Hidden Content Control Panel: Reveal All | Hide All

S6.7 Chaos

This section is included for general interest. It is not part of the examinable programme of the course.

A tour

Reveal
Hide

A tour

Preamble: keeping it short and simple

Of necessity this will be a brief excursion into chaos. It falls into two parts.

We will set the scene through a case study of one particular system –an interesting pendulum.

Why this system?

Why this system?

We have chosen this particular system for the case study because we built it here a few years ago and it is available to use as a ‘real’ lecture demonstration: it is important to appreciate that the phenomena of interest here are visisble in real systems, not just computer simulations. We also wanted to remind you that, contary perhaps to your eperience to date, a pendulum can actually be interesting!
 

We will then set out some of the key ideas by considering some of the FAQs (frequently asked questions) about chaos.

[A] Chaos: a case study

  • The system is a ’spherical’ pendulum.
  • It is free to swing in x and y directions at natural frequency ω
  • It is damped
  • It is driven in the x direction, at tunable frequency ωD
  • Its amplitude of oscillation is not always small
Graphic - No title - spherpend

What we find from observation:

  • The non-resonant behaviour (that is, the behaviour when ωD is far from ω) is simple: it is just damped SHM in the x direction
  • The near-resonant behaviour is complex:
    Graphic - No title - spherpendorbits
    Graphic - No title - spherpendorbitsfull
    • The amplitude of motion is bigger
    • The motion in the x direction is unstable  –even if we start the system off oscillating in the x direction it begins to exhibit y-oscillations also, spontaneously.
    • The resulting motion is a complex mixture of x and y oscillations.
    • This behaviour changes in an irregular fashion – a series of snapshots of the orbits of the bob in the x-y plane show seemingly random changes in sense, beween clockwise and anticlockwise.
    • There is extreme sensitivity to the initial conditions –even a small change in how we start the pendulum off results in completely different forms of behaviour after a few cycles of the driving force.

This is an instance of chaos

Examples of Chaos in Pendulums

[B] Chaos: some FAQs

Frequently Asked Question: What is chaos?

Answer:

Chaos is the complex irregular-looking behaviour exhibited by systems which might be expected to behave simply

Frequently Asked Question: Where do we find chaos?

Answer:

In systems which evolve in time according to non-linear equations.

Example: the EOM of a planar pendulum

  • For small amplitudes one may use
    \[ \frac{d^2\theta}{dt^2} \simeq -\frac {g}{l} \theta \]
    …a linear equation
  • For larger amplitudes one must use

    \[ \frac{d^2\theta}{dt^2} = -\frac{g}{l} \times \sin \theta =-\frac{g}{l} \times [ \theta -\frac{1}{6} \theta ^3 + \ldots ] \]

    …a non-linear equation

Frequently Asked Question: How wide-spread is ’non-linearity’?

Answer:

It is the rule rather than the exception

Referring to the area of Non-linear Physics is like referring to Zoology as the study
of Non-elephant Animals

[Stanislaw Ulam]

Frequently Asked Question: Why did it take so long to ‘discover’ chaos?

Answer:

We had to wait for computers! Most pen-and paper methods start from the assumption that the effects of non-linearity are small, and may be treated as a correction to some underlying linear behaviour. Such methods fail to signal their own limitations. It needed the arrival of computers (a calculator will do, when you know where to look) to see that non-linearity changes the picture entirely.

Frequently Asked Question: In what areas of science has chaos been found?

Answer:

Virtually everwhere …

Frequently Asked Question: In what sense are chaotic systems ’unpredictable’?

Answer:

The future is ‘unpredictable’ because the ‘present’ is not knowable with infinite precision.

Frequently Asked Question: How does a system become chaotic?

Answer:

A typical scenario:

  • We begin with system exhibiting ’simple’ behaviour
  • We change a parameter enhancing non-linearity
  • The simple behaviour evolves through sequence of instabilities into chaos

Examples

Frequently Asked Question: How can we characterize the future of a chaotic system?

Answer:

In the language of probability

Graphic - No title - presents