Table 1.
The qualitative behavior of the piecewise linear map shown in Fig. 4 is described for each parameter regime. The top section describes why some parameter values are irrelevant to our analysis. The bottom section describes relevant parameter ranges that can fully account for the dynamics witnessed in prior multirhythmic bursting models.
| s1 | s2 | Dynamics |
|---|---|---|
| ⋯ | |s2|<1 | s2 must have modulus greater than 1 |
| |s1|<1 | s2<−1 | Nonbiological because sav<0 |
| s1>1 | s2>1 | Nonbiological because sav>1 indicating that the system is a repeller |
| 0<s1<1 | s2<−1 | Map is not well defined |
| s1<−1 | s2>1 | Chaotic and possibly multirhythmic since the full map is formed by a chain of sawtooth maps with only unstable fixed points |
| −1<s1<0 | s2>1 | Attractors are oscillatory fixed points and the system is possibly multirhythmic |
| 0<s1<1 | s2>1 | Attractors are nonoscillatory fixed points and the system is possibly multirhythmic |
| s1>1 | s2<−1 | Chaotic and possibly multirhythmic since the full map is formed by a chain of tent maps with only unstable fixed points |