# 8: Bifurcations

- Page ID
- 7809

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

- 8.1: 8.1 What are Bifurcations?
- One of the important questions you can answer by mathematically analyzing a dynamical system is how the system’s long-term behavior depends on its parameters. Most of the time, you can assume that a slight change in parameter values causes only a slight quantitative change in the system’s behavior too, with the essential structure of the system’s phase space unchanged. However, sometimes you may witness that a slight change in parameter values causes a drastic, qualitative change in the system’s

- 8.2: Bifurcations in 1-D Continuous-Time Models
- For bifurcation analysis, continuous-time models are actually simpler than discrete-time models (we will discuss the reasons for this later). So let’s begin with the simplest example, a continuous-time, ﬁrst-order, autonomous dynamical system with just one variable:

- 8.3: Hopf Bifurcations in 2-D Continuous-Time Models
- For dynamical systems with two or more variables, the dominant eigenvalues of the Jacobian matrix at an equilibrium point could be complex conjugates. If such an equilibrium point, showing an oscillatory behavior around it, switches its stability, the resulting bifurcation is called a Hopf bifurcation.

- 8.4: Bifurcations in Discrete-Time Models
- The bifurcations discussed above (saddle-node, transcritical, pitchfork, Hopf) are also possible in discrete-time dynamical systems with one variable: