean? Well, let's begin with the notion of functions
More informationMATH 521, WEEK 2: Rational and Real Numbers, Ordered Sets, Countable Sets
MATH 521, WEEK 2: Rational and Real Numbers, Ordered Sets, Countable Sets 1 Rational and Real Numbers Recall that a number is rational if it can be written in the form a/b where a, b Z and b 0, and a number
More information