Classification of Bursting Mappings

Stable Node: $x_{n+1}=\frac12x_n$

Unstable Node: $x_{n+1}=2x_n$

Stable spiral $x_{n+1}=-\frac12x_n$

Unstable spiral $x_{n+1}=-2x_n$

Stable Periodic Solutions: $x_{n+1}=-x_n$

Stable Periodic Solution: $x_{n+1}=\left\{\begin{array}{ll} 1 & x_n<0 \\ 0 & x_n=0 \\ -1 & x_n>0 \end{array}\right.$

"Node" bursting: $x_{n+1}=\left\{\begin{array}{ll} 4x_n(1-x_n) & x_n<.98 \\ .000098x_n & x_n > .98 \end{array} \right.$

"Focus" bursting: $x_{n+1}=\left\{\begin{array}{ll} 1-4x_n(1-x_n) & x_n<.98 \\ .25+.00003x_n & x_n > .98 \end{array} \right.$

"Fold" bursting: $x_{n+1}=\left\{\begin{array}{ll} -.25+2x_n-.3(x_n-1)^3 & x_n<3.7 \\ x_n-4 & x_n > 3.7 \end{array} \right.$