Extended Phase Space

 

To continue with the analysis of planar string vibrations, we now turn our attention to the forced Duffing equation in dimensionless variables (from eqs. (3.11 and 3.14)),

 

where F is the forcing amplitude and is the normalized forcing frequency.

It is often useful to rewrite an nth-order differential equation as a system of first-order equations, and to recall the geometric interpretation of a differential equation as a vector field. To this end, consider the change of variable v = x', so that

 

where is the autonomous , or time-independent, term of v' and is the time-dependent term of v'. The phase space for the forced Duffing equation is topologically a plane, since each dependent variable is just a copy of , and the phase space is formally constructed from the Cartesian product of these two sets, .

A vector field is obtained when to each point on the phase plane we assign a vector whose coordinate values are equal to the differential system evaluated at that point. The vector field for the unforced, undamped Duffing equation is shown in Figure 3.7(a).

  
Figure 3.7: Extended phase space for the Duffing oscillator.

This vector field is static  (time-independent). In contrast, the forced Duffing equation has a time-dependent vector field since the value of the vector field at (x,v) at is

In Figure 3.7(b) we show what the integral curves look like when plotted in the extended phase space , which is obtained by introducing a third variable,

 

With this variable the differential system can be rewritten as

 

By increasing the number of dependent variables by one, we can formally change the forced (time-dependent) system into an autonomous (time-independent) system.

Moreover, since the vector field is a periodic function in z, it is sensible to introduce the further transformation

 

thereby making the third variable topologically a circle, . With this transformation the forced Duffing equation becomes

 

One last reduction is possible in the topology of the phase space of the Duffing equation. It is usually possible to find a trapping region  that topologically is a disk, a circular subset . In this last instance, the topology of the phase space for the Duffing equation is simply , or a solid torus  (see Figure 3.7(c)).