Khorozov-Takens normal form

Find a Lyapunov function for Khorozov-Takens normal form... 

C.7. 81. Following the same steps as in the study of the generic Bogdanov-Takens normal form, analyze the structure of the bifurcation set near the origin /xi = /i2 = 0 in the Khorozov-Takens normal form with reflection symmetry ... 

On this interval, the homoclinic-8 bifurcates in the same way as in the Khorozov-Takens normal form. Both loops, which form the homoclinic-8 are orientable. The dimension of the center homoclinic manifold is equal to 2. The third dimension does not yet play a significant role. Therefore, it follows from the results in Sec. 13.7 that on the right of HS, there are two unstable cycles cycles 1 and 2 in Fig. 13.7.9). To the left of HS, a symmetric saddle periodic orbit (cycle 12) bifurcates from the homoclinic-8 (see also Fig. C.7.5). 

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