1. 1.1 Dynamical Systems MODELING CHANGE
MA354
1.1 Dynamical Systems
MODELING CHANGE

2. Introduction to Dynamical Systems

3. Modeling Change: Dynamical Systems
A dynamical system is a changing system.
Definition
Dynamic: marked by continuous and productive activity or change
(Merriam Webster)

4. Modeling Change: Dynamical Systems
A dynamical system is a changing system.
Definition
Dynamic: marked by continuous and productive activity or change
(Merriam Webster)

5. Historical Context Implicit relation: xn+1 = f(xn)
the term ‘dynamical system’ originated from the field of Newtonian mechanics
the evolution rule was given implicitly by a relation that gives the state of the system only a short time into the future.
system: x1, x2, x3, … (states as time increases)
Implicit relation: xn+1 = f(xn)
17th century
Source: Wikipedia

6. Dynamical Systems Cont.
To determine the state for all future times requires iterating the relation many times—each advancing time a small step.
The iteration procedure is referred to as solving the system or integrating the system.
Source: Wikipedia

7. Dynamical Systems Cont.
Once the system can be solved, given an initial point it is possible to determine all its future points
Before the advent of fast computing machines, solving a dynamical system was difficult in practice and could only be accomplished for a small class of dynamical systems.
Source: Wikipedia

8. A Classic Dynamical System
The double pendulum
The model tracks the velocities and positions of the two masses.
Evidences rich dynamical behavior, including chaotic behavior for some parameters.
Motion described by coupled ODEs.
Source: math.uwaterloo
Source: Wikipedia

9. The Double Pendulum
Chaotic: sensitive dependence upon initial conditions
These two pendulums start out with slightly different initial velocities.
Source: math.uwaterloo

10. Wikipedia page on the double pendulum

11. State and State Space
A dynamical system is a system that is changing over time.
At each moment in time, the system has a state. The state is a list of the variables that describe the system.
Example: Bouncing ball
State is the position and the velocity of the ball

12. State and State Space
Over time, the system’s state changes. We say that the system moves through state space
The state space is an n-dimensional space that includes all possible states.
As the system moves through state space, it traces a path called its trajectory, orbit, or numerical solution.

13. Dimension of the State Space
n-dimensional
As n increases, the system becomes more complicated.
Usually, the dimension of state space is greater than the number of spatial variables, as the evolution of a system depends upon more than just position – for example, it may also depend upon velocity.

14. The double pendulum State space: 4 dimensional
(What are the static parameters
of the system?)
What are the
4 changing variables (state
variables) that the system
depends upon?
Must completely describe the system at time t.

15. https://twitter.com/NewsfromScience/status/817443381412556800
“mechanics of how animals use tails as fly swatters”

16. Mathematical Description of Dynamical Systems

17. Modeling Change: Dynamical Systems
From your book:
‘Powerful paradigm’

18. Modeling Change: Dynamical Systems
Powerful paradigm:
future value = present value + change
equivalently:
change = future value – current value

19. Modeling Change: Dynamical Systems
Powerful paradigm:
future value = present value + change
equivalently:
change = future value – current value

20. Modeling Change: Dynamical Systems
Powerful paradigm:
future value = present value + change
equivalently:
change = future value – current value
change = current value – previous value

21. Descriptions of Dynamical Systems
Discrete versus continuous
Implicit versus explicit
As nth term of a sequence versus nth difference between terms

22. Descriptions of Dynamical Systems
Discrete versus continuous
Implicit versus explicit
As nth term of a sequence versus nth difference between terms

23. Describing Change (Discrete verses Continuous)
Discrete description: Difference Equation
Continuous description: Differential Equation

24. Descriptions of Dynamical Systems
Discrete versus continuous
Implicit versus explicit
As nth term of a sequence versus nth difference between terms

25. Implicit Equations
Since dynamical systems are defined by defining the change that occurs between events, they are often defined implicitly rather than explicitly. (Example: differential equations are implicit, describing how the function is changing, rather than the function explicitly)

26. Explicit Verses Implicit Equations
Implicit Expression:
Explicit Expression:
To find the nth term, you must calculate the first (n-1) terms.
First 10 terms:
{1,1,2,3,5,8,13,21,34,55}
To find the nth term, you simply plug in n and make a single computation.
First 10 terms:
{1,1,2,3,5,8,13,21.0,34.0,55.0}

