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Introductory article on the physics of Chaos

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THE CHAOTIC ORDER
“In all chaos there is a cosmos, in all disorder a secret order.”
- Carl Jung
Chaos
–
a word, rather an ancient word srcinally denoting a complete lack of form or systematic arrangement, but now too often misused, is understood to imply the absence of some kind of order that ought to be present; conjuring up images of real world examples where
nothing but the ‘chaotic order’ prevails. But what is this chaotic order and what exactly, for the
sake of classical physics, is chaos?
Let’s
see where this hackneyed view of chaos falls short and establishes order out of chaos.
The Butterfly Effect
“A butterfly sitting on the coast of Brazil flaps its wings and 5,000 miles away, a tornado sets off in Texas.”
Seems a bunch of baloney, isn’t it?
Inspired by this idea, in 1971 Edward Lorenz presented his ground-
breaking paper titled “
Does
the flap of a butterfly’s wings in Brazil set off a tornado in Texas?”
to spark life back into the chaotic notions in science. A popular slogan in Science, Butt
erfly Effect, simply put is “minor details having major impacts”,
or an effect by virtue of which a proverbial minor detail can contribute to a world differing massively from the one without it.
In mathematics, one can call this “sensitive dependence” on i
nitial conditions, where a small change at one place in a deterministic non-linear system can result in large differences in a later state.
Deterministic System
A deterministic system is a system whose present state is in principle fully determined by its initial conditions. These kinds of systems also involve no randomness to predict the future states of the system. Hence, if one knows a certain set of initial conditions then the time-evolution of the system can be fully predicted using these sets of initial conditions. In quantum mechanics, the Schrodinger wave equation, which describes the continuous time
evolution of a system’s wave function, is deterministic whereas the nature of quantum
mechanics (quantum indeterminacy) is in itself, and is non-deterministic.
Quantification of Sensitivity to Initial Conditions
Before we quantize the sensitivity of a deterministic system to initial conditions, let us understand the notions of dynamical Hamiltonian systems. A dynamical system
is a ‘rule’ for time evolution on a state space (the set of all possible states
in a dynamical system). It consists of a phase space or state space, whose coordinates describe the state at any instant, and a dynamical rule that specifies the immediate future of all state variables, given only the present values of these state variables. For example,
the state of a pendulum is its angle and angular velocity, and the evolution rule is Newton's equation
. A Hamiltonian system is simply a dynamical syste
m governed by Hamilton’s equations of
motions. Canonical Coordinate
()
where these are obtained from the generalized coordinates of the Lagrangian formalism by a Legendre Transform. Then, the time-evolution of the system is uniquely defined by Hamilt
on’s equations:
, where
()
is the Hamiltonian of the system, and which reduces to the sum of kinetic energy and potential energy for a closed system.
Liouville’s Theorem:
The phase-space distribution function is constant along the trajectories of the system
—
i.e. that the density of system points in the vicinity of a given system point travelling through phase-space is constant with time. This distribution in phase-space can be described by the density
()
.
Then, according to Liouville’s Theorem,
()
The distribution of points in phase space evolves like an incompressible fluid.
In chaotic dynamics, we are interested in Hamiltonian dynamical systems, i.e. non dissipative systems where, by Liouville's theorem, the volume in phase space of some initial distribution is conserved. If we have a system with N degrees of freedom, then its trajectories move through a2N dimensional phase space consisting of N coordinates and N corresponding canonically conjugate momenta. The rate of trajectories drifting apart in phase space can be measured by the Lyapunov exponent
, formally
()
[|()()||()()|]
The 2N dimensional vectors u (t), v (t) describe the values of all the coordinates
*
+
and momenta
*
+
at time t starting from two neighbouring initial conditions u(0) and
v(0), evolving according to Hamilton’s equations of motions.
Inverting the defining equation for the Lyapunov Exponent produces the equation for the growth of the distance in phase space:
|()()|
|()()|
Defining
()()()
and
()()()
, we have
|()|
|()|
Where
, the mean rate of separation of trajectories of the system is called the Lyapunov Exponent. For any finite accuracy,
()
of the initial data, the dynamics is predictable only up to a finite Lyapunov Time.
In the limit of inﬁnite time, the Lyapunov exponent is a global measure of the rate at which
nearby trajectories diverge, averaged over the strange attractor. The Lyapunov exponent
λ *lambda+ is useful for distinguishing among the various types of
orbits. It works for discrete as well as continuous systems.
λ
< 0 [lambda] The orbit attracts to a stable fixed point or stable periodic orbit. Negative Lyapunov exponents are characteristic of dissipative or non-conservative systems (the damped harmonic oscillator for instance). Such systems exhibit asymptotic stability; the more negative the exponent, the greater the stability. Super stable fixed points
and super stable periodic points have a Lyapunov exponent of
. This is something akin to a critically damped oscillator in that the system heads towards its equilibrium point as quickly as possible.
λ
= 0 [lambda] The orbit is a neutral fixed point (or an eventually fixed point). A Lyapunov exponent of zero indicates that the system is in some sort of steady state mode. A physical system with this exponent is conservative. Such systems exhibit Lyapunov stability. Take the case of two identical simple harmonic oscillators with different amplitudes. Because the frequency is independent of the amplitude, a phase portrait of the two oscillators would be a pair of concentric circles. The orbits in this situation would maintain a constant separation, like two flecks of dust fixed in place on a rotating record.
λ
> 0 [lambda] The orbit is unstable and chaotic. Nearby points, no matter how close, will diverge to any arbitrary separation. All neighborhoods in the phase space will eventually be visited. These points are said to be unstable. For a discrete system, the orbits will look like snow on a television set. This does not preclude any organization as a pattern may emerge. Thus the snow may be a bit lumpy. For a continuous system, the phase space would be a tangled sea of wavy lines like a pot of spaghetti. A physical example can be found in Brownian motion. Although the system is deterministic, there is no order to the orbit that ensues. Some orbits with Lyapunov Exponents Conclusion: If a deterministic system is locally unstable (Positive Lyapunov Exponent) and globally mixing (Positive Entropy), it is said to be chaotic.
Lorenz Equations
In 1963, Edward Lorenz derived a three-dimensional system from a drastically simplified model of convection rolls in the atmosphere.

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