## Complex Eigenvalues of Second Order Autonomous Systems

The system with linearisation matrix given byhas eigenvalues which are the solution to

These eigenvalues are

The eigenvectors corresponding to these eigenvectors will be complex when solved using

We can however transform coordinates. When we transform to polar coordinates, the system becomes much simpler. If the eigenvalues arethen define

Similarlyand differentiation gives

Hence

We obtain the simple equationsand

We can solve these to obtainand

Elimination ofbetween these two equations gives

Ifandorandthenincreases exponentially asincreases. Ifandhave the same sign thendecreases exponentially asincreases.