27. Example
Given the following sequence, find the explicit and implicit descriptions:

28. More Examples of Implicit Relations
ak+1 = ak ∙ ak
ak = 5
ak+2 = ak + ak+1
Constant Sequence
Fibonacci Sequence

29. Exercise I
Generate the first 5 terms of the sequence for rule I given that a1=1.
A(k+1)=A (k)*A (k)

30. Exercise I
Generate the first 5 terms of the sequence for rule I given that a1=1.
ak+1 = ak ∙ ak

31. Exercise I
Generate the first 3 terms of the sequence for rule I given that a1=3.
ak+1 = ak ∙ ak

32. Role of an ‘Initial Condition’
These are called different trajectories of the dynamical system.
An interesting problem in dynamical systems is describing the type of trajectories that are possible with any specific system.
Consider the implications of a memoryless system…

33. Exercise II Generate the first 5 terms of the sequence for rule II.
II ak=5

34. Exercise III
Generate the first 8 terms of the sequence for rule III given that a1=1 and a2=1.
III. ak+2 = ak + ak+1

35. Exercise III
Generate the first 8 terms of the sequence for rule III given that a1=1 and a2=-1.
III. ak+2 = ak + ak+1

36. Descriptions of Dynamical Systems
Discrete versus continuous
Implicit versus explicit
As nth term of a sequence versus nth difference between terms

37. Modeling Change: Dynamical Systems
Difference equation:
describes change (denoted by ∆)
equivalently:
change = future value – current value
change=future value-present value
 = xn+1 – xn

38. The set of first differences is a0= a1 – a0 , a1= a2 – a1 ,
… consider a sequence
A={a0, a1, a2,…}
The set of first differences is
a0= a1 – a0 ,
a1= a2 – a1 ,
a2= a3 – a1, …
where in particular the nth first difference is
an+1= an+1 – an.

39. Homework Assignment 1.1
Problems 1-4, 7-8.

40. Homework Assignment 1.1 Problems 1-4, 7-8. Example
(3a) By examining the following sequences, write a difference equation to represent the change during the nth interval as a function of the previous term in the sequence.

41. Example 3(a)
(3a) By examining the following sequences, write a difference equation to represent the change during the nth interval as a function of the previous term in the sequence.

42. Example 3(a)
(3a) By examining the following sequences, write a difference equation to represent the change during the nth interval as a function of the previous term in the sequence.
We’re looking for a description of this sequence in terms of the differences between terms:
an = change = new – old = xn+1 – xn

43. Example 3(a)
(3a) By examining the following sequences, write a difference equation to represent the change during the nth interval as a function of the previous term in the sequence.
We’re looking for a description of this sequence in terms of the differences between terms:
an = change = new – old = xn+1 – xn
Find implicit relation for an+1 in terms of an
Solve an = an+1 – an

44. Example 3(a)
(3a) By examining the following sequences, write a difference equation to represent the change during the nth interval as a function of the previous term in the sequence.
We’re looking for a description of this sequence in terms of the differences between terms:
an = change = new – old = xn+1 – xn
an+1 = an+2
Find implicit relation for an+1 in terms of an
Solve an = an+1 – an
an = 2

45. More Examples of Implicit Relations
ak+1 = ak ∙ ak
ak = 5
ak+2 = ak + ak+1
Find the difference equation
description of each.
Constant Sequence
Fibonacci Sequence

46. Related: “Markov Chain”
A markov chain is a dynamical system in which the state at time t+1 only depends upon the state of the system at time t. Such a dynamical system is said to be “memory-less”. (This is the ‘Markov property’.)
Counter-example: Fibonacci sequence

47. Class Project: Dynamical System in Excel
In groups of 3, we’ll create a dynamical system using the “fill down” function in Excel.
In groups, decide on an interesting dynamical system that is described by a simple rule for the state at time t+1 that only depends upon the current state. (Markov Chain) Describe your system to the class.
Model your dynamical system in Excel by producing the states of the system in a table where columns describe different states and rows correspond to different times. (You may need to modify your system in order to implement it in Excel.)