# Bifurcations due to a nonlinearity parameter

Suppose we want to analyze the behavior of the system $$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},t;\varepsilon),\quad \mathbf{x}\in\mathbb{R}^n,\quad t\in\mathbb{R}^+,\quad\varepsilon\in\mathbb{R}^+,$$ in the case where $$\mathbf{f}(\mathbf{x},t;\varepsilon)$$ is smooth and depends in some way on the variable $$x_i^{p+\varepsilon}$$ for some $$1\leq i \leq n,\ p\in\mathbb{N}.$$ Have bifurcations due to changes in $$\varepsilon$$ in systems such as these (perhaps also in the discrete analogue) been studied extensively, and if so, where can I find literature on this?