Complex Rhythms - The Physics of Birdsong - page 139

that is part of an unstable solution can grow with time, taking us away from
the solution. Therefore, the claim that, for a wide region of the parameter
space ( T,A ), a given model shows solutions locked in a particular way refers
to “stable” solutions, i.e., those toward which the system will evolve for all
initial conditions within a wide region of the phase space. This requires a
condition additional to the one expressed in (9.9), namely one that guarantees
stability. The condition is easy to understand: if we expand the map P by
calculating its first derivative DP ,

φ n +1 + δ = P ( φ n + )
∼ P ( φ n )+ DP ( φ n ) , (9.10)

the condition that δ is smaller than is guaranteed if the value of DP is of
modulus smaller than one. Now we are able to find the region of parameter
space for which stable solutions of period one can exist: we have to show that
the cuves φ n +1 = φ n cross for some value of φ , and that the slope of the
curve φ n +1 = P ( φ n ) at the point of intersection is smaller than one.

9.3.4 Locking Organization

What about other temporal patterns? After all, we became involved in this
tour of nonlinear phenomena in order to explain “complex” rhythmic pat-
terns. The procedure to find the regions where other solutions exist is very
simple: we define a new map which is the second iterate of the one we have
used so far. The set of fixed points of the second-iterate map, P 2 ,

φ n +2 = P 2 ( φ n )
= P [ P ( φ n )] , (9.11)

is the solutions to the following equation:

φ n +2 = φ n . (9.12)

This will contain, on the one hand, the period-one fixed points that we have
already met (a solution that repeats itself after one period of the forcing will
keep on repeating itself after n times as well). But on the other hand, this set
can also contain solutions that repeat themselves after twice the period of the
forcing, without repeating themselves in one period. These are “period-two”
solutions. Working with the map (9.11), we can again impose the conditions
of existence and stability of fixed points that we discussed before, and find
the region of parameter space where these conditions are met.
The result of performing this study for solutions of different periods is
displayed in Fig. 9.5, where the “tongues” enclose the set of parameter values
( T,A ) for which solutions of different period occur. Moreover, we can advance
in our description of the system by defining the rotation number r ,aratioof