MA354
1.1 Dynamical Systems
MODELING CHANGE
Introduction to
Dynamical Systems
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
17th century
• 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)
Source: Wikipedia
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
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
A Classic Dynamical System
The double pendulum
Evidences rich dynamical
behavior, including chaotic
behavior for some
parameters.
Motion described by
coupled ODEs.
The model tracks the velocities and positions of
the two masses.
Source:Source:
math.uwaterloo
Wikipedia
The Double Pendulum
These two pendulums
start out with slightly
different initial
velocities.
Chaotic: sensitive dependence upon initial
conditions
Source: math.uwaterloo
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 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.
Mathematical
Description of
Dynamical Systems
Modeling Change: Dynamical Systems
From your book:
‘Powerful paradigm’
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:
f ( x  x)  f ( x)  f
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
Descriptions of Dynamical Systems
• Discrete versus continuous
• Implicit versus explicit
• As nth term of a sequence versus nth difference
between terms
Descriptions of Dynamical Systems
• Discrete versus continuous
• Implicit versus explicit
• As nth term of a sequence versus nth difference
between terms
Describing Change
(Discrete verses Continuous)
• Discrete description: Difference Equation
f ( x  x)  f ( x)  f
 f  f ( x  x)  f ( x)
• Continuous description: Differential Equation
f ( x  t )  f ( x)
f ( x)  lim
t 0
t
Descriptions of Dynamical Systems
• Discrete versus continuous
• Implicit versus explicit
• As nth term of a sequence versus nth difference
between terms
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)
Explicit Verses Implicit Equations
• Implicit Expression:
a (1)  1,
a (2)  1,
a (k )  a (k  1)  a (k  2)
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}
• Explicit Expression:

1  5   1  5 
f (k ) 
k
2
k
k
5
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}
Example
• Given the following sequence, find the explicit
and implicit descriptions:
1, 3, 5, 7, 9, 11,  
More Examples of Implicit Relations
I.
ak+1 = ak ∙ ak
II.
ak = 5
III. ak+2 = ak + ak+1
Constant Sequence
Fibonacci Sequence
Exercise I
Generate the first 5 terms of the sequence for
rule I given that a1=1.
I.
A(k+1)=A (k)*A (k)
Exercise I
Generate the first 5 terms of the sequence for
rule I given that a1=1.
I.
ak+1 = ak ∙ ak
Exercise I
Generate the first 3 terms of the sequence for
rule I given that a1=3.
I.
ak+1 = ak ∙ ak
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…
Exercise II
Generate the first 5 terms of the sequence for
rule II.
II.
ak=5
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
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
Descriptions of Dynamical Systems
• Discrete versus continuous
• Implicit versus explicit
• As nth term of a sequence versus nth
difference between terms
Modeling Change: Dynamical Systems
Difference equation:
describes change (denoted by ∆)
change=future value-present value
equivalently:  = x
n+1 – xn
change = future value – current value
… 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.
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.
2, 4, 6, 8,10,
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.
2, 4, 6, 8,10,
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.
2, 4, 6, 8,10,
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.
2, 4, 6, 8,10,
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.
2, 4, 6, 8,10,
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
an+1 = an+2
an = 2
More Examples of Implicit Relations
I.
ak+1 = ak ∙ ak
II.
ak = 5
III. ak+2 = ak + ak+1
Find the difference equation
description of each.
Constant Sequence
Fibonacci Sequence
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
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.)