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   :format: latex


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=============
Elegant Chaos
=============


Introduction
============

One of the most intriguing books on chaotic systems is
:raw-latex:`\cite{sprott}` which is a true treasure trove for everybody
fascinated by these mathematical objects. In the following, we will
implement two of these systems on an analog computer.

A jerk system
=============

So-called *jerk system*\ s are special cases of *autonomous dissipative
systems*.The system under consideration in the following is described by
the following differential equation:

.. math:: \dddot{x}=-\ddot{x}+9\dot{x}-x^2\dot{x}-5x

Scaling chaotic systems for implementation on an analog computer is
typically a bit nasty which holds true for this particular case, too. A
few numerical calculations using a simple Euler integration routine show
that the term :math:`x^2\dot{x}` has a domain exceeding
:math:`[-10^3:10^3]` etc. Scaling is done by scaling down all inputs of
one integrator by the same scale factor :math:`\lambda_i` until no
overload occurs anymore. Its output signal is then scaled back by
:math:`\frac{1}{\lambda_i}` before being fed to the next integrator,
which is then scaled as well etc. Table 1 shows the values
for the six coefficient potentiometers required in the computer setup
shown in figure 1. The left
column shows the parameters obtained by scaling “by the book” while the
right column shows a parameter set that was obtained by tweaking some of
the parameters manually. It is remarkable that the behaviour of the
analog simulation shown in figure 2 is much more tame over a rather
wide range of parameters than the numerical results shown in
[Sprott 2010, p. 77]. Nevertheless, it is possible to find
regions that exhibit chaotic behavior by some manual parameter
variation.

.. table:: Table1: Potentiometer settings for the jerk system

   ============= ========== =================
   Potentiometer Calculated Manually obtained
   ============= ========== =================
   P1            0.1        0.1
   P2            0.75       0.75
   P3            0.12       0.17
   P4            0.2        0.2
   P5            0.333      0.1
   P6            0.333      1
   ============= ========== =================

.. list-table::
   :widths: 75 75
   :header-rows: 0

   * - .. figure:: circuit01.png
   	      :width: 400
              :alt: Setup for the jerk system
     - .. figure:: jerk_result.jpg
   	      :width: 400
              :alt: Typical behavior of the jerk system
     
   * - Figure 1: Setup for the jerk system

     - Figure 2: Typical behavior of the jerk system

		

Nosé-Hoover Oscillator
======================

Another beutiful chaotic system is the Nosé-Hoover oscillator, an
example of an autonomous conservative system which is described by
the following set of coupled differential equations:

.. math::

   \begin{aligned}
      \dot{x}&=y\\
      \dot{y}&=yz-x\\
      \dot{z}&=1-y^2
     \end{aligned}

Scaling this system is quite mean, too, and yields the computer setup
shown in figure 3, the results of
which are shown in the mesmerizing oscilloscope screen capture shown in
figure 4.
 		
                   
.. list-table::
   :widths: 75 75
   :header-rows: 0

   * - .. figure:: circuit02.png
   	      :width: 400
              :alt: Setup for the Nosé-Hoover oscillator
     - .. figure:: nose_hoover_result.jpg
   	      :width: 400
              :alt: Phase state plot of the Nosé-Hoover oscillator
     
   * - Figure 3: Setup for the Nosé-Hoover oscillator

     - Figure 4: Phase state plot of the Nosé-Hoover oscillator


References
==========

[SPROTT 2011] Julien Clinton Sprott, *Elegant Chaos – Algebraically Simple Chaotic
Flows*, World Scientific Publishing Co. Pte. Ltd, 2010