are time-periodic functions with finite amplitudes. This means the high-order corrections are non-convergent unless the coefficient functions of the growing function $z_{i2}^{0}$ are equal to zero. So the general motion is unstable periodic oscillations, the necessary-sufficient conditions for stable oscillations are $$\lim_{t\rightarrow \pm \infty }\int\limits_{C_{1}}^{t}z_{i1}^{0}\epsilon
_{i}dt=0.$$ The above conditions are non-integrable, clearly, they contain the following necessary conditions $$\int\limits_{-\infty }^{+\infty }z_{i1}^{0}\epsilon _{i}dt=0.$$ Apparently, the first integral $(i=1)$ of the necessary conditions is the Melnikov function of the system, the necessary conditions indicate that the Melnikov function is equal to zero; this means that the stable oscillations are Melnikov chaotic. But because of the non-sufficient property of the above condition, not all chaotic oscillations are stable. Integrating the above equations, one can obtain the necessary conditions are a series of relations of the initial conditions and parameters, for fixed initial conditions, modifying the parameters can control the instability of the chaotic oscillations. Substituting the expressions of $z_{11}^{0}$ and $%
\epsilon _{1}$ into the necessary condition, integrating it yields the first-order condition $$-\frac{8\eta (\Lambda H-1)\sqrt{\Lambda H-1}}{3\Lambda ^{2}}-\frac{2\Delta
E_{1}H}{\sqrt{\Lambda ^{2}}}\omega \pi \cos \frac{\omega C_{0}}{\sqrt{%
\Lambda H-1}}\sec h\frac{\omega \pi }{2\sqrt{\Lambda H-1}}$$ $$+\frac{2\Delta E_{1}(\Lambda H-1)\sqrt{\Lambda H-1}}{\Lambda \sqrt{\Lambda
^{2}}}(1+\frac{\omega ^{2}}{\Lambda H-1})\omega \pi \cos \frac{\omega C_{0}}{%
\sqrt{\Lambda 