Dynamical Billiards

A Demonstration of Chaos Theory

By the Billiard Ballers

Chaotic Systems History

• Began in 1880s, with Henri Poincare study of 3-body systems (found there could be non-periodic motion that does not tend towards a fixed point)
• Edward Lorenz was an early pioneer, studying weather systems
• Began to increase in popularity when electronic computers were invented
• Gained interest in field because computer simulations’ roundoff errors caused large variation in simulation results

For Phoenix.

Overview of Chaotic Systems: A Descent into Chaos

In order for a Dynamical system to be classified as chaotic it must:

1. Be sensitive to initial conditions
2. Be topologically mixing
3. Have dense periodic orbits

Stephen for next few slides

• Topologically transitive?

Linear -> non-linear -> chaotic

Sensitivity to Initial Conditions

Non-chaotic System

• A driven damped linear oscillator approaches a unique attractor regardless of the initial conditions.
• The difference in solutions Z(x,t), with differing initial conditions, can be expressed as:$$\mid \Delta Z \mid=e^{ƛ t} \mid \Delta Z_0 \mid$$
• As $t \rightarrow \infty$ the transient solutions die out, so ƛ must be negative
• Any errors in the initial conditions will decay

Sensitivity to Initial Conditions

• Nonlinear dynamic systems can have many attractors, each of which depends on the initial conditions.
• The difference in solutions of nonlinear systems with differing initial conditions, can also be expressed as:$$\mid \Delta Z \mid=e^{ƛ t} \mid \Delta Z_0 \mid$$
• As $t \rightarrow \infty$ the, ƛ no longer must be negative
• Chaotic systems are all nonlinear

Sensitivity to Initial Conditions

• ƛ is called the Lyapunov exponent
• ƛ is positive in a chaotic system
• So solutions with different initial conditions will diverge exponentially as time passes
• So any errors in the initial conditions will grow exponentially with time; this is what is meant by sensitivity to initial conditions

Topologically Mixing

Definition: A dynamical system $f$ on a phase space $X$ is said to be topologically mixing if for every two open sets $A$ and $B$ in $X$, there is $N$ sufficiently large such that $f^n(A)\cap B \neq \varnothing$ for all $n \geq N$.

• By iterating $f$ over any subset of the phase space we can make two different subsets eventually overlap
• This means that any two different trajectories will become arbitrarily close regardless of initial conditions

Dense Periodic Orbits

• Any orbits appearing to be periodic are actually arbitrarily close to, but not repeating
• The union of the sets of all these orbits forms the phase space

Billiards

Physics of the Simulation

• Can be represented as by using a infinite potential well
• Infinite on the boundary and zero inside
• No loss of speed due to friction

• Specular reflection

• Can use either Hamiltonian or Lagrangian

Physics of the Simulation: Example of a Square

• We can approximate the potential of square well as $U(x,y)=x^n+y^n$ ,$\lim_{n\to\infty}$ with $T=\frac{1}{2}m(\dot{x}^2+\dot{y}^2)$

Lagrangian

• This makes the Lagrangian: $\mathcal{L}=\frac{1}{2}m(\dot{x}^2+\dot{y}^2)-(x^n+y^n)$ which gives two independant equations: $$\ddot{x}=\frac{1}{m}nx^{n-1}$$ $$\ddot{y}=\frac{1}{m}ny^{n-1}$$

Hamiltonian

• From Lagrangian can define Hamiltonian $\mathcal{H}=\frac{1}{2m}(P_x^2+P_y^2)+x^n+y^n$

• This gives four coupled equations: $$\dot{x}=\frac{P_x}{m}, \dot{y}=\frac{P_x}{m}$$ $$\dot{P_x}=nx^{n-1}, \dot{P_y}=ny^{n-1}$$

• Both of these methods can be solved either analytically or numerically for some large $n$

Physics of the Simulation: Example of a Square

• Using Matlab these solution spaces were found with the Lagrangian formalism $n$=12

Physics of the Simulation: Example of a Square

• Using Matlab these solution spaces were found with the Lagrangian formalism $n$=24

Simulation Solutions

• Instead of solving the equations of motion in our simulation, graphing methods were used
• Plot a straight line and solve for the intersection with the boundary
• Determine the velocity vectors in the tangent and normal basis
• Flip the sign of the normal velocity component
• Repeat

Simulation Solutions: Example of Square

• A square of a side length $2a$ centered at the origin will be our potential
• If the particle starts at the origin with velocity $V_x$ and $V_y$, the motion is represented as: $$y=\frac{V_y}{V_x}x$$
• If $V_y>V_x$ then the next position will be: $$x_1=\frac{V_x}{V_y}a$$
• This can be done iteratively to find the graphical solutions: $$x_6=-V_xt+a$$
  

Ryan, Henry, Phoenix, Ryan?

Rectangle Stadium

Periodic Orbits

Rational Velocities

Rectangle Stadium

Dense Periodic Orbits

Irrational Velocities

Rectangle Stadium

Sensitivity to Initial Conditions

Rectangle Stadium

Topological Mixing

Circle Stadium

Non-periodic Orbits

Circle Stadium

Periodic Motion

Circle Stadium

Sensitivity to Initial Conditions

Circle Stadium

Topological Mixing

Bunimovich Stadium

Sensitivity to Initial Conditions

Bunimovich Stadium

Topological Mixing

Lorentz Gas

The billiard arises from studying the behavior of two interacting disks bouncing inside a square, reflecting off the boundaries of the square and off each other.

Sensitivity to Initial Conditions

Lorentz Gas

Topological Mixing

Possibly move GUI to near beginning?

GUI