Synchronization in Chaos

$$\begin{array}{rl}\frac{m{d}^2{x}_1}{d{t}^2}& =-m{w}_0^2{x}_1-K(x_1-x_2)\\ \frac{m{d}^2{x}_2}{d{t}^2}& =-m{w}_0^2{x}_2-K(x_2-x_1)\end{array}$$

Since the solutions are likely to be oscillating, we substitute trial functions ( some Phasors ):

$$x_1(t)=A_1{e}^{i\omega t}{x}_2(t)=A_2{e}^{i\omega t}$$

with frequency $\omega$ into equations.

$$\begin{array}{rl}{A}_1\left({\omega }^2-{\omega }_0^2-\frac{K}{m}\right)& =-\frac{K}{m}{A}_2\\ A_2\left({\omega }^2-{\omega }_0^2-\frac{K}{m}\right)& =-\frac{K}{m}{A}_1\end{array}$$

multiplying together:

$$\begin{array}{rl}{\left({\omega }^2-{\omega }_0^2-\frac{K}{m}\right)}^2{A}_1{A}_2& ={\left(\frac{K}{m}\right)}^2{A}_1{A}_2\\ {\left({\omega }^2-{\omega }_0^2-\frac{K}{m}\right)}^2& ={\left(\frac{K}{m}\right)}^2\end{array}$$

leaving a quadratic equation in frequency $\omega$ with two solutions.

$${\omega }_1=w_0$$ $${\omega }_2={\omega }_0\sqrt{1+\frac{2K}{m{\omega }_0^2}}$$