*3.5. Mixed Geometric Control*

More striking, parameter *c* almost has a positive correlation with the offset of signal *z*, almost without changing other signals, and also has a negative correlation with the amplitude of variable *x* and positive correlation with the amplitude of variable *y*. Figure 14 shows the typical phase trajectories and waveforms. Figure 15 shows the corresponding Lyapunov exponent spectra and average value of the *x*, *y* and *z* signals. Therefore, all in all, there are five parameters, *a*, *b*, *c*, *m* and *n*, rescaling the system variables, some of which are restricted in a specific region, as shown in Table 2.

**Figure 14.** Typical chaotic oscillation of system (2) with *a* = *5, b* = 4, *k* = 0.5, *m* = 1 under initial conditions [1, −1, −1, 1]: (**a**) phase trajectory in *x*-*z*, (**b**) signal *z*(*t*), (**c**) phase trajectory in *x*-*y*, (**d**) signal *x*(*t*).

**Figure 15.** Dynamical evolution of system (2) with *a* = 5, *b* = 4, *k* = 0.5, *m* = 1 under initial conditions [1, −1, −1, 1]: (**a**) Lyapunov exponent spectra of *c*, (**b**) average values of the signals *x*, *y* and *z*.


**Table 2.** Five independent parameters in system (2) for geometric control.
