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?


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