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%% 7.1 Chaos Theory
% There are basically two types of chaotic systems
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% (7‑1)
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% A simple Hamiltonian system is the free motion of a ball in a limited stadium-shaped area. At the boundary, the ball is elastically reflected. The smallest change in the start position leads to a completely different trajectory after about 10 reflections.
% A simple dissipative system is a driven harmonic oscillator with friction. Regardless of the starting position, the movement ends on a phase space ellipse.
% Such systems may result in attractors, fractals, and chaos.
% Example: Duffing oscillator (dissipative)
% This is a driven oscillator with friction and a two-trough potential.
%
% (7‑2)
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% Depending on the drive amplitude f, the oscillator moves in the phase space in one cycle, or in 2 cycles, 4 cycles, 8 cycles, etcetera. At a certain drive amplitude, adjacent trajectories diverge exponentially, as in billiards above.
% Deterministic chaos
% Deterministic chaos is when these two conditions are met:
%
%  (7‑3)
% Exponential divergence refers to the temporal behavior of the distance between two adjacent trajectories
%
% (7‑4)
% Attractors in dynamic systems
% The following definitions can be used to formulate the three properties of an attractor:
%
%
% (7‑5)
% Fractals
% A fractal structure consists of itself. For example, in the Cantor set, each line consists of two lines and a space in between. This applies to every sublevel, so that something arises between lines with spatial dimension 1 and points with dimension 0. The dimension is broken or fractal.
% Example: Cantor set
% The dimension D can be expressed by the scaling s and the number N of the substructures, in the case of the Cantor set by 3 and 2:
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% Strange attractor
% A strange attractor has all of the following properties:
%
% (7‑6)
% In the Duffing Oscillator for a sufficiently large drive amplitude f an example can be found. The fractal structure can be recognized qualitatively by zooming into the attractor more and more.
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% Different magnifications of the duffing oscillator in the space of
% position, speed and drive amplitude
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%% Time evolution of the phase space
% The temporal behavior of the phase space volume shows the essential difference between Hamiltonian and dissipative systems.
%
% (7‑7)
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%
% (7‑8)
%% Hamiltonian systems
% Invariance of the volume
% The invariance of the volume originates from Hamilton's equations of motion:
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% (7‑9)
% Integrability (no chaos)
% Hamiltonian systems are called integrable if they can be fully determined by the calculation of one-dimensional integrals.
% Under certain circumstances, multidimensional systems can be separated and therefore made integrable by choosing suitable generalized coordinates.
% For example, the Kepler problem
%
% (7‑10)
% Integration
%
% (7‑11)
% Other examples: Spherical pendulum, circular cone, heavy axis on an inclined plane, central force motion, heavy spinner, two-centre problem, swinging string of pearls.
%% Chaotic behavior in Hamiltonian systems
% Non-separable systems may show an exponential divergence of neighboring trajectories. So, they are chaotic.
% A sufficient condition for this behavior is the divergence of the classical perturbation theory. This explains for example the structure of the Saturn rings.
% One can also numerically show the exponential divergence of adjacent trajectories, for example for the stadium problem (free motion with boundary conditions).
%
% In the stadium problem, two closely adjacent trajectories diverge exponentially with
% time. On the left, the trajectories are shown in different time intervals. On the right, the
% logarithmic distance of the trajectories is shown as a function of time.
% Further examples
% Three-body problem (asteroids around sun, rings of Saturn, double pendulum, ...)