Differential Equations from a Dynamical Systems Point of View
This research concerns the chaos involved in nonlinear dynamical systems. To gain
some understanding of nonlinear dynamical systems, we began our study with linear
systems including homogeneous and non-homogeneous LRC (inductor-resistance-capacitor) circuits. In the latter case, we explored the phenomenon of resonance
in a forced system. We then used MATHEMATICA to develop a complete classification
including graphical representations of all possible dynamics to second order
linear systems. With this information, we were able to depict the bifurcation
diagram for these solutions. Then we moved on to LRC circuits including a
semiconductor to add an element of nonlinearity. This closely modeled the Van
der Pol equation. With the use of Jacobian and other methods, we linearized
these systems near the equilibrium points to get a qualitative concept of the
systems. The next nonlinear system that we explored was the forced pendulum which
has chaotic behaviour in certain regions. We again used MATHEMATICA to understand
the periodic and chaotic behavior of the system and find the Lyapunov exponent.
Our future research will involve methods of controlling the chaos by steering it
into perodic orbits and use the model to control the noise of communication system.
This research is funded by NASA LEWIS CENTER.
