Introduction

This project consists of the analysis of a four dimensional model. The model that we analyzed is supposed to represent the oscillations of a shock absorber that is being forced with an oscillating forcing term. Although it is possible to represent such a system with a system of two non-autonomous, we choose to represent it as a system of four dimensional autonomus equations. The first two variables (x and y) are analagous to the system for an unforced oscillator (i.e. x is the position and y is the velocity of the oscillator). The other two variable (z and w) represent the forcing term. The parameter A is the amplitude an omega is the angular frequency of the forcing oscillation. The parameters a, b, and lambda depend on the forcing term of the oscillator while c1 and c2 are constants which affect the damping of the oscillator.

In our analysis, we found that the equations for z'(t) and w'(t) could be decoupled from the first two equations and that the solutions to this uncoupled system was:
for lambda > 0 and that z(t) and w(t) decay as t increases for lambda < 0.

We also discovered that the origin acted as a source when lambda > 0 and c1 < 0 and as a sink when lambda < 0 and c1 > 0. When c1 and lambda have the same sign, interesting behavior occurs. In these cases, the origin acts as a saddle with four eigenvectors, two of which are stable. We also determined how varying the constants c1 and lambda affected the corresponding phase portraits.

The rest of this report is split into three sections: the background section describes the model. The results of our analysis are given in the third section as well as numerous calculations, graphs, and phase-portraits. Finally, the