Chaos Theory
Chaos theory is the study of a particular type of systems that evolved from some initial conditions. A small perturbation in the initial setup of a chaotic system may lead to drastically different behavior, a concept popularly referred to as the butterfly effect from the idea that the actions of a butterfly may dramatically alter the physical state of the rest of the world. Although the behavior of chaotic systems may seem scattered and random, chaotic systems are strictly defined to be deterministic, meaning that a particular set of initial conditions always evolves in the same way.
Chaotic maps can be either discrete or continuous functions where slightly different initial values are gradually mapped further and further apart over time. Typically, they are given either in terms of recurrence relations for discrete maps, or in the time domain for continuous maps.
Applications of chaos theory are widespread across biology, chemistry, physics, economics, and mathematics, among other fields. Often, systems with a large number of coupled variables exhibit chaotic behavior, including weather systems, job markets, population dynamics, and celestial mechanics.
Conditions for Chaos
There are three required mathematical properties for a system to be classified as chaotic^{[1]}.
1) Sensitivity to initial conditions
2) Topological mixing
3) Density of periodic orbits
In certain cases, the second two imply the first mathematically. However, as discussed below, each of the three conditions captures different qualitative aspects of chaotic systems in general. Each is defined as a condition on the phase space of a dynamical system. In one dimension, the phase space is the two-dimensional space whose axes are the position \(x\) and velocity \(\dot{x}\) of a point; in higher dimensions, the axes are the positions and velocities in each possible direction.
Sensitivity to Initial Conditions
Suppose one has two sets of initial conditions \(z_0\) and \(z_0^{\prime}\) for a dynamical system, separated by a distance of \(\Delta z (t)\) in phase space, which may increase or decrease as the system evolves in \(t\). As the system evolves, the separation between the two initial states evolves in time as
\[\Delta z(t) = e^{\lambda t} \Delta z(0).\]
The exponent \(\lambda\) is called a Lyapunov exponent. In \(d\) spatial dimensions, the phase space is \(2d\)-dimensional, so there are \(2d\) Lyapunov exponents: one for each direction of separation of the states in phase space. Systems are often characterized by the largest of these exponents; if the largest exponent is positive, the separation grows exponentially in time (at least locally) and the system is chaotic. If the largest exponent is negative, the trajectories of the initial conditions \(z_0\) and \(z_0^{\prime}\) stay close in phase space, so closely separated trajectories are good approximations for each other and the system is not chaotic.
Topological Mixing
Sensitivity to initial conditions alone is not enough to make a map chaotic. For instance, consider the dynamical system generated by the map:
\[z_{n+1} = f(z_n) = 1.5z_n+1.\]
Consider two starting points \(z_0 = 0\) and \(z_0^{\prime} = 0.1\). The evolution of each point is displayed in the diagram below:
Since the map generating the system multiplies by \(1.5\) and then adds one, any small difference between starting points is magnified by a factor of \(1.5\) at each step. However, the system is not chaotic: regardless of the starting point, every point approaches positive or negative infinity, so the asymptotic behavior given a set of initial conditions is very predictable.
The topological mixing condition is designed to exclude such cases. It essentially says that given any possible set of states for the dynamical system, a given set of initial conditions for the system will eventually evolve to at least some of the states in the set. Topologically, this can be stated as follows: any open set in the phase space of the dynamical system eventually intersects any other given open set in the phase space.
Density of Periodic Orbits
Density of periodic orbits in a dynamical system means that any given point in phase space is arbitrarily close to a set of initial conditions that leads to a periodic orbit. This is an interesting condition because combined with topological mixing it implies sensitivity to initial conditions. Take two close initial conditions, and draw open sets around each initial condition in phase space such that the two open sets are disjoint. By topological mixing, these open sets eventually evolve to intersect any other given open set, i.e. they "smear out" over the rest of phase space over time. But if there are arbitrarily many periodic orbits within each of these open sets, this "smearing out" is only possible if the periodic orbits look drastically different. One way of looking at this is that any given point looks like it ought to be well-approximated by a nearby periodic orbit, but it is not, because the full set of periodic orbits must evolve to intersect any open set in phase space. If this is true of arbitrarily close initial conditions, the trajectories in phase space must diverge, since the nearby periodic orbits don't converge to the trajectories of the initial conditions.
