# Background

The shock model is essentially a forced osillator model (with nonlinear damping). The model represents that of the shock absorbers in an automobile. It has some similarities to the well-known van der Pol oscillator.

The model contains 4 state variables x,y,z,w, and seven parameter values, c1,c2,,a,b,,A(mp). x is the position, y is the velocity, and z and w are forcing functions. c2, a, b, , and A are positive constants and , c1 are constants that can take any value.

The model equations are given below.

``` x' = y```

``` y' = - x - (c1 + c2(x - y)2)y + aw + bz```

``` z' = z - w - A(z2 + w2)z```

``` w' = z + w - A(z2 + w2)w```

` ` To focus our investigation of this model, 5 parameter values were fixed, at:

parameter value
c2 3
0.3
a 3
b 0
A 1

The two remaining parameters are varied to observe the behavior of the system.

The equations for x and y are coupled together. x and y also depend upon the values of z and w. z and w only depend on each other. z and w can be decoupled from the first two equations. Using polar coordinates, the asymptotic solutions for z(t) and w(t) are given by

z(t) = 0,
w(t) = 0 when < or = 0;

z(t) = ,
w(t) = when is > 0.

where is a phase angle that depends on the initial conditions of z and w.