MA354

Dynamical SystemsT H 2:30 pm– 3:45 pm

Dr. Audi Byrne

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 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

{5, 5, 5, 5, 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)

{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.

Difference Equation

= change

= new – old

= xn+1 – xn

… 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.

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.)