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

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.

system: x1, x2, x3, … (states as time increases)

Implicit relation: xn+1 = f(xn)

Source: Wikipedia

17th century

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

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

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 the static parametersof the system?)

What are the4 changing variables (state variables) that the systemdepends upon?

Must completely describe the system at time t.

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:

change = future value – current valuefxfxxf )()(

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

Difference equation:

describes change (denoted by ∆)

equivalently:

change = future value – current value

change=future value-present value

= xn+1 – xn

Describing Change (Discrete verses Continuous)

• Discrete description: Difference Equation

• Continuous description: Differential Equation

)()( xfxxff

fxfxxf )()(

t

xftxfxf

t

)()(lim)(

0

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)

Explicit Verses Implicit Equations

• Implicit Expression:

• Explicit Expression:

52

5151)(

k

kk

kf

)2()1()(

,1)2(

,1)1(

nanana

a

a

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.

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}

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}

Example

• Given the following sequence, find the explicit and implicit descriptions:

,11,9,7,5,3,1

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 rule I given that A (1)=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)

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)

Exercise II

Generate the first 5 terms of the sequence for rule II.

II. A(k)=5

Exercise II

Generate the first 5 terms of the sequence for rule II.

II. A(k)=5

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)

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.)

MA354

Difference Equations(Homework Problem Example)

… 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.

,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”. (This is the ‘Markov property’.)

Counter-example: Fibonacci sequence