- Project Background:
Dynamical systems generated by difference equations can exhibit a
range of dynamical behaviors, from fixed points to periodic orbits to
strange attractors to chaos.
Part of the definition of chaos is the idea known as sensitive
dependence to initial conditions.
In brief this means that if the same dynamical system is given two
different, but close to one another, initial conditions then as the
two orbits evolve they will separate by an arbitrarily large distance.
Yet the numerical precision of calculations done in the service of an
orbit can yield dynamics which appear non-chaotic while if the same
calculation were carried out exactly, a chaotic orbit would result.
- Project Description:
The student will study the behavior of difference equations such as
the logistic equation.
In particular the investigation will address the issue of the
relationship between sensitive dependence to initial conditions,
numerical precision, and coalescence (the behavior that results when
two orbits which have different initial conditions eventually have the
same orbit).
- Desired Student Background:
Ability to program in a high-level language, preferably Mathematica, C/C++, or FORTRAN, and mathematical knowledge at the
level of MATH 365 Ordinary Differential Equations.