Growth in size and period of limit cycle

At the point of Hopf bifurcation, the emerging limit cycle has zero amplitude and an oscillatory period given by 2n/a 0. As we begin to move away from the bifurcation point the amplitude A and period T grow in a form we can calculate according to the formulae 

In these equations th term (p — p ) represents how far we have moved away from the bifurcation point, in terms of the dimensionless concentration of reactant. There are two new quantities p2 and t2 which tell us a number of things. The amplitude A grows as the square root of the distance from the bifurcation point (p — p ), and so the term (p — p )/p2 must be positive. If lx2 turns out to be positive, then the limit cycle must grow as p is increased beyond p if fx2 is negative, the limit cycle grows as p decreases below fx. The growth (or decrease) in oscillatory period is linear in (p — p ) and depends on the ratio t2jp2. 

Note that this involves the derivative of the real parts -of the eigenvalues Alt 2 or, equivalently, of the trace of J evaluated at the Hopf bifurcation point. We know that Reflj 2) is passing through zero at this point. 

If d tr(J)/d/i is positive at p, then tr(J) will be negative (corresponding to a stable stationary state) for p p and positive (corresponding to an unstable stationary state) for p p. In these circumstances p2 will have the opposite sign to / 2 thus if P2 is negative, p2 will be positive. The limit cycle 

For t2, therefore, we need the derivative of the imaginary part of the eigenvalues evaluated at the Hopf bifurcation point. We may also note that the sign of the quotient t2//r2 is of less immediate significance than those of / 2 and n2. 

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