MLAP 31500

**Natural
Sciences Elective**

**Order and
Chaos in the Natural World**

Spring Quarter 2014

Glossary
of Terms

Adapted from Moon, F. C.
1987, *Chaotic
Vibrations* (New York: John Wiley & Sons).

**Attractor:**
A set of points or a subspace in phase space toward which a time history
approaches after transients die out.
For example, equilibrium points or fixed points in maps, limit cycles,
or a toroidal surface of section for quasiperiodic motions, are all classical attractors.

**Cantor set:**
Formally, a set of points obtained on a unit interval by throwing out the
middle third and iterating this operation on the remaining intervals. This operation, when carried to the
limit leads to a fractal set of points on the line with dimension (ln2/ln3).

**Chaotic:**
Denotes a type of motion that is sensitive to changes in initial
conditions. A
motion for which trajectories starting from slightly different conditions
diverge exponentially with time.
A motion with positive Lyapunov
exponent.

**Equilibrium point:** In a continuous dynamical
system, a point in phase space toward which a solution may approach as
transients decay. In mechanical
systems, this usually means a state of zero acceleration and velocity. For maps, equilibrium points may come
in a finite set where the system visits each point in a sequential manner as
the map or difference equation is iterated. Also called a fixed point.

**Feigenbaum**** number:** A
property of a dynamical system related to the period-doubling sequence. The ratio of successive differences
between period-doubling bifurcation parameters approaches the number
4.669É. This property and the Feigenbaum number have been discovered in many physical
systems in the prechaotic regime.

**Fractal:** A geometric property of a set of points
in an *n*-dimensional
space having the quality of self-similarity at different length scales and
having noninteger fractal dimension less than *n*.

**Fractal dimension:** The fractal dimension is a
quantitative property of a set of points in an *n*-dimensional space
which measures the extent to which the points fill a subspace as the
number becomes very large.

**Henon**** map:** A set of two coupled
difference equations with one quadratic nonlinearity. When one parameter is set equal to
zero, the equations resemble the logistic or quadratic map.

**Horseshoe map:** A map of the plane onto
the plane. Points in the lower
half of a rectangular domain are stretched and contracted and mapped into a
vertical strip in a section of the left-hand plane, while points in the upper
half are stretched and contracted and mapped onto a strip in the right
half-plane. The process is like
transforming a rectangular domain into a horseshoe shaped set of points, hence
the name. Similar
to the baker's transformation.
Repeated iterations can yield a fractal-like set of points.

**Intermittency:** A type of chaotic motion
in which long time intervals of regular, periodic or stationary dynamical
motion are followed by short bursts of randomlike
motion. The time between bursts is
not fixed but is unpredictable.

**KAM theory:** The initials stand for the theorists Kolmogorov, Arnold, and Moser who developed a theory
regarding the existence of periodic or quasiperiodic
motions in nonlinear Hamiltonian systems (i.e., systems that have no
dissipation and in which the forces can be derived from a potential). The theory states that if small
nonlinearities are added to a linear systems, the
regular motions will continue to exist.

**Limit cycle:** In the engineering literature, a
periodic motion that arises from a self-excited or autonomous system as in
electrical oscillations. In the dynamical systems literature, it
also includes forced periodic motions.

**Lorenz equations:** A set of three first-order
autonomous differential equations that exhibit chaotic solutions. The equations were derived and studied
by E. N. Lorenz as a model of atmospheric convection. This set of equations is one of the principal paradigms for
chaotic dynamics.

**Mandelbrot set:** If *z* is a complex variable, the quadratic
map *z*
-> *z*^{2}
+ *c* has
more than one attractor. Fixing
the initial conditions, one can vary the complex parameter *c* to determine the basin of attraction
as a function of *c*. The basin boundary is fractal, and the
basin is known as the Mandelbrot set.

**Map, mapping:** A mathematical rule that
takes a collection of points in some *n*-dimensional space and maps them into another
set of points. When the rule is
iterated, a map is similar to a set of difference equations.

**Period doubling:** Refers to a sequence of
periodic vibrations in which the period doubles as some parameter in the system
is varied. In the classic model,
these frequency-halving bifurcations occur at smaller and smaller intervals of
the parameter. Beyond a critical
value of the parameter, chaotic vibrations occur. This scenario of chaos has been observed in many physical
systems, but it is not the only road to chaos.

**Phase Space:** In mechanics, phase space is an abstract
mathematical space whose coordinates are generalized coordinates and
generalized momenta. In dynamical systems governed by a set of first-order
evolution equations, the coordinates are the state variables or components of
the state vector.

**Poincare section (map):** The
sequence of points in phase space generated by the penetration of a continuous
evolution trajectory through a generalized surface of plane in the space. For a periodically forced, second-order
nonlinear oscillator, a Poincare map can be obtained by stroboscopically
observing the position and velocity at a particular phase of the forcing
function.

**Quasiperiodic****:** A vibration motion
consisting of two or more incommensurate
frequencies. (Two frequencies are
said to be incommensurate if their ratio can is not equal to the ratio of two
integers.)

**Rayleigh-Benard convection:**
Circulatory patterns of motion in a fluid produced by a thermal gradient
and gravitational forces. The
chaos model of Lorenz attempted to simulate some of the dynamics of thermal
convection.

**Renormalization:** A mathematical theory in
functional analysis (a branch of mathematics) in which properties of some
mathematical set of equations at one scale can be related to those at another
scale by a suitable change of variables.
Developed by Nobel Prize winning physicist K. Wilson. Used in the theory of quadratic maps to
derive the Feigenbaum number.

**Self-similarity:** A property of a set of
points in which geometric structure on one length scale is similar to that on another
length scale.

**Strange attractor:** Refers to the attracting
set in phase space on which chaotic orbits move. An attractor that is not an equilibrium
point nor a limit cycle, nor a quasiperiodic
attractor. An attractor in phase space with fractal dimension.

**Surface of section:** See *Poincare section*.

**Taylor-Couette flow:** The
flow of a fluid between two rotating, concentric cylinders.

**Torus (invariant); The** coupled motion of two undamped
oscillators is imagined to take place on the surface of a torus, with the
circular motion around the small radius representing the oscillatory vibration
of one oscillator and motion around the large radius representing the other
oscillator. If the motion is
periodic, then a closed helical trajectory will wind around the torus. If the motion is quasiperiodic,
then the orbit will come close to all points on the torus.

**Universal property (universality):** A
property of a dynamical system that remains unchanged for a certain class of
nonlinear problems. For example,
the Feigenbaum number relating the sequence of
bifurcation parameters in period doubling is the same for a certain class of
nonlinear, noninvertible, one-dimensional maps.

**Van der Pol
equation:** A second-order differential equation with linear restoring
force and nonlinear damping which exhibits a limit cycle behavior. The classical
mathematical paradigm for self-excited oscillations.

**LINKS:**

Return to Course Page: mla315spring2013.html

Return to Peter Vandervoort's Home Page: pov.html

Go to the home page of the Department of Astronomy and Astrophysics

of the
University of Chicago: __ http://astro.uchicago.edu/__