The Bykov model

One of the mechanisms considered by Bykov is the following sequence of reactions  

We shall begin with the substantiation of a certain general property of the system (6.115). It follows from equations (6.115) that for small positive x, y the inequalities x 0, y 0 are satisfied, i.e. x, y increase. On the other hand, for (x + y) 1 it follows from (6.115) that x 0, y 0 and hence x, y decrease. More detailed considerations reveal that phase trajectories of the system (6.115) remain, for t- oo, within the limit set 

Note that when k3 0, k 3 = 0, the y-variable may be regarded as a parameter, y = A, A = const. The system (6.115) is then reduced, by virtue of the Tikhonov theorem, to the equation 

We shall now examine the possibility of occurrence of oscillations in the system (6.115). Stationary states of the system (6.115) satisfy the equations  

By linearizing the system of equations (6.115) we obtain the characteristic equation, 

---

Note in the end that the Takens-Bogdanov bifurcation may occur in the Bykov model, see Section 5.5.3.4. The sensitive state of the Takens-Bogdanov bifurcation, Xt = X2 = 0, may be obtained by imposing in (6.121a), (6.121b) the following requirements on the constants k t, k2, k 3... 

---