Discrete Dynamical Systems
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1 Discrete Dynamical Systems Justin Allman Department of Mathematics UNC Chapel Hill 18 June 2011
2 What is a discrete dynamical system? Definition A Discrete Dynamical System is a mathematical way to describe changes in a system over time. Two items are always present:
3 What is a discrete dynamical system? Definition A Discrete Dynamical System is a mathematical way to describe changes in a system over time. Two items are always present: (i) specified initial state
4 What is a discrete dynamical system? Definition A Discrete Dynamical System is a mathematical way to describe changes in a system over time. Two items are always present: (i) specified initial state (ii) a function which specifies how to obtain the state of the system, given the previous state(s).
5 What is a discrete dynamical system? (cont.) Let s see that definition again, but this time in mathese. Definition A Discrete Dynamical System is a sequence of real numbers {A(0), A(1), A(2),...} obtained by specifying
6 What is a discrete dynamical system? (cont.) Let s see that definition again, but this time in mathese. Definition A Discrete Dynamical System is a sequence of real numbers {A(0), A(1), A(2),...} obtained by specifying (i) A(0)
7 What is a discrete dynamical system? (cont.) Let s see that definition again, but this time in mathese. Definition A Discrete Dynamical System is a sequence of real numbers {A(0), A(1), A(2),...} obtained by specifying (i) A(0) (ii) a function f : R R such that A(n + 1) = f (A(n)).
8 What is a discrete dynamical system? (cont.) Let s see that definition again, but this time in mathese. Definition A Discrete Dynamical System is a sequence of real numbers {A(0), A(1), A(2),...} obtained by specifying (i) A(0) (ii) a function f : R R such that A(n + 1) = f (A(n)). Definition A Discrete Dynamical System is a mathematical way to describe changes in a system over time. Two items are always present:
9 What is a discrete dynamical system? (cont.) Let s see that definition again, but this time in mathese. Definition A Discrete Dynamical System is a sequence of real numbers {A(0), A(1), A(2),...} obtained by specifying (i) A(0) (ii) a function f : R R such that A(n + 1) = f (A(n)). Definition A Discrete Dynamical System is a mathematical way to describe changes in a system over time. Two items are always present: (i) specified initial state
10 What is a discrete dynamical system? (cont.) Let s see that definition again, but this time in mathese. Definition A Discrete Dynamical System is a sequence of real numbers {A(0), A(1), A(2),...} obtained by specifying (i) A(0) (ii) a function f : R R such that A(n + 1) = f (A(n)). Definition A Discrete Dynamical System is a mathematical way to describe changes in a system over time. Two items are always present: (i) specified initial state (ii) a function which specifies how to obtain the state of the system, given the previous state(s).
11 A simple example Let s consider a simple example of supply and demand. We ll write a dynamical system which tells us how the price of corn might evolve over time. We ll make a few assumptions.
12 A simple example Let s consider a simple example of supply and demand. We ll write a dynamical system which tells us how the price of corn might evolve over time. We ll make a few assumptions. (1) The quantity of corn demanded decreases as the price increases.
13 A simple example Let s consider a simple example of supply and demand. We ll write a dynamical system which tells us how the price of corn might evolve over time. We ll make a few assumptions. (1) The quantity of corn demanded decreases as the price increases. (2) The quantity of corn supplied increases as the price increases.
14 A simple example Let s consider a simple example of supply and demand. We ll write a dynamical system which tells us how the price of corn might evolve over time. We ll make a few assumptions. (1) The quantity of corn demanded decreases as the price increases. (2) The quantity of corn supplied increases as the price increases. (3) Each year, the price of the product is adjusted so that the supply matches the demand. This is the so-called market-clearing price.
15 A simple example Let s consider a simple example of supply and demand. We ll write a dynamical system which tells us how the price of corn might evolve over time. We ll make a few assumptions. (1) The quantity of corn demanded decreases as the price increases. (2) The quantity of corn supplied increases as the price increases. (3) Each year, the price of the product is adjusted so that the supply matches the demand. This is the so-called market-clearing price. (4) When the price is zero, no corn is supplied and only a finite quantity is demanded.
16 A simple example (cont.) Let s make up some numbers! We ll write the simplest type of model that fits these assumptions; a linear one.
17 A simple example (cont.) Let s make up some numbers! We ll write the simplest type of model that fits these assumptions; a linear one. D = 1.2P + 20
18 A simple example (cont.) Let s make up some numbers! We ll write the simplest type of model that fits these assumptions; a linear one. D = 1.2P + 20 S = 0.8P
19 A simple example (cont.) Let s make up some numbers! We ll write the simplest type of model that fits these assumptions; a linear one. D = 1.2P + 20 S = 0.8P Now let s build in the dynamics. How does the price change in time? We need to ask ourselves, when do the demander and the supplier make their decisions?
20 A simple example (cont.) Let s make up some numbers! We ll write the simplest type of model that fits these assumptions; a linear one. D = 1.2P + 20 S = 0.8P Now let s build in the dynamics. How does the price change in time? We need to ask ourselves, when do the demander and the supplier make their decisions? D(n) = 1.2P(n) + 20
21 A simple example (cont.) Let s make up some numbers! We ll write the simplest type of model that fits these assumptions; a linear one. D = 1.2P + 20 S = 0.8P Now let s build in the dynamics. How does the price change in time? We need to ask ourselves, when do the demander and the supplier make their decisions? D(n) = 1.2P(n) + 20 S(n + 1) = 0.8P(n)
22 A simple example (cont.) We have already used assumptions (1), (2), and (4). We need to make use of (3). That is, we must always have that D(n) = S(n).
23 A simple example (cont.) We have already used assumptions (1), (2), and (4). We need to make use of (3). That is, we must always have that D(n) = S(n). D(n + 1) = 1.2P(n + 1) + 20 S(n + 1) = 0.8P(n)
24 A simple example (cont.) We have already used assumptions (1), (2), and (4). We need to make use of (3). That is, we must always have that D(n) = S(n). D(n + 1) = 1.2P(n + 1) + 20 S(n + 1) = 0.8P(n) Thus, 1.2P(n + 1) + 20 = 0.8P(n)
25 A simple example (cont.) We have already used assumptions (1), (2), and (4). We need to make use of (3). That is, we must always have that D(n) = S(n). D(n + 1) = 1.2P(n + 1) + 20 S(n + 1) = 0.8P(n) Thus, 1.2P(n + 1) + 20 = 0.8P(n) and a little more algebra gives P(n + 1) = 2 50 P(n)
26 A simple example (cont.) Formula P(n + 1) = 2 50 P(n) We have some interesting questions to ask:
27 A simple example (cont.) Formula P(n + 1) = 2 50 P(n) We have some interesting questions to ask: (i) Is there an equilibrium price? That is, a price which will remain fixed for all eternity?
28 A simple example (cont.) Formula P(n + 1) = 2 50 P(n) We have some interesting questions to ask: (i) Is there an equilibrium price? That is, a price which will remain fixed for all eternity? (ii) If we begin with an initial price P(0) which is near the equilibrium, will the price eventually reach the equilibrium?
29 A simple example (cont.) To answer (i), we need to find a special price P such that if P(0) = P, then P(n) = P for all values of n.
30 A simple example (cont.) To answer (i), we need to find a special price P such that if P(0) = P, then P(n) = P for all values of n. We must solve the equation: P = 2 3 P
31 A simple example (cont.) To answer (i), we need to find a special price P such that if P(0) = P, then P(n) = P for all values of n. We must solve the equation: P = 2 3 P The solution is P = 10. We call this the equilibrium value.
32 A simple example (cont.) To answer (ii) we ll need a little more sophistication. My favorite method is geometric and called cobwebbing. To understand this, we ll try to write an algorithm to evaluate the sequence of prices {P(0), P(1), P(2),..., P(n),...}.
33 A simple example (cont.) To answer (ii) we ll need a little more sophistication. My favorite method is geometric and called cobwebbing. To understand this, we ll try to write an algorithm to evaluate the sequence of prices {P(0), P(1), P(2),..., P(n),...}. First we ll write our DDS in function notation. We have that P(n + 1) = f (P(n)), where f (x) = 2 3 x
34 A simple example (cont.) To answer (ii) we ll need a little more sophistication. My favorite method is geometric and called cobwebbing. To understand this, we ll try to write an algorithm to evaluate the sequence of prices {P(0), P(1), P(2),..., P(n),...}. First we ll write our DDS in function notation. We have that P(n + 1) = f (P(n)), where f (x) = 2 3 x (1) Set x = P(0).
35 A simple example (cont.) To answer (ii) we ll need a little more sophistication. My favorite method is geometric and called cobwebbing. To understand this, we ll try to write an algorithm to evaluate the sequence of prices {P(0), P(1), P(2),..., P(n),...}. First we ll write our DDS in function notation. We have that P(n + 1) = f (P(n)), where f (x) = 2 3 x (1) Set x = P(0). (2) Set y = f (x). This gives P(next).
36 A simple example (cont.) To answer (ii) we ll need a little more sophistication. My favorite method is geometric and called cobwebbing. To understand this, we ll try to write an algorithm to evaluate the sequence of prices {P(0), P(1), P(2),..., P(n),...}. First we ll write our DDS in function notation. We have that P(n + 1) = f (P(n)), where f (x) = 2 3 x (1) Set x = P(0). (2) Set y = f (x). This gives P(next). (3) Set y = x.
37 A simple example (cont.) To answer (ii) we ll need a little more sophistication. My favorite method is geometric and called cobwebbing. To understand this, we ll try to write an algorithm to evaluate the sequence of prices {P(0), P(1), P(2),..., P(n),...}. First we ll write our DDS in function notation. We have that P(n + 1) = f (P(n)), where f (x) = 2 3 x (1) Set x = P(0). (2) Set y = f (x). This gives P(next). (3) Set y = x. (4) Go back to step (2).
38 A simple example (cont.) Let s translate the last slide to geometry:
39 A simple example (cont.) Let s translate the last slide to geometry: Method (Cobwebbing) Let P(n + 1) = f (P(n)) be a D.D.S. To determine the stability of equilibrium values, begin by drawing the graphs of y = f (x) and y = x. Their intersections are the equilibrium points. Then:
40 A simple example (cont.) Let s translate the last slide to geometry: Method (Cobwebbing) Let P(n + 1) = f (P(n)) be a D.D.S. To determine the stability of equilibrium values, begin by drawing the graphs of y = f (x) and y = x. Their intersections are the equilibrium points. Then: (1) Put your pencil on the x-axis at x = P(0).
41 A simple example (cont.) Let s translate the last slide to geometry: Method (Cobwebbing) Let P(n + 1) = f (P(n)) be a D.D.S. To determine the stability of equilibrium values, begin by drawing the graphs of y = f (x) and y = x. Their intersections are the equilibrium points. Then: (1) Put your pencil on the x-axis at x = P(0). (2) Trace a line vertically to y = f (x).
42 A simple example (cont.) Let s translate the last slide to geometry: Method (Cobwebbing) Let P(n + 1) = f (P(n)) be a D.D.S. To determine the stability of equilibrium values, begin by drawing the graphs of y = f (x) and y = x. Their intersections are the equilibrium points. Then: (1) Put your pencil on the x-axis at x = P(0). (2) Trace a line vertically to y = f (x). (3) Trace horizontally to y = x.
43 A simple example (cont.) Let s translate the last slide to geometry: Method (Cobwebbing) Let P(n + 1) = f (P(n)) be a D.D.S. To determine the stability of equilibrium values, begin by drawing the graphs of y = f (x) and y = x. Their intersections are the equilibrium points. Then: (1) Put your pencil on the x-axis at x = P(0). (2) Trace a line vertically to y = f (x). (3) Trace horizontally to y = x. (4) Go back to step (2).
44 A simple example (cont.) Let s translate the last slide to geometry: Method (Cobwebbing) Let P(n + 1) = f (P(n)) be a D.D.S. To determine the stability of equilibrium values, begin by drawing the graphs of y = f (x) and y = x. Their intersections are the equilibrium points. Then: (1) Put your pencil on the x-axis at x = P(0). (2) Trace a line vertically to y = f (x). (3) Trace horizontally to y = x. (4) Go back to step (2). If an equilibrium point attracts nearby cobwebs it is called stable. Otherwise, it is called unstable.
45 Now lets play! Here are some examples!
46 Now lets play! Here are some examples! (a) P(n + 1) = 3 2 P(n) [f (x) = 3 2 x ]
47 Now lets play! Here are some examples! (a) P(n + 1) = 3 2 P(n) [f (x) = 3 2 x ] (b) A(n + 1) = 1.8A(n) 0.8A 2 (n). [f (x) = 1.8x 0.8x 2 ]
48 Now lets play! Here are some examples! (a) P(n + 1) = 3 2 P(n) [f (x) = 3 2 x ] (b) A(n + 1) = 1.8A(n) 0.8A 2 (n). [f (x) = 1.8x 0.8x 2 ] (c) H(n + 1) = 1.8H(n) 0.8H 2 (n) [f (x) = 1.8x 0.8x ]
49 Now lets play! Here are some examples! (a) P(n + 1) = 3 2 P(n) [f (x) = 3 2 x ] (b) A(n + 1) = 1.8A(n) 0.8A 2 (n). [f (x) = 1.8x 0.8x 2 ] (c) H(n + 1) = 1.8H(n) 0.8H 2 (n) [f (x) = 1.8x 0.8x ] (d) K(n + 1) = 1.8K(n) 0.8K 2 (n) [f (x) = 1.8x 0.8x ]
50 Now lets play! Here are some examples! (a) P(n + 1) = 3 2 P(n) [f (x) = 3 2 x ] (b) A(n + 1) = 1.8A(n) 0.8A 2 (n). [f (x) = 1.8x 0.8x 2 ] (c) H(n + 1) = 1.8H(n) 0.8H 2 (n) [f (x) = 1.8x 0.8x ] (d) K(n + 1) = 1.8K(n) 0.8K 2 (n) [f (x) = 1.8x 0.8x ] (e) L(n + 1) = 1.8L(n) 0.8L 2 (n) 0.2. [f (x) = 1.8x 0.8x 2 0.2]
51 Now lets play! Here are some examples! (a) P(n + 1) = 3 2 P(n) [f (x) = 3 2 x ] (b) A(n + 1) = 1.8A(n) 0.8A 2 (n). [f (x) = 1.8x 0.8x 2 ] (c) H(n + 1) = 1.8H(n) 0.8H 2 (n) [f (x) = 1.8x 0.8x ] (d) K(n + 1) = 1.8K(n) 0.8K 2 (n) [f (x) = 1.8x 0.8x ] (e) L(n + 1) = 1.8L(n) 0.8L 2 (n) 0.2. [f (x) = 1.8x 0.8x 2 0.2] (f) Q(n + 1) = 3.2Q(n) 0.8Q 2 (n). [f (x) = 3.2x 1.8x 2 ]
52 What do we notice? Question Under what circumstances is an equilibrium value stable?
53 What do we notice? Question Under what circumstances is an equilibrium value stable? Theorem Given the discrete dynamical system A(n + 1) = f (A(n)), the equilibrium value A is stable if and only if the slope of the tangent line to the graph of f (x) at x = A has absolute value less than one.
54 Bifurcations Now we ll look closely at a very special system, which can actually be used to measure the population of bacterial cultures.
55 Bifurcations Now we ll look closely at a very special system, which can actually be used to measure the population of bacterial cultures. Formula B(n + 1) = rb(n)(1 B(n)) where r is a positive real number and reflects the growth rate of the bacterial population.
56 Bifurcations Now we ll look closely at a very special system, which can actually be used to measure the population of bacterial cultures. Formula B(n + 1) = rb(n)(1 B(n)) where r is a positive real number and reflects the growth rate of the bacterial population. Definition Given a DDS with a parameter r, a bifurcation point is a value of r at which the behavior of the system is altered qualitatively. In practice, this means that if r is a bifurcation point, then the stability or number of equilibrium values changes as r varies past r.
57 Computations We go to Excel.
58 Chaos Definition (????) A DDS is said to exhibit chaotic behavior if the following conditions are met:
59 Chaos Definition (????) A DDS is said to exhibit chaotic behavior if the following conditions are met: A sensitive dependence on initial conditions
60 Chaos Definition (????) A DDS is said to exhibit chaotic behavior if the following conditions are met: A sensitive dependence on initial conditions Although there is a specified rule to know the next step in the sequence, long term predictive ability is essentially non-existent.
61 Chaos Definition (????) A DDS is said to exhibit chaotic behavior if the following conditions are met: A sensitive dependence on initial conditions Although there is a specified rule to know the next step in the sequence, long term predictive ability is essentially non-existent. Other examples:
62 Chaos Definition (????) A DDS is said to exhibit chaotic behavior if the following conditions are met: A sensitive dependence on initial conditions Although there is a specified rule to know the next step in the sequence, long term predictive ability is essentially non-existent. Other examples: The weather.
63 Chaos Definition (????) A DDS is said to exhibit chaotic behavior if the following conditions are met: A sensitive dependence on initial conditions Although there is a specified rule to know the next step in the sequence, long term predictive ability is essentially non-existent. Other examples: The weather. Lorenz attractors.
64 Further Resources Discrete Dynamical Systems: Theory and Applications, James Sandefur, Oxford Press. Chaos, James Glieck, Penguin edition.
65 Further Resources Discrete Dynamical Systems: Theory and Applications, James Sandefur, Oxford Press. Chaos, James Glieck, Penguin edition. THE END.
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