On first page of chapter 3 of his book "Elements of Applied Bifurcation Theory", Kuznetsov considers the system
$$\dot{x} = f(x,\alpha), \quad x \in \mathbb{R}^{n}, \alpha \in \mathbb{R}^{1}$$
Let $x_{0}$ be hyperbolic equilibrium for $\alpha = \alpha_{0}$. One way hyperbolicity may be broken is if a $ \textit{pair}$ of simple complex eigenvalues becomes $\lambda_{1,2} = \pm i \omega_{0}, \omega_{0}>0$ for some value of $\alpha$. He states that "it is obvious that one needs more parameters to allocate extra eigenvalues on the imaginary axis".
I don't understand why one needs more than one parameter in the ode to have more eigenvalues moving through the imaginary axis. Could someone explain the reason for this?
Thanks.