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Fast phase change correction

Upon integrating Eqns. (39-40) numerically, one discovers that the system oscillates very quickly, on the order of one or more oscillations per unit time scale. This is bad for computations, because very small time steps will be needed to accurately resolve the behavior of the system. Secondly, the important dynamics of the system will be happening ``on top of'' these oscillations.

One solution to this problem is to essentially cancel the change in phase of the system. A change of variables can be made in order to do this. The effects of b in our analysis will be neglected with the assumption that the added injection will do little to change the frequency at which the system oscillates.

The original e in Eqn. (1) will be transformed into a new variable E by


where exp represents the exponential function. Substituting this into Eqn. (1), one obtains an expression for the rate of change of the new variable E of the form


Essentially, what has been done is to add a term to Eqn. (1) which will allow one to ``follow'' the system around through its changes in phase. The new variable E will now be called e.

Now a value must be found for tex2html_wrap_inline1512 . Essentially, it is the frequency at which the system would like to naturally oscillate. The imaginary terms in Eqn. (42) are


So, in order to suppress the imaginary part of Eqn (42), it is neccesary to set


Though n varies with time, the value tex2html_wrap_inline1634 can be substituted for n in Eqn. (43) where tex2html_wrap_inline1638 is the value of n at the fixed point of the system. The assumption made here is that the expression for the frequency thus obtained will still hold for values of n away from the fixed point. This argument is reinforced by the fact that oscillations of n, if any, tend to be around this fixed point tex2html_wrap_inline1638 and have small amplitude (see Figure [1]).

Setting Eqn. (39) equal to zero and ignoring tex2html_wrap1603 , tex2html_wrap1613 can be found to be


This allows Eqn. (39) to be written


next up previous
Next: Fixed Points Up: Analysis Previous: Analysis