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MA354. 1.1 Dynamical Systems MODELING CHANGE. 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. - PowerPoint PPT PresentationTRANSCRIPT
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MA354
1.1 Dynamical Systems
MODELING CHANGE
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Modeling Change: Dynamical Systems
A dynamical system is a changing system.
Definition
Dynamic: marked by continuous and productive activity or change
(Merriam Webster)
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Modeling Change: Dynamical Systems
A dynamical system is a changing system.
Definition
Dynamic: marked by continuous and productive activity or change
(Merriam Webster)
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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.
system: x1, x2, x3, … (states as time increases)
Implicit relation: xn+1 = f(xn)
Source: Wikipedia
17th century
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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
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• 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.
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A Classic Dynamical System
The double pendulum
The model tracks the velocities and positions of the two masses. Source: Wikipedia
Evidences rich dynamical behavior, including chaotic behavior for some parameters.
Motion described by coupled ODEs.
Source: math.uwaterloo
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The Double Pendulum
Chaotic: sensitive dependence upon initial conditions
Source: math.uwaterloo
These two pendulums start out with slightly different initial velocities.
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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
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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.
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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.
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The double pendulum
State space: 4 dimensional
(What are the static parametersof the system?)
What are the4 changing variables (state variables) that the systemdepends upon?
Must completely describe the system at time t.
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Modeling Change: Dynamical Systems
From your book:
‘Powerful paradigm’
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Modeling Change: Dynamical Systems
Powerful paradigm:
future value = present value + change
equivalently:
change = future value – current value
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Modeling Change: Dynamical Systems
Powerful paradigm:
future value = present value + change
equivalently:
change = future value – current valuefxfxxf )()(
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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
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Modeling Change: Dynamical Systems
Difference equation:
describes change (denoted by ∆)
equivalently:
change = future value – current value
change=future value-present value
= xn+1 – xn
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Describing Change (Discrete verses Continuous)
• Discrete description: Difference Equation
• Continuous description: Differential Equation
)()( xfxxff
fxfxxf )()(
t
xftxfxf
t
)()(lim)(
0
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Inplicit 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 describe how the function is changing, rather than the function
explicitly)
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Explicit Verses Implicit Equations
• Implicit Expression:
• Explicit Expression:
52
5151)(
k
kk
kf
)2()1()(
,1)2(
,1)1(
nanana
a
a
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Explicit Verses Implicit Equations
• Implicit Expression:
• Explicit Expression:
52
5151)(
k
kk
kf
)2()1()(
,1)2(
,1)1(
kakaka
a
a To find the nth term, you must calculate the first (n-1) terms.
To find the nth term, you simply plug in n and make a single computation.
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Explicit Verses Implicit Equations
• Implicit Expression:
• Explicit Expression:
52
5151)(
k
kk
kf
)2()1()(
,1)2(
,1)1(
kakaka
a
a To find the nth term, you must calculate the first (n-1) terms.
To find the nth term, you simply plug in n and make a single computation.
First 10 terms:{1,1,2,3,5,8,13,21,34,55}
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Explicit Verses Implicit Equations
• Implicit Expression:
• Explicit Expression:
52
5151)(
k
kk
kf
)2()1()(
,1)2(
,1)1(
kakaka
a
a To find the nth term, you must calculate the first (n-1) terms.
To find the nth term, you simply plug in n and make a single computation.
First 10 terms:{1,1,2,3,5,8,13,21,34,55}
First 10 terms:{1,1,2,3,5,8,13,21.0,34.0,55.0}
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Example
• Given the following sequence, find the explicit and implicit descriptions:
,11,9,7,5,3,1
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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
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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)
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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)
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Exercise I
Generate the first 3 terms of the sequence for rule I given that A (1)=3.
I. A(k+1)=A (k)*A (k)
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Exercise II
Generate the first 5 terms of the sequence for rule II.
II. A(k)=5
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Exercise II
Generate the first 5 terms of the sequence for rule II.
II. A(k)=5
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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)
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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)
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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.)
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MA354
Difference Equations(Homework Problem Example)
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… 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.
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Homework Assignment 1.1
• Problems 1-4, 7-8.
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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.
,10,8,6,4,2
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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
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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
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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
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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
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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