Chaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions.

1. Simulations

1.1 Infinite-period Bifurcation

\[\begin{align*} & \dot{r} = r(1 - r^{2}) \\ & \dot{\theta} = \mu - sin\theta \end{align*}\]

1.2 Lorenz Attractor

\[\begin{align*} & \dot{x} = a(y - x) \\ & \dot{y} = x(b - z) - y \\ & \dot{z} = xy - cz \end{align*}\]

One commonly used set of constants is a = 10, b = 28, c = 8 / 3. Another is a = 28, b = 46.92, c = 4. “a” is sometimes known as the Prandtl number and “b” the Rayleigh number.

  • $ 0 < \rho \le 1$ : 0 is the asymptotically stable critical point - the system converges to zero.
  • $ 1 < \rho < 13.926 $ : Depending on the initial values, the solution will quickly converge to one of the critical points.
  • $ \rho = 13.926 $ : homoclinic explosions. a complicated invariant set is born.
  • $ 13.926<\rho<24.06 $ : transient chaos, the system is not “chaotic”, because long-term behavior is not aperiodic. On the other hand the dynamics do exhibit sensitive dependence on initial conditions.
  • $ 24.06<\rho<24.74 $ : two types of attractors : fixed points and a strange attractor. (large perturbation can knock from one to another).
  • $ \rho > 24.74 $ : The famous Lorenz attractor!
  • $ \rho ≈ 99.65 $ : There is a stable 6-period orbit in the form $a^{2}ba^{2}b$.
  • $ \rho ≈ 100.75 $ : There is a stable 3-period orbit $a^{2}b$, that is it spirals around one critical point once and the other twice.
  • $ \rho > 313 $ : has a globally attraction limit cycle.

exhibit stretching and folding of trajectories

1.3 Logisitic Map

\[x_{n+1} = rx_{n}(1 - x_{n})\]

If the system’s Lorenz map is nearly one-dimensional and unimodal, then the universality theory applies.

2. Fractal

Demos:

  1. C has structure at arbitrarily small scales.
  2. C is self-similar.
  3. The dimension of C is not an integer.

3. Philosophy

Chaos - Stanford Encyclopedia of Philosophy

3.1 Quantum Chaos

Quantum Chaology: “the study of semiclassical, but nonclassical, phenomena characteristic of systems whose classical counterparts exhibit chaos”.

classical chaotic dynamics quantum dynamics
the state space supports fractal structure the state space does not support fractal structure
bounded macroscopic systems, continuous energy spectrum associated with its motion bounded, isolated systems, discrete energy spectrum associated with its motion

Some limitations :

  • Schrödinger’s equation is linear, quantum states starting out initially close remain just as close (in Hilbert space norm) throughout their evolution - no stretching and folding mechanism in the system space.
  • Phenomena such as SDIC (sensitive dependences on initial conditions) could only be possible in quantum systems that appropriately mirror classical system behaviors
  • Semi-classical quantum systems could be expected to mirror the behavior of their corresponding classical systems only up to the Ehrenfest time, of the order ln(2π/h) secs. (Berry et al. (1979))
    • two effects at work in semi-classical systems over time: (1) the coalescing of classical chaotic trajectories and (2) the spreading of quantum wave packets.

Some Broader Implications of Chaos:

  • Chaos and Determinism. chaos : unpredictability in deterministic system.
  • Free Will and Consciousness.
    • local context (particle trajectories) : quantum effects affecting the outcomes of human choices.
      • but how quantum mechanics itself and measurements are interpreted as well as the status of indeterminism?
    • global context : arrangements of the particles are the fundamental explanatory elements and not the individual particles and trajectories.
      • which are generally irreversible with respect to time.
      • nonlinear nonequilibrium models also exhibit SDIC.
  • Human and Divine Action in a Nonlinear World, apply NDS theory to complex behavior.