Examples and Applications
Complex quadratic polynomials
A complex quadratic polynomial is a standard quadratic equation where the variable involved can be a complex number. A particularly simple example of this is the polynomial
\[f(z) = z^2 + c\]
for some constant \(c\). One can define a dynamical system from this map via the recursion \(z_{n+1} = f(z_n)\). In this case, the dynamical system defined is chaotic. In fact, given \(z_0 = 0\), the system diverges as \(n \to \infty\) except when \(c\) takes values on a fractal set, the Mandlebrot set.
Rössler attractor
The Rössler attractor refers to a dynamical system given by the following set of first-order ODEs in three dimensions:
\[\begin{array} &\frac{dx}{dt} = -y-z, &\frac{dy}{dt} &=x + ay, &\frac{dz}{dt} &= b+z(x-c)\end{array}\]
for \(a,b,c\) real parameters. It is similar to the well-known chaotic map called the Lorenz attractor, although mathematically simpler. Notably, the Rössler attractor has been used to study equilibria in reaction chemistry.
Double pendulum
The double pendulum is one of the simplest scenarios in physics where chaotic behavior is manifest. Hamilton's equations of motion for the double pendulum yield four coupled first-order ordinary differential equations, which is a sufficient condition for chaos. The below animation shows the highly unpredictable evolution of the double pendulum given a particular initial configuration.
Hénon map
The Hénon map is a discrete map on the plane given by the set of recursion relations
\[ \begin{align} x_{n+1} &= 1 - ax_n^2+y_n \\ y_{n+1} &= bx_n \end{align} \]
for some parameters \(a\) and \(b\). The choice of parameters \((a,b) = (1.4,0.3)\) is called the classical Hénon map, which exhibits chaotic behavior.
Coupled map lattice
In a map lattice, a discrete array of points are indexed and arranged in a lattice, and each site is able to evolve according to a particular type of recursion relation. In an uncoupled map lattice, each site evolves independently, e.g. by
\[x_{n+1} = rx_n ( 1-x_n),\]
where \(r\) is some parameter. In a coupled map lattice, the recursion for the \(i^\text{th}\) site depends not only on the previous value at that site but also the adjacent site:
\[(x_{n+1})_i = \epsilon \big(rx_n ( 1-x_n)\big)_i + (1 -\epsilon)\big(rx_n ( 1-x_n)\big)_{i-1}.\]
The parameter \(\epsilon\) gives the degree of the coupling; as \(\epsilon \to 1\) the map lattice becomes uncoupled, and as \(\epsilon \to 0\) the map is maximally coupled. Map lattices are interesting first because both uncoupled and coupled map lattices are chaotic even though coupled map lattices display much richer structure. Secondly, however, they have been used to model interactions between adjacent chemicals in space as well as electrical circuits.
Chaotic mixing
In chaotic mixing, quantities such as density, viscosity, or temperature that track the flow of a fluid mix in a fractal-like way. Such fluids are governed by a system of first-order ODEs. For a fluid governed by the Navier-Stokes equations in three dimensions, there are sufficient degrees of freedom for the fluid flow to be chaotic.
Ikeda map
The Ikeda map is a discrete complex map given by the recursion
\[z_{n+1} = A + Bz_n e^{i\big(|z_n|^2 + C\big)},\]
for some parameters \(A,B\), and \(C\). It is a model used in physics for a series of pulses of laser light interacting in a nonlinear medium called an optical resonator. The parameter \(B\) characterizes how lossy the resonator is.
Standard map
The standard map is a dynamical system defined as a recursion relation on the square, that is
\[ \begin{align} p_{n+1} &= p_n + K \sin (\theta_n) \\ \theta_{n+1} &= \theta_n + p_{n+1} \end{align} \]
with \(p_n\), \(\theta_n\) modulo \(2\pi\) and \(K\) some constant. It describes the momentum and angle of a particle constrained to a ring which experiences periodic kicks in a particular direction of strength \(K\), that is, a particle obeying the Hamiltonian
\[H = \frac{p^2}{2} + K\cos (x) \sum_n \delta (t - n)\]
with the extra constraint that the momentum be periodic. This system is called the kicked rotator and the constant \(K\) is called the kicking strength. It arises in the study of any periodically kicked systems and is thus particularly useful in physics with particles confined to a ring, such as accelerator or plasma physics.
References
- Hasselblatt, B., & Katok, A. (2003). A First Course in Dynamics: With a Panorama of Recent Developments. Cambridge University Press.