Stability of solar systems

Game of rock, paper, scissor, lizard, spock.

Fractals

Motion of a charge particle in the field of two electrostatic waves.

This is to corroborate the previous answer with respect to solar system orbits and a zero-sum two player game

http://www.math.colostate.edu/~shipman/47/volume3b2011/M640_LaneHarvard_Schwagger.pdf

Coupled oscillators:

H=0.5*(px^2+py^2)+0.25*(x^4+y^4)+epsilon*x^2*y^2, where e

Another interesting Hamilitonian system is the three-particle Toda lattice. This system describes three particles moving on a ring with exponential interactions. If the particle masses are all the same, the system is integrable and both first integrals are known analytically. One is the total energy and the other is a fairly complicated quantity that reduces to the angular momentum in the limit of small displacements. If the masses are unequal, chaos can appear. This system is described in Chapter 4 of "Chaos and Integrability in Nonlinear Dynamics" by Michael Tabor.

Nose-Hoover oscillator. System without equilibria and hidden chaotic oscillations.

The three-body gravitational problem is one venerable example.

Perhaps the simplest such system, even simpler than Henon-Heiles, is the following:

x'' = xy

y'' = -y - x^2

(Eq. (6.22) in my "Elegant Chaos" book.)

The double pendulum system is one of my favorites.  It is simple to write down the equations, easy to build, and provides rich chaotic dynamics that are easy to visualize.  

Interestingly, the double pendulum equations are mathematically very similar to the three-body problem, where the trajectories are largely mediated by rank-1 saddle points.

Dear Sir,

Apart from double pendulum system and reversed pendulum, I recommend you the papers of dr. Ch. Skokos about chaos detection tools like SALI and GALI. One of the used models is a N-dimensional Hamiltonian, called Fermi - Pasta - Ulam lattice.  

I attach one of my papers about double pendulum and a lesser known indicator of order and chaos, called Fast Norm Vector Indicator.

Best regards.

Preprint The Universal Fractal Chirality of Riemann Zeta-Function: On...