Abstract

The aim of the project is to obtain a new approach to bifurcation theory. We study dynamical systems, in particular singular perturbations, fractal geometry, complex analysis, and numerical modeling of dynamical systems. Qualitative theory studies solutions of equations without explicitly solving the equations. These equations naturally depend on parameters, so we study the bifurcation theory, where bifurcation means a qualitative change in the behavior of a system. A system undergoing a bifurcation can change stability, which is a crucial problem in mathematical modeling.
The famous unsolved 16th Hilbert problem asks for a uniform upper bound on the number of limit cycles for a given polynomial vector field. Limit cycles can be born from bifurcations, and also in singular perturbations of the system. In this project our innovative approach is reading properties of bifurcations of dynamical systems from the measure A(fⲗ,ε), and from the fractal zeta functions ζ(fⲗ,ε) of ε-neighborhoods of orbits of the family fⲗ, depending on parameter ⲗ. A fractal dimension called the box dimension, is determined from the leading term of the asymptotic expansion of ε –> A(fⲗ,ε),) but other terms also give important information about the system. Complex analysis and the Mellin transform are the main tools in the construction of zeta functions ζ(fⲗ,ε).
Furthermore, new numerical methods for detecting bifurcations based on the fractal approach will be developed. These methods will include GPU based numerical computations, nonlinear optimization, and fine sampling of the bifurcation parameter space.
Limit cycles born in singular perturbations, often called canards, will be considered using box dimension and fractal zeta function methods to obtain the cyclicity of the canard cycles. The Canard phenomenon is the phenomenon where for a small parameter the limit cycle created in a Hopf bifurcation stays of “small size” for a while, but then it changes very rapidly to a “big size” cycle.