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Modeling Change: Dynamical Systems
‘Powerful paradigm’
future value = present value + change
equivalently:
change = future value – current value
(change = current value – previous value)
Modeling Change: Dynamical Systems
‘Powerful paradigm’
future value = present value + change
equivalently:
change = future value – current value
Modeling Change: Dynamical Systems
‘Powerful paradigm’
future value = present value + change
equivalently:
change = future value – current value
(change = current value – previous value)
Modeling Change: Dynamical Systems
A dynamical system is a changing system.
Definition
Dynamic: marked by continuous and productive activity or change
(Merriam Webster)
Modeling Change: Dynamical Systems
A dynamical system is a changing system.
Definition
Dynamic: marked by continuous and productive activity or change
(Merriam Webster)
Historical Context
• 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.
Implicit relation: xn+1 = f(xn)
Source: Wikipedia
Some Examples of Implicit Relations
I. A(k+1)=A (k)*A (k)
II. A(k)=5
III. A(k+2)=A (k)+A (k+1)
Constant Sequence
Fibonacci Sequence
Exercise I
Generate the first 5 terms of the sequence for rules I given that A (1)=1 and A (2)=1.
I. A(k+1)=A (k)*A (k)
Exercise I
Generate the first 5 terms of the sequence for rule I given that A (1)=1.
I. A(k+1)=A (k)*A (k)
{1, 1, 1, 1, 1}
Exercise III
Generate the first 5 terms of the sequence for rule III given that A (1)=1 and A (2)=1.
III. A(k+2)=A (k)+A (k+1)
Exercise III
Generate the first 5 terms of the sequence for rule III given that A (1)=1 and A (2)=1.
III. A(k+2)=A (k)+A (k+1)
{1, 1, 2, 3, 5}
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
• 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
Dynamical Systems Cont.
A Classic Dynamical System
The double pendulum
Source: Wikipedia
Evidences rich dynamical behavior, including chaotic behavior for some parameters.
Motion described by coupled ODEs.
Source: math.uwaterloo
The Double Pendulum
Chaotic: sensitive dependence upon initial conditions
Source: math.uwaterloo
These two pendulums start out with slightly different initial velocities.
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
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.
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.
The double pendulum
State space: 4 dimensional
What are the4 parametersthat the systemdepends upon?
Must completely describe the system at time t.
Describing Change
• Discrete description: Difference Equation
• Continuous description: Differential Equation
Difference Equation
A dynamical system may be described by a difference equation.
WRITE THIS DOWN
A difference equation is a rule that relates the state of a dynamical system at a future time to the state of a dynamical system at an earlier time.
… 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.
Homework Assignment 1.1
• Problems 1-4, 7-8.• Due Wednesday 1/21.
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.
,10,8,6,4,2
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.
,10,8,6,4,2
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.
,10,8,6,4,2
We’re looking for a description of this sequence in terms of the differences between terms:
an = change = new – old = xn+1 – xn
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.
,10,8,6,4,2
We’re looking for a description of this sequence in terms of the differences between terms:
an = change = new – old = xn+1 – xn
(1) Find implicit relation for an+1 in terms of an
(2) Solve an = an+1 – an
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.
,10,8,6,4,2
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(1) Find implicit relation for an+1 in terms of an
(2) Solve an = an+1 – an
an = 2
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”.
Class Project: Dynamical System in Excel
In groups of 3, we’ll create a dynamical system using the “fill down” function in Excel.
I. 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.
II. 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